• Generate, through investigation using technology, a table of values showing the instantaneous rate of change of apolynomial function, f(x), for various values of x (e.g., construct a tangent to the function, measure its slope, and create aslider or animation to move the point of tangency), graph the ordered pairs, recognize that the graph represents a function
called the derivative, or and make connections between the graphs of f(x) and or and [e.g., when f(x) is
linear, is constant; when f(x) is quadratic, is linear; when f(x) is cubic, is quadratic] [2.1]
• Determine the derivatives of polynomial functions by simplifying the algebraic expression and then
taking the limit of the simplified expression as h approaches zero [i.e., determining ] [2.1]
• Verify the power rule for functions of the form where n is a natural number [e.g., by determining the equations of
the derivatives of the functions and algebraically using andgraphically using slopes of tangents] [2.2]
• Verify the constant, constant multiple, sum, and difference rules graphically and numerically [e.g., by using the function and comparing the graphs of and by using a table of values to verify that
given and ], and read and interpret proofs of the constant,
constant multiple, sum, and difference rules (student reproduction of the development of the general case is not required) [2.2]
• Determine algebraically the derivatives of polynomial functions, and use these derivatives to determine the instantaneousrate of change at a point and to determine point(s) at which a given rate of change occurs [2.2]
• Verify that the power rule applies to functions of the form where n is a rational number [e.g., by comparing valuesof the slopes of tangents to the function with values of the derivative function determined using the power rule],and verify algebraically the chain rule using monomial functions [e.g., by determining the same derivative for by using the chain rule and by differentiating the simplified form, ] and the product rule using polynomial functions[e.g., by determining the same derivative for by using the product rule and by differentiating theexpanded form ] [2.3]
• Solve problems, using the product and chain rules, involving the derivatives of polynomial functions, sinusoidal functions,
exponential functions, rational functions [e.g., by expressing as the product
radical functions [e.g., by expressing as the power and other simple combinations of
functions [2.3, 2.4, 2.5]
• Solve problems, using the derivative, that involve instantaneous rates of change, including problems arising from real-worldapplications (e.g., population growth, radioactive decay, temperature changes, hours of day-light, heights of tides), giventhe equation of a function. Sample problem: The size of a population of butterflies is given by the function
where t is the time in days. Determine the rate of growth in the population after five days using the
derivative, and verify graphically using technology. [2.2, 2.3, 2.4, 2.5]
What “big ideas” should students develop in this chapter?Students who have successfully completed the work of this chapter and who understand theessential concepts and procedures will know the following:
• The derivative of a function f at a point is , or
if the limit exists.
• A function is said to be differentiable at a if exists. A function is differentiable on aninterval if it is differentiable at every number in the interval.
• The derivative function for any function is given by if thelimit exists.
• The value of the derivative at a point can be interpreted as the slope of the line tangent to thepoint or the instantaneous rate of change at the point.
• The derivative function can be determined using various rules, including: the constantfunction rule, the power rule, the constant multiple rule, the sum rule, and the difference rule.
• The rules for calculating derivatives of functions make the process easier than determining thederivative from first principles.
• The derivative of a product of differentiable functions is not the product of their derivatives.• There are rules for determining the derivative of a function that is a product or a power of
another function.• The derivative of a quotient of two differentiable functions is not the quotient of their
derivatives.• The quotient rule for differentiation simplifies the process for determining the derivative of a
function that is written as a quotient.• The chain rule is used to determine the derivative of a composite function. If
, then h 9(x) 5 f 9(g(x)) ? g 9(x).h(x) 5 f (g(x))
f 9(x) 5 limh→0
f (x 1 h) 2 f (x)h ,f (x)
f 9(a)
f 9(a) 5 limx→a
f (x) 2 f (a)x 2 a ,
f 9(a) 5 limh→0
f (a 1 h) 2 f (a)h(a, f (a))
Review of Prerequisite Skills Use concepts and skills developed 1 day Diagnostic Testpp. xx–xx prior to this chapter.
Section 2.1: The Derivative Introduce the definition of the derivative 1 day graphing calculator Function pp. xx–xx function. Section 2.1 Extra Practice
Section 2.2: The Derivatives of Begin to develop some rules for 1 day graphing calculatorPolynomial Functions pp. xx–xx differentiation. Section 2.2 Extra Practice
Section 2.3: The Product Rule Introduction of the product rule for 1 day graphing calculatorpp. xx–xx differentiation. Section 2.3 Extra Practice
Section 2.4: The Quotient Rule Introduction of the quotient rule for 1 day Section 2.4 Extra Practicepp. xx–xx differentiation.
Section 2.5: The Derivatives of Introduce the chain rule to deal with 1 day graphing calculatorComposite Functions pp. xx–xx composite functions. Section 2.5 Extra Practice
Mid-Chapter Review: pp. xx–xx 5 days Mid-Chapter Review Chapter Review: pp. xx–xx Extra Practice;Chapter Test: p. xx Chapter ReviewCareer Link: p. xx Extra Practice
Pacing10–15 min Using the Review35–40 min Exercises
Student Book Pages 00–00Using the ReviewDiscuss with students the review concepts listed on page XX of the studentbook. Be sure to emphasize the following points.• When taking the product of exponential expressions with the same base,
add the exponents, i.e., When taking the quotient ofexponential expressions with the same base, subtract the exponents, i.e.,
When taking an exponential expression to a power, multiplythe powers, i.e.,
• When simplifying radicals, the square root distributes over multiplication, i.e., Students should be able to thenfactor out perfect squares from radical expressions, i.e.,
Students should also be ableto rationalize denominators of numerical expressions by multiplying thenumerator and denominator by the conjugate.
• Parallel lines have the same slope. Perpendicular lines have slopes that areopposite reciprocals; equivalently, two lines are perpendicular if theproduct of their slopes is
• When two rational expressions are being multiplied, multiply thenumerators and multiply denominators. When two rational expressionsare being divided, rewrite the problem as multiplication by the reciprocalof the denominator. When adding or subtracting rational expressions, acommon denominator must first be obtained. After the first operation,combine like terms, factor, and cancel out common factors to simplify.
• Students should be able to expand the product of two polynomials.Students should also be able to factor quadratic trinomials, expressionsthat involve finding a greatest common factor first, and expressions thatinvolve special forms, such as perfect square trinomials, the difference ofsquares, and the sum and difference of cubes.
• Students should also be able to factor polynomial expressions of degreethree or greater using various techniques, including the Factor Theorem.
• Students should know the difference quotient, how to evaluate andsimplify it with various functions, and its connection to the slope of asecant line.
Initial Assessment What You Will See Students Doing…
When students understand… If students misunderstand…
Students can simplify complicated expressions involvingexponents and radicals by applying the various rules.
Students can find equations of lines given multiplecombinations of conditions, i.e., slopes, perpendicular lines,points on the line, y-intercept, etc.
Students can methodically factor a variety of differentpolynomials.
Students can simplify rational expressions, particularlyadding or subtracting by obtaining a common denominator.
Students can rewrite numerical expressions involvingradical denominators as equivalent expressions with wholenumber denominators.
Students can determine an expression for average rate ofchange using the difference quotient and use it to makeestimates of instantaneous rates of change.
Students misapply the rules of exponents, for instanceadding exponents when they should be multiplying.
Students can only find equations of lines when given basicinformation such as slope and y-intercept, but cannot doslightly more complicated examples.
Students either factor incorrectly, or obtain correctfactorizations through a tedious guess and check method.
Students try to “add across” when adding rationalexpressions.
Students do not simplify the resulting rational expression(not cancelling common factors or combining like terms)after applying the first operation.
Students struggle to rewrite numerical expressionsinvolving radical denominators as equivalent expressionswith whole number denominators because they havedifficulty identifying the conjugate radical and do notmultiply two radicals together correctly.
Students have difficulty determining an expression foraverage rate of change using the difference quotientbecause they substitute incorrectly and have difficulty withthe algebra involved in simplification. Students struggle touse their expression to make estimates of instantaneousrates of change.
New Vocabulary/Symbolsderivativenormaldifferentiable
Nelson Websitehttp://www.nelson.com/math
dydxf 9(x),
Prerequisite Skills/Concepts• Determining an expression for the difference quotient for a given
function and point• Estimating instantaneous rates of change/slopes of tangent lines• Evaluating limits
Specific Expectations • Generate, through investigation using technology, a table of values
showing the instantaneous rate of change of a polynomial function, f (x),for various values of x (e.g., construct a tangent to the function, measureits slope, and create a slider or animation to move the point of tangency),graph the ordered pairs, recognize that the graph represents a function
called the derivative, or and make connections between the
graphs of f (x) and or and
• Determine the derivatives of polynomial functions by simplifying
the algebraic expression and then taking
the limit of the simplified expression as h approaches zero
Mathematical Process Focus• Connecting• Representing• Reflecting
Draw the graph of on the board, overhead or graph and displayit using The Geometers Sketchpad. Have the students determine the slopeof the tangent line at by using techniques from the previous chapterand techniques they remember from MHF4U, e.g., determining the limitof the difference quotient. When the class has finished, have each of thestudents pick an additional x value and determine the slope of the tangentline. Collect the data from the class and see if they can make any generaliza-tions and/or predictions as to the slope of the tangent line at Students should see that the slope can be found by multiplying the x valueby two. Tell the students that they will be investigating these types of gener-alizations in this section. If using Sketchpad, plot the point (2, 4) on thegraph and construct a second point on the graph of . Use these twopoints to construct a secant line and measure its slope. Drag the pointcloser to the point of tangency and discuss the idea of the limiting processof the slope as the distance between the two points gets smaller and smaller.Define the derivative of f at the point x 5 a.
In addition to presenting this section using the solved examples, afterintroducing students to the derivative at a point, give the students thecubic and have them discover the deriva-tive function using a quadratic regression on the graphing calculator. Havestudents form groups of three. Each group should be assigned a differentpoint at which to determine the derivative either by successive approxima-tions or from first principles (limits). Have each group present their x value and the slope of the tangent line on the board. As a whole class,gather the data and draw a scatter plot. Students should see that the pointsform what looks to be a parabola. Have the students run a quadraticregression on the data to determine the derivative function. Use the func-tion to predict an additional point, and confirm this point algebraically.Finally, determine the derivative from first principles (limit) to confirm theregression model.
Example 1 presents a function and a point at which to determine the deriva-tive. Present this problem to the whole class, and emphasize that the derivativeis the slope of the tangent line at the given point. Draw a picture of the graphand its tangent line to reinforce this point. At the end of the example, thestudents are introduced to an alternate definition of the derivative at a point.Ask students to think about why the definitions are equivalent, and have themconfirm the answer to this example by working with the alternate definitionin their small groups. You can use the graphing calculator to confirm thecalculated value by graphing the function and drawing a tangent line at
The calculator will display the equation of the tangent line in theform where m is the slope of the tangent line.
Example 2 introduces the notion of the derivative function. Much of the work for this was done in the introduction, so the teacher should beable to go over it relatively quickly. Have the students work on theInvestigation in their small groups. This investigation allows students todiscover the power rule that will be formally presented in the next section.Take up any questions with the whole class.
Answers to InvestigationA. a.
b.c.
B. The coefficient of the derivative function is the same as the originalpower, and the power of the derivative function is one less than theoriginal power.
Example 3 has students determine the derivative of from firstprinciples (limit). It is essential that students can do this using the limitdefinition and continue to connect it to the slope of the graph. Havestudents work this problem in their small groups without using their text-books. Check their work and clear up and discrepancies. Have studentsdiscuss whether the pattern fits with what they discovered in the investiga-tion. Students should see that it does. Emphasize to the whole class thedomain of the original function and domain of the derivative function.Ask students to think about why the derivative does not exist at even though the function has a value at this point.
Example 4 has students determine the equation of the tangent line at apoint. The small groups should be able to handle this problem with booksclosed. The teacher may have to circulate to ensure that students can fol-low the steps in order, i.e., determine the derivative at the point, then usethe point on the graph, together with the slope, to write the equation ofthe line. Have each group compare their work with that of the textbookwhen they are finished. Also, have each group graph the function and thetangent line on their calculator for further confirmation.
Example 5 is very similar to Example 4 but asks students to determine anormal line instead of a tangent line. After defining normal, have thegroups follow the same procedure as in Example 4.
Before doing Example 6, discuss what can cause a function to not be dif-ferentiable at a point. Draw each of the three situations on the board:cusp, vertical tangent, and discontinuity. Emphasize that a function can becontinuous but not differentiable, but if a function is differentiable, itmust be continuous. Have the students discuss the differentiability of the function in Example 6 in their small groups. Take up any questionswith the whole class. This example also illustrates that corners are pointswhere a function is not differentiable.
Students can determine the derivative at a point by usingthe limit definition.
Students can determine the derivative function by usingthe limit definition.
Students can recognize when a graph is notdifferentiable and confirm that fact using one sided limitson either side of the point in question.
Students can determine the equation of a tangent lineand the equation of a normal line by using derivatives.
Students can only estimate the derivative at a point byusing successive approximations of average rate ofchange.
Students only guess at the derivative function by lookingat patterns in the derivative at several points.
Students equate differentiability with continuity.
Students determine the slope of the tangent line ornormal line to a graph, but cannot arrive at the equationof such a line.
Differentiating Instruction How You Can Respond
When students understand… If students misunderstand…
EXTRA SUPPORT
1. Some students will struggle applying the limit definition to determine the derivative function right away. Havethese students determine several derivatives at specific points (still using the limit definition) and look forpatterns in their presentation. Students should see that where there was once a numerical value (3, 4, 5, etc.),we can simply replace it with a.
2. Some students will be able to determine the slope of a tangent line, but not the equation. Give these studentssome review problems in which they are given the slope and a point on the graph in order to strengthen theirequation-writing skills.
EXTRA CHALLENGE
1. Have students examine the derivative at several points on plot them, and attempt to determine thederivative function.
2. Have students examine the derivative at several points on plot them, and attempt to determine thederivative function.
f (x) 5 2x,
f (x) 5 sin x,
Key Assessment Question #7
Students accurately determine all derivatives with propernotation from first principles.
Students attempt to determine the derivatives byapplying patterns discovered in the section.
Students have difficulty with the substitutions in thedifference quotient and have difficulty evaluating theresulting limit using techniques from Chapter 1.
Students attempt to determine derivatives by determiningthe derivatives at points and examining patterns.
2.2 THE DERIVATIVES OF POLYNOMIAL FUNCTIONSSection at a Glance
Begin to develop some rules for
differentiation.
GOAL
Prerequisite Skills/Concepts• The ability to express simple rational functions as powers using negative
exponents• The ability to express simple radical functions as powers using rational
exponents• Evaluating algebraic expressions for given values• Determining the equation of a line based on given information
Specific Expectations• Verify the power rule for functions of the form , where n is a
natural number [e.g., by determining the equations of the derivatives ofthe functions and
algebraically using and graphically using slopes of
tangents]• Verify the constant, constant multiple, sum, and difference rules
graphically and numerically [e.g., by using the function andcomparing the graphs of and by using a table of values toverify that given and
], and read and interpret proofs of the
constant, constant multiple, sum, and difference rules (studentreproduction of the development of the general case is not required)
• Determine algebraically the derivatives of polynomial functions, and usethese derivatives to determine the instantaneous rate of change at a pointand to determine point(s) at which a given rate of change occurs
Mathematical Process Focus• Connecting• Representing• Problem Solving• Reasoning and Proving
limh→0
f (x 1 h) 2 f (x)hg(x) 5 3x
f (x) 5 xf 9(x) 1 g 9(x) 5 ( f 1 g)9(x),kf 9(x);g 9(x)
g(x) 5 kf (x)
limh→0
f (x 1 h) 2 f (x)h
f (x) 5 x 4f (x) 5 x 3,f (x) 5 x 2,f (x) 5 x,
f (x) 5 x n
Preparation and Planning
Pacing5–10 min Introduction15–20 min Teaching and Learning25–30 min Consolidation
New Vocabulary/Symbolsconstant function rulepower functionpower ruleconstant multiple rulesum ruledifference rule
Nelson Websitehttp://www.nelson.com/math
MATH BACKGROUND SECTION OVERVIEW
• Students should already be familiar with determining the derivative function from first principles.
• Students will learn that the derivative of a constant function is zero.
• Students will justify then learn to apply the constant function, power, constant multiple, sum, anddifference rules in order to determine derivatives of polynomial functions in an efficient manner.
• Students will use the derivative to determine slopes and equations of tangent lines of polynomial functions.
• Students will determine when the slope of a polynomial function is zero through algebraic means.
• Students will use the derivative to determine instantaneous rate of change in contextual situations andinterpret the results.
Have a student begin at one end of the room and walk to the other end.Instruct the student to begin slowly, speed up, slow to a stop at thehalfway point, then move at a steady rate to the other end of the room.Have each student attempt to draw a displacement vs. time graph thatmodels the student’s distance from their starting point. It may be necessaryto have the student repeat the motion several times. Take up suggestionsand discuss the results until a fairly accurate graph is drawn. Finally focuson the portion where the student was at rest in the middle of the room.What is the slope of the graph during this interval? Write a function forjust that portion of the graph. Students should see that this piece of thegraph is a constant function and the slope is zero. Ask the students tothink about the connection between slope and rate of change and to dis-cuss the fittingness of the zero slope to the context of the demonstration(i.e., the rate of change, or speed, was zero).
Teaching and Learning(15 to 20 min)
Instead of presenting the properties followed by the proofs, students candiscover these properties for themselves. For instance, have the studentsrely on their work from the previous section in order to determine thederivative function for using first principles. Once they havecompleted this, have them discuss the non-significance of the number five.Students will see that the derivative of any constant function is zero. Thiscan be further emphasized by graphing several constant functions andshowing that these result in horizontal lines whose slopes are always zero at all points on the function. The work for the power rule was done inthe previous section’s Investigation, so it need not be repeated here. Todiscover the constant multiple rule, have students determine the derivativefunctions for and using first principles. Finally,have students determine the derivative of a polynomial such as
using first principles. Students should seethat they could have applied the power and constant multiple rules to eachindividual term, which is essentially using the sum and difference rules.
Students can also practice their understanding by writing questions thatmodel Example 5 and Example 6. Have each pair of students write a qua-dratic function that has “nice” solutions. This quadratic will serve as thederivative function. (See Example 6 .) Have the students take the derivativefunction and create the original function using what they know about thevarious rules in this section. The pairs can then switch their problem withanother group, solve the other group’s problem, and switch back for peerevaluation.
Technology-Based Alternate LessonIf graphing software or graphing calculators are available, students cangraph polynomials, draw tangent lines, and obtain their slopes via technology.Data can be collected very quickly in this manner. Once several (x, slope)pairs are obtained, students can create a scatter plot of the data, make aneducated guess on the type of function that appears, and run regressionwith their graphing calculators in order to discover derivatives. Anotheralternative is to use The Geometer’s Sketchpad. Polynomial functions canbe created by choosing the Graph menu and selecting New Function and itthen can be graphed by selecting the function, going to the Graph menuand choosing Plot Function. Selecting the function and choosing the Graphmenu allows you to choose Derivative. This displays the derivative functionof the selected function and its graph can then also be plotted. In this way,students can do many problems in a short amount of time and discuss thepatterns that arise.
Consolidation(25 to 30 min)
Using the Solved Examples (Pairs)
Example 1 gives students the constant function rule. This can and shouldbe presented to the whole class in a very short amount of time. Graphicalconfirmation will help emphasize the meaning.
Example 2 is the most important example in the section for foundationalpurposes. Ask students to try to remember their results from the investiga-tion in the previous section, i.e., the derivative function for Once students have recalled it correctly, present the formal proof and havethe class follow along in their textbook. Finally, have the class work in pairsat applying the power rule to the problems in Example 2. When studentsare finished, they should check their work with the solution in the book. Itmay be useful to go through a couple of additional questions with the classof the type given in b and c.
Example 3 deals with the constant multiple rule, while Example 4 deals withthe sum and difference rules. Present all three of these rules to the wholeclass, and give the pairs all of the problems in both examples. The pairsshould work on determining the derivatives without using their textbook.Have them check their work with another pair, and then finally by lookingin the book. Take up any questions with the whole class.
Example 5 has students determine the equation of a tangent line by com-puting the derivative using the various rules in this section. This problemtends to give students some difficulties. Have the students begin the prob-lem in pairs, but take up the example with the whole class when they arefinished. Emphasize using graphing technology to check the work, but besure that students can arrive at the answer without using the calculator.
Example 6 uses the function from Example 5 and has students determinewhere the graph has a slope of zero. Having just found the derivative func-tion, this is as simple as setting the derivative equal to zero. This process is
Students can determine derivatives of polynomialswithout using the limit definition.
Students can determine the equation of a tangent line ata given point on a polynomial graph.
Students can determine algebraically where the slope ofa tangent to a polynomial function is zero.
Students can follow the algebraic proofs of the rulesdiscussed in this section and work through examplesfrom first principles to demonstrate the rules.
Students do not apply the appropriate rules to determinederivatives of polynomials in a timely manner.
Students can determine the slope of a tangent line, butcannot apply the knowledge to determining the equationof the line.
Students must examine a graph in order to guess wherethe slope is zero.
Students can apply the various rules, but cannot explainwhy they are true.
Differentiating Instruction How You Can Respond
When students understand… If students misunderstand…
EXTRA SUPPORT
1. Some students will have trouble expressing simple rational functions and radical functions as powers. Have
students work on several review questions that deal with the rules and
2. Some students may have no trouble getting the derivative function using the rules, but will not be able to set thederivative to zero in order to determine where the slope is zero. Have these students work on several reviewproblems involving determining the zeros of polynomials.
EXTRA CHALLENGE
1. Have students investigate what effect linear transformations have on the derivative functions. For instance, beginwith the graph of which has derivative What is the derivative of What if thedegree of the parent function was something other than 2?
2. Have students prove that the derivative function of is not Have them investigate this graphand try to develop the derivative function.
y 9 5 x2x21.y 5 2x
y 5 a(x 2 b)2 1 c?y 9 5 2x.y 5 x2,
xmn 5 Ïn
xm.x2n 51x n
Key Assessment Question #17
Students methodically determine both tangent lines thatinclude the given points.
Students guess and check in order to determine thetangent lines.
Students incorrectly use the given point as the point oftangency.
What You Will See Students Doing…
critical to understanding. Emphasize the connection between slope of atangent and derivatives, then have the students read through the examplein pairs and discuss the solution. Discuss the problem with the whole class.
Answer to the Key Assessment Question17. a. and
b. and 4x 1 y 2 1 5 020x 2 y 2 47 5 016x 2 y 2 29 5 0y 5 3
Prerequisite Skills/Concepts• Expanding and simplifying polynomial expressions• Constant, constant multiple, sum, and difference rules for derivatives of
functions
Specific Expectations• Verify that the power rule applies to functions of the form ,
where n is a rational number [e.g., by comparing values of the slopes of
tangents to the function with values of the derivative functiondetermined using the power rule], and verify algebraically the chain ruleusing monomial functions [e.g., by determining the same derivative for
by using the chain rule and by differentiating the
simplified form, ] and the product rule using polynomialfunctions [e.g., by determining the same derivative for
by using the product rule and bydifferentiating the expanded form ]
• Solve problems, using the product and chain rules, involving thederivatives of polynomial functions, sinusoidal functions, exponential
functions, rational functions [e.g., by expressing as the
product radical functions [e.g., by
expressing as the power and other
simple combinations of functions
Mathematical Process Focus• Connecting• Selecting Tools and Computational Strategies• Reasoning and Proving
Give students the following function:and ask them to determine
the derivative. Students should remember from the last section that theycan combine the functions before determining the derivative. Ask the stu-dents if a similar strategy would hold for the product of two polynomials.Ask students to think about how they would prove or disprove such aproperty. Finally, give the students the function written as
If the derivative of the product is the product of thederivatives, then the derivative should be not 2x.
Teaching and Learning(10 to 15 min)
The product rule can be taught in the reverse manner to how the textbookpresents it. In other words, give the students two polynomials, and
and have them compute the expression Tellthe students that the resulting expression is a derivative of some simplecombination of the two polynomials. It will not take students long to realize that the function under consideration is
The power of a function rule is very easily discovered by giving students afunction such as Have the students determine thederivative by expanding the function using the binomial theorem andusing methods from the last section. Next have the students apply thederivative in the “wrong” way, i.e., Have studentscompare the two expressions to first notice that they are not the same. Askstudents to think about the difference between the two. If they cannot seethe difference, have them factor the actual derivative. Many students willnotice that the extra factor, is the derivative of Fromhere, students can deduce the power of a function rule.
Technology-Based Alternate LessonIf a Computer Algebra System (CAS) is available, students can use it tocalculate many derivatives of products in a very short amount of time.These systems are either available as computer software or as part of someadvanced calculators that have symbolic manipulation. Give studentsexamples of simple products at first, such as Have the students change the 2 to other values to observe patterns.The students can then move on to more complicated products such as
Another alternative is touse The Geometer’s Sketchpad. The product of two functions can becreated by choosing the Graph menu and selecting New Function and itthen can be graphed by selecting the function, going to the Graph menuand choosing Plot Function. Selecting the function and choosing theGraph menu allows you to choose Derivative. This displays the derivativefunction of the selected function and its graph can then also be plotted.
The use of CAS or Sketchpad allows for many examples to be examined ina very timely manner.
Example 1 asks students to prove that the derivative of a product is notequal to the product of the derivatives. Present the product function fromthe example. Ask students to determine the derivative by expanding theoriginal product. Then ask them to proceed as if one could simply multi-ply the separate derivatives. Finally, discuss with students why one counterexample is sufficient to prove that a statement is false.
Preceding Example 2 is the statement and proof of the product rule.Students will need to see this proof done by the teacher. Once the studentshave been introduced to the idea, Example 2 asks them to apply it to a verystraightforward situation. Have the students use the Think-Pair-Share strate-gy to solve the problem without using the textbook. Go over the example asa whole class when they are finished.
Example 3 is only different in that the polynomials are a little morecomplex and that it asks students to evaluate the derivative at Have students proceed as they did in Example 2.
Example 4 asks students to develop a product rule for a product of threefunctions by using the product rule for two functions twice. Studentsshould be able to develop this rule using Think-Pair-Share. If they struggle,go over the example with the whole class. Have each pair come up with asimple example that they can use to confirm their rule.
The development of the Power of a Function Rule occurs prior toExample 5 and is critical because it is a precursor to the general chain rule.It has students investigate where Students shouldbe led through the derivation for Then let them determine thesolution for and in their pairs. In Example 6 students areasked to determine the derivative of a rational function by using the prod-uct and power rules. Once the power of a function rule has been devel-oped, have each member of the pair select one of Example 5 and 6. Theindividual students should study the assigned example, take notes on thesolution, and present the problem and solution to his or her partner with-out using the textbook. Take up any questions with the whole class.
Example 7 is an application problem involving derivatives and velocity. Usethis as an opportunity to reinforce the connection between a derivative, theslope of a tangent line, and instantaneous rate of change. Students shoulduse a Think-Pair-Share strategy to solve this problem with books closed.
Students use the product rule and power of a functionrule as a shortcut for determining derivatives of products.
Students can determine the derivative of a quotient byrewriting it as a product.
Students can present an algebraic argument withsupporting examples for each of the rules in this section.
Students can use derivatives to interpret real lifesituations involving velocity.
Students insist on multiplying the polynomials out inorder to determine the derivative.
Students forget to write the power when rewriting aquotient as a product.
Students move a power down and decrease it by 1 todetermine the derivative, i.e., they forget to multiply bythe “derivative of the inside”.
Students do not see the connection between velocity(rate of change) and derivatives.
21
Differentiating Instruction How You Can Respond
When students understand… If students misunderstand…
EXTRA SUPPORT
1. Some students will have difficulty not simply multiplying derivatives. These students should spend more timeexpanding polynomials in order to determine the derivatives.
2. Some students will have a hard time keeping the various derivative rules straight. Encourage these students tomake a summary sheet of derivative rules that they refer to as they work through the exercises until theybecome comfortable with them.
EXTRA CHALLENGE
1. Have students investigate the derivative functions for and by calculating it at enough pointsto guess the function from a scatter plot. Then have them use the derivatives and the product rule in order todetermine the derivative of
2. Have students investigate the power of a function rule with exponents that are not integers and conjecturewhether or not it is true.
y 5 tan x.
y 5 cos xy 5 sin x
Key Assessment Question #5
Students use the product and power rules to correctlydetermine all derivatives.
Students determine derivative by multiplying thepolynomials out.
Students determine the derivatives of the product bydetermining the product of the individual derivativesand/or misapply the power rule.
• The derivative of a function f at a point is or if the limit exists.
• A function is said to be differentiable at a if exists. A function is differentiable on an interval if it is differentiable atevery number in the interval.
• The derivative function for any function f (x) is given by if the limit exists.
• The value of the derivative at a point can be interpreted as the slope of the line tangent to the point or the instantaneousrate of change at the point.
• The derivative function can be determined using various rules, including: the constant function rule, the power rule, theconstant multiple rule, the sum rule, and the difference rule.
• The rules for calculating derivatives of functions make the process easier than determining the derivative from firstprinciples.
• The derivative of a product of differentiable functions is not the product of their derivatives.
• There are rules for determining the derivative of a function that is a product or a power of another function.
f 9(x) 5 limh→0
f(x 1 h) 2 f(x)h ,
f 9(a)
f 9(a) 5 limx→a
f (x) 2 f (a)x 2 a ,f 9(a) 5 lim
h→0
f (a 1 h) 2 f (a)h ,(a, f (a))
Using the Mid-Chapter Review
Ask students if they have any questions about any of the topics covered so far in the chapter. Review any topics thatstudents seem to be having trouble with. Have students complete as many of the exercises in class as possible and thencomplete any unfinished questions for homework.
In order to gain greater insight into students’ understanding of the material covered so far in the chapter, you may want toask them questions such as the following:
• Can you think of a reason to calculate a derivative using first principles? For a particular function? What about a generaltype of function? (The point is that the limit definition is useful for establishing a rule. Once the rule is determined, then thevalue of the limit approach is diminished because it is a far more tedious method for determining derivatives.)
• What does the sign of tell you about the function at
• What does the derivative of a function represent?
x 5 a?f ′(a)
For review of the material in these exercises ... Refer to ...
Prerequisite Skills/Concepts• Expanding and simplifying polynomial expressions• Constant, constant multiple, sum, difference, product and power of a
function rules for derivatives of functions
Specific Expectations• Solve problems, using the product and chain rules, involving the
derivatives of polynomial functions, sinusoidal functions, exponential
functions, rational functions* e.g., by expressing as the
product radical functions e.g., by
expressing as the power and other
simple combinations of functions
*Even though the curriculum does not specify the development of thequotient rule, students who will take a postsecondary Calculus coursewill benefit from its exposure.
Mathematical Process Focus• Connecting• Reasoning and Proving• Selecting Tools and Computational Strategies
c e.g., f (x) 5 x sin x, f (x) 5sin xcos x d
df (x) 5 (x 2 1 5)12f (x) 5 Ïx 2 1 5
cf (x) 5 (x 2 1 1)(x 2 2 1)21 d ,
f (x) 5x 2 1 1x 2 2 1c
MATH BACKGROUND SECTION OVERVIEW
• Students should already be familiar with derivatives of polynomials, the product rule and the power of afunction rule.
• Students will use the product rule to develop the quotient rule.
• Students will use the quotient rule to determine derivatives of rational functions.
• Students will determine equations of tangent lines to rational functions.
• Students will apply the quotient rule to problems involving instantaneous rates of change.
Student Book Pages 00–00
Preparation and Planning
Pacing5–10 min Introduction10–15 min Teaching and Learning30–35 min Consolidation
Remind students that they have been working on derivatives of variouscombinations of polynomials. They have already developed rules for thederivative of a product, sum, and difference of two polynomials. Ask thestudents to think about the one basic operation that is left. Present the
students with the function Ask the students if they already
have enough tools to evaluate this derivative. Students should recall fromthe previous section that they can write this as a product. Have thestudents determine the derivative individually, and check their workas a whole class.
Teaching and Learning(10 to 15 min)
Students can be taught to discover the quotient rule by rewriting thedenominator using a power and applying a version of the power of afunction rule. This is the strategy used in the previous section. Emphasizeto students that the power of a function rule was only stated for positiveintegers, but this derivation strategy may be more straightforward withstudents.
This section also provides a good opportunity to talk about vertical asymp-totes of rational functions and the non-differentiability of the function atthese points. Give students a rational function to graph and ask them todetermine the slope of the graph at various points. What happens to thederivative at the vertical asymptote? Students should see that the denomi-nator of the original function shares a factor in common with the denomi-nator of the derivative, and hence, where one function is undefined, so isthe other.
Students can also determine the derivatives of rational functions by doinglong division first on the quotient. This does not avoid using the quotientrule, but it does provide a good opportunity to review long division. Givethe students an example and have them determine the derivative by firstusing the quotient rule, and then by using long division. Ask them to dis-cuss the relationship between the two answers as well as the pros and consof each method.
After presenting the derivation of the quotient rule with the whole class,have the class work in groups of 3 to apply the rule to the function inExample 1. This example is a straightforward application of the rule.Students may check their work with that in the book.
Example 2 has students determine the equation of a tangent line thatrequires the quotient rule. Students should work in their groups to solvethis question. When they are finished, go over the technology solution as awhole class. Emphasize to students the importance of being able to solvethis problem by using an algebraic approach.
Example 3 has the students use derivatives to determine where the tangentline is horizontal. Students may need to be reminded that a horizontal tan-gent line has slope zero. From there, the small groups should be able tosolve the problem and check their solution with other groups. Go over thisexample as a whole class when they are finished, and take up any questionsor discrepancies.
Students can explain the relationship between thequotient rule and the product rule.
Students can apply the quotient rule to efficientlydetermine the derivatives of rational functions.
Students can determine the equation of a tangent line toa rational function.
Students can apply the quotient rule to problemsinvolving rates of change.
Students cannot apply the quotient rule, but must rewritethe function as a product.
Students mix up the order in the numerator when usingthe quotient rule.
Students can determine the slope of a function using thequotient rule, but cannot determine the equation of thetangent line.
Students cannot interpret rates of change in real lifeapplications in terms of derivatives.
Differentiating Instruction How You Can Respond
When students understand… If students misunderstand…
EXTRA SUPPORT
1. Some students will have difficulty remembering the quotient rule. Encourage these students to make a summarysheet of derivative rules that they refer to as they work through the exercises until they become comfortable withthem. Give these students daily quizzes asking them to recall the formula until they can do it without looking.
2. Some students will consistently get an answer opposite to the correct answer when using the quotient rule. Thisis a result of subtracting in the numerator incorrectly. Show them the error they are making using a numericalexample (e.g. These students would benefit from memorizing the product and quotient rulespresented using the same order in both.
EXTRA CHALLENGE
1. Have students determine when the equation of the tangent line to at the point crosses its
graph. Does it always cross the graph? How many times?
2. Have students develop a quotient rule for and for . Are the two expressions different?
f(x)
Qg(x)h(x)R
Q f(x)g(x)Rh(x)
x 5 3f (x) 5ax 1 bcx 1 d
5 2 3 Þ 3 2 5).
Key Assessment Question #5
Students determine all derivatives by hand using thequotient rule.
Students use the product rule instead of the quotient ruleto solve the problems.
Students attempt to determine the answers by usingtechnology instead of algebraic means.
Prerequisite Skills/Concepts• Creating a composite function from two given functions• Recognizing when a given function is a composite function and identifying
the outer and inner functions• Constant, constant multiple, sum, difference, product, power of a function
and quotient rules for derivatives of functions• Factoring
Specific Expectations • Solve problems, using the product and chain rules, involving the derivatives
of polynomial functions, sinusoidal functions, exponential
functions, rational functions e.g., by expressing as the
product radical functions
e.g., by expressing as the power
and other simple combinations of functions
Mathematical Process Focus• Connecting• Reflecting• Selecting Tools and Computational Strategies• Reasoning and Proving
ce.g.,�f(x) 5 x sin x, f (x) 5sin xcos xd
f (x) 5 (x 2 1 5)12 Tf (x) 5 Ïx 2 1 5S
f (x) 5 (x 2 1 1)(x 2 2 1)21T,
f (x) 5x 2 1 1x 2 2 1
S
Preparation and Planning
Pacing5 min Introduction5–10 min Teaching and Learning35–40 min Consolidation
Ask the students to determine the derivative ofby expanding the polynomial and
then using various rules they learned in previous sections. Ask students if they can shortcut the derivative process by writing the following
Many students will recognize thatthe power of a function rule forbids the problem from being this simple.Student should notice that there is a missing factor of two, which happensto be the derivative of from the shortcut. Tell students that theywill be investigating rules for this kind of function.
Teaching and Learning(5 to 10 min)
Students can be led to discover the chain rule by using what they knowabout the power of a function rule. The power of a function rule is aspecial case of the chain rule. Start the students with the function
and have them determine the derivative using whatthey know from previous sections. Students should have no trouble seeingthat Then have students write as
where and Have students determineand Finally, express in terms of g, h, and Point
out to students that This is the chain rule. Notethat this is not a proof, but is evidence based on what the students alreadyknow.Once students know the chain rule, you can have them express a singlefunction as multiple compositions to apply the chain rule and observe the
resulting equality. For instance, given the function students can write
(1) where and or
(2) where and
Have students come up with even more creative compositions for thisfunction, then apply the chain rule to see if the derivative is the same.
g(x) 51x,h(x) 5 x 2 1 3f (x) 5 g(h(x) )
g(x) 51
x 1 3,h(x) 5 x 2f (x) 5 g(h(x) )
f (x) 51
x 2 1 3,
f 9(x) 5 g 9(h(x) )h 9(x).h 9.g 9,f 9(x)g 9(x).h 9(x)
Example 1 is a simple review of composite functions. Students will need verylittle help with this, but may need to be reminded the order of application inthe composition of two functions. Present this example to the class on theoverhead and allow them to work out one or two of the parts. If time is anissue, this example can also serve as an introduction to the section.
Example 2 and Example 3 are preceded by the statement and proof of thechain rule. Begin by presenting the chain rule together with its proof. Besure to present both notations. Then have each member of the pair chooseeither Example 2 or Example 3 to study. When they are finished studyingthe problems, have them explain their example to their partner. Call on onestudent to present the solution on the board for each of the examples, andfield any questions that arise.
Example 4 is an application of the chain rule to a rate of change problemfrom environmental science. Give the students the statement of the problemin their pairs, but do not allow them to use their textbook. The pairs shouldwork on the problem, then check their work against the book. Take up anyquestions with the whole class.
Example 5 is an example involving the power of a function. Students canwork this problem in their pairs. However, if time does not permit, giventhat this is a problem that could have been in the section on the productrule, this is an example that could be skipped.
Example 6 has students determine the equation of a tangent line to a graphby using the chain rule to determine the derivative. Have the pairs studythis example, then complete the work with the whole class. Emphasize theuse of technology in this example, having students follow along with theircalculators.
Example 7 asks the students to determine the deriviative using the chainrule. Have the students solve the problem without the use of their textbook,then trade their work with another pair for peer evaluation. If there is a dis-crepancy amongst pairs, have the group use the book’s solution to settle theissue.
Example 8 involves using multiple rules to determine a derivative. Studentswill need some extra time to work this out. The textbook presents a solutionusing the product rule and a solution using the quotient rule. This examplewould make a good exit problem. Ideally, have students solve the problemin their pairs, then check their work by first deciding which of the two booksolutions was closest to their own strategy. Then take up any unique solu-tions with the whole class.
Students can apply the chain rule in the correct order todifferentiate a composite function.
Students can determine the slope and equation oftangent lines by use of the chain rule.
Students can determine the derivative of a complicatedfunction by applying more than one rule.
Students can follow and/or explain the proof of thechain rule.
Students can use the chain rule to solve real worldapplications involving rates of change.
Students rely on the power of a function rule and cannotuse the chain rule in order to differentiate compositefunctions.
Students are still attempting to determine derivativesfrom first principles.
Students can determine derivatives, but cannot writeequations of tangent lines.
Students can only differentiate functions that involve onerule.
Students cannot interpret real life applications involvingrates of change in terms of derivatives.
Differentiating Instruction How You Can Respond
When students understand… If students misunderstand…
EXTRA SUPPORT
1. Some students will have trouble determining complicated derivatives. Break these problems up into smallersteps for these students; this may require more individualized attention.
2. Some students will try to avoid the chain rule and simply use the power of a function rule. Have these studentswrite a paragraph of explanation as to how the power of a function rule is a special case of the chain rule anddemonstrate this using an example.
EXTRA CHALLENGE
1. Have students who excel at the chain rule develop a rule for where
2. Ask students to investigate functions such that How many examples of such functions can theydetermine?
f 9(x) 5 f (x).
f (x) 5 g(h( j(x))).f 9(x)
Key Assessment Question #9
Students determine the slope of the tangent line byapplying the chain rule for determining the derivative.
Students cannot determine the derivative algebraically,but instead attempt to determine the slope byapproximation techniques.
Students only determine the derivative of the outerfunction and forget to multiply by the derivative of theinner function.
Students determine the derivative but do not substitutethe x value to determine the slope.
This chapter has introduced students to the methods used to determine thederivatives of polynomial and rational functions. These methods includethe constant, power, sum-and-difference, product, and quotient rules fordifferentiation. This summary includes three vehicles to help you assess theachievement level of your students.
1. an additional set of review questions that can be used in a variety ofways with your students
2. a test that gauges students’ knowledge and performance capabilities
3. an achievement rubric that is included for your convenience in assessingstudents’ level of achievment
Suggested Time: 70–75 minIn this guided discovery activity the elasticity of demand or the influence ofprice on the demand of a product or service is utilized to illustrate theneed for obtaining more efficient methods of determining derivatives. Thelearning will be accomplished through utilizing the learners’ prior knowl-edge as consumers of products and services and their prior knowledgefrom Chapter 1. The learning provides the basis for a constructivistapproach proceeding from the development of a context (brainstormingabout elasticity) to a specific example (calculating elasticity) to developingthe need for the general case (finding an algebraic expression for instanta-neous elasticity of demand).
Introducing the Career Link (Whole Class)As a class, introduce the Career Link on Student Book page XX. Discussthe difference between elastic and inelastic demand. Have the studentswork individually at discussion question 1 on page XX. After a few min-utes, ask for examples and make a list on the board. Introduce the studentsto the calculation of the elasticity of a product. As a class, have the stu-dents calculate the elasticity of the movie rental in the second discussionquestion. Tell the students that they will be investigating the elasticity ofdemand in various products during this economics-based Career Link.
Using the Career Link (Groups)Students should work in groups of three on the prompts. The studentsshould be reminded to answer all parts, complete with all calculations andsupporting explanations. Students may need help understanding the ideaof approximating a curve with a linear function (tangent line). After thestudents have had a chance to get started on the prompts, stop the classand go over a simpler example. For instance, have students approximatethe curve with a tangent line at For what values isthis a good approximation? For what values is this a bad approximation?Once students understand this concept, have them return to the prompts.If students finish early, have them check their work with another group.Field questions with the whole class as they arise.
Adapting the Task• For struggling students, have the groups only deal with one of the two
demand functions. It may also be necessary to do parts b and c as awhole class with student input. The two demand functions are quitedifferent in nature. Have the students write a paragraph that discusses thesimilarities and differences in the functions and what real life phenomenamight account for the differences.
• Have the students come up with a revenue function and use graphingtechnology to determine the maximum points. Use this work to confirmthe answers to the previous prompts.
Suggested Organization as an Assessment TaskA suggested sequence for implementing this task for assessment purposesin the classroom is as follows:1. Review the concept of elasticity, specifically addressing what inelastic
(increase the price to obtain higher revenues) and elastic (decrease theprice to obtain higher revenues) demands mean in the context of set-ting price levels. (5 min)
2. Review and discuss how student’s work will be evaluated (e.g., present arubric on an overhead projector). (5 min)
3. Have students brainstorm in small groups how this task can be solved.They do not actually perform any calculations (i.e., determine deriva-tives). During this phase the students could be required to keep sepa-rate brainstorming notes that are handed in immediately following thebrainstorming session. (10 min)
4. Students work on the task independently, completing all calculationsand a written summary of their results and methods. (up to 50 min)
Assessment and EvaluationIt is suggested that a task-specific rubric be developed by adapting theGeneric Career Link Rubric provided on page XX of this Teacher’s Guide.To evaluate Learning Skills, students can complete a self assessment, oranecdotal notes can be made by the teacher. For ease of evaluation, studentsubmissions may be organized into the following format. Each reportincludes two sections. The first section, “Communication of Findings”summarizes the results of the task and clearly explains the methods used(i.e., justifying reasoning). The second part, “Supporting Calculations,”shows all calculations, TI-83 Plus screen captures, hand drawn- graphs, etc.
Generic Career Link Wrap-Up RubricThis rubric can be adapted to help you assess student achievement on each chapter’s Career Link Wrap-Up performance task.
Assessment of Learning-What to Look for in Student Work…
A critical problem facing the retail industry is the establishment of pricepoints for products and services. The goal is to set a price that maximizesrevenue. This is accomplished via an understanding of elasticity ofdemand. As elasticity is based upon the rate of change of demand withrespect to price, this pricing problem can be solved using differentialcalculus. In this Authentic Performance Task students will apply the toolsof calculus to predict elasticity and then make recommendations aboutincreasing or decreasing price levels in order to increase revenue.
This Authentic Performance Task Affords Students an opportunity to:• utilize an algebraic mathematical model in a real-world context to make
predictions and decisions using the tools of differential calculus;• demonstrate successful use of the power, quotient, and chain rules;• consolidate their understanding of functions supported by calculus
(i.e., why is very closely approximated by
);• sequence and select mathematical tools as part of the problem solving
process;• communicate their work clearly by justifying their reasoning and
demonstrating proper mathematical terminology and form;• utilize graphing-calculator technology as a tool in the problem solving
• The derivative of a function f at a point is or if the limit exists.
• A function is said to be differentiable at a if exists. A function is differentiable on an interval if it is differentiable atevery number in the interval.
• The derivative function for any function is given by if the limit exists.
• The value of the derivative at a point can be interpreted as the slope of the line tangent to the point or the instantaneousrate of change at the point.
• The derivative function can be determined using various rules, including: the constant function rule, the power rule, theconstant multiple rule, the sum rule, and the difference rule.
• The rules for calculating derivatives of functions make the process easier than determining the derivative from firstprinciples.
• The derivative of a product of differentiable functions is not the product of their derivatives.• There are rules for determining the derivative of a function that is a product or a power of another function.• The derivative of a quotient of two differentiable functions is not the quotient of their derivatives.• The quotient rule for differentiation simplifies the process for determining the derivative of a function that is written
as a quotient.• The chain rule is used to determine the derivative of a composite function. If then h 9(x) 5 f 9(g (x)) ? g 9(x).h (x) 5 f (g(x)),
f 9(x) 5 limh→0
f(x 1 h) 2 f(x)h ,f (x)
f 9(a)
f 9(a) 5 limx→a
f(x) 2 f(a)x 2 a ,f 9(a) 5 lim
h→0
f(a 1 h) 2 f(a)h ,(a, f (a))
Using the Chapter Review
Ask students if they have any questions about any of the topics covered in the chapter. Review any topics that studentsseem to be having trouble with. Have students complete as many of the Practice Questions in class as possible and thencomplete any unfinished questions for homework.
In order to gain greater insight into students’ understanding of the material covered in the chapter, you may want to ask themquestions such as the following:• What is the value in the quotient rule if it is always possible to write a quotient as a product and then use the product rule?• Which notation do you prefer, the prime notation or the Leibniz notation? Why? How does the notation help you think about
what the derivative means?
For review of the material in these exercises ... Refer to ...
5. Find the equation of the normal line to the curve at the point (2, 0).
6. If and , find when
7. Find the points on the graph where the slope of the tangent line is parallel to .
8. An environmental study of a suburban community suggests that t years from now, the average level of carbonmonoxide in the air will be parts per million.a. At what rate will the carbon monoxide level be changing with respect to time one year from now?
b. By now how much will the carbon monoxide level change in the first year?
c. By now how much will the carbon monoxide level change over the next (second) year?
9. If and find where g(x) 5 Ïx f (x).g 9(4)f 9(4) 5 25,f (4) 5 3
1. Answers may vary. For example, you need to use the quotient rule when the function given as a quotientcannot be simplified to a polynomial. While it is possible to use the product rule after rewriting a quotient as aproduct with one factor having a negative exponent, this method may require more simplification than wouldbe required by using the quotient rule.
2. is the graph that starts in the third quadrant and ends in the fourth quadrant. f starts in the first quadrantand ends in the third quadrant.
3.
4. a.
b.
c.
d.
e.
5.6.
7. and
8. a. parts per million per yearb. 0.15 parts per millionc. 0.25 parts per million
1. For each function, determine the value of thederivative for the given value of a.a.b.c.d.e.f.
2. Use the product rule or the power of a function ruleto determine for each function.a.b.c.d.
e.
f.
3. In each case, determine using the product rule orthe power of a function rule.a.b.c.d.
e.
f.
4. Determine the slope of the tangent line tofor each of the
following values of x.a.b.c.d.e.f.
5. For each of the following functions, determine theequation of the tangent line at the given x value.a.
b.
c.d.e.f.
6. For each of the following functions, determine theslope of a line perpendicular to the tangent line at thegiven x value.a.b.c.
d.e.f.
7. For each of the following functions, determine theequation of a line that is perpendicular to the tangentline at the given x value.a.b.c.
d.
e.
f.
8. For each of the following functions, determine thevalue(s) of x for which the function has a horizontaltangent line.a.b.c.d. f (x) 5 (x 1 1)(x 1 2)(x 1 3)
