My class notes for my Calculus class--a one semester course. Includes all notes and handwritten answers to all questions. Includes basic Derivatives and Integrals.Hope this helps anyone in need of some math help.
Lectu re Secti ons Objective Assignment 1 1.1, 1.2 Distance, midpoint formula, translation of points 1.1: 3, 9, 11, 15, 21, 23, 25, 27 Graphs and intercepts of functions, standard form of equations of circles 1.2: 3, 5, 13, 17, 19, 41, 45, 55, 57, 74 b,c Understanding Goals: 1. To understand the rectangular coordinate system. 2. To understand how to find the distance between two points and midpoints of line segments connecting two points. 3. To understand how to translate points in a coordinate plane. 4. To understand how to sketch graphs of equations by hand. 5. To understand how to find the x- and y- intercepts of equations and points of intersection of two graphs. 6. To understand how to complete the square and write the standard forms of equation of circles. 7. To understand the use of mathematical models and to use it to solve real-life problems. Cartesian Plane (rectangular coordinate system) is formed by x-axis and y-axis. The point of intersection of these two axes is the origin. Example 1: Plot the points , , and . 1
Transcript
Lecture Sections Objective Assignment
1 1.1, 1.2Distance, midpoint formula, translation of
points1.1: 3, 9, 11, 15, 21, 23, 25, 27
Graphs and intercepts of functions, standard form of equations of circles
1.2: 3, 5, 13, 17, 19, 41, 45, 55, 57, 74 b,c
Understanding Goals:1. To understand the rectangular coordinate system.2. To understand how to find the distance between two points and midpoints of line
segments connecting two points.3. To understand how to translate points in a coordinate plane.4. To understand how to sketch graphs of equations by hand.5. To understand how to find the x- and y- intercepts of equations and points of
intersection of two graphs.6. To understand how to complete the square and write the standard forms of
equation of circles.7. To understand the use of mathematical models and to use it to solve real-life
problems.
Cartesian Plane (rectangular coordinate system) is formed by x-axis and y-axis. The point of intersection of these two axes is the origin.
Example 1: Plot the points , , and .
The Distance FormulaThe distance d between the points and in the plane is
.
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Example 2: Use the Distance Formula to show that the points , and are vertices of a right triangle.
The Midpoint FormulaThe midpoint of the segment joining the points and is
Midpoint
Example 3: Find the midpoint of the line segment joining the points and .
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Example 4: Find the vertices of the parallelogram [ ] after it has been translated 1 unit to the left and 2 units down.
x-intercept is in the form of (a,0) if it is a solution point of the equation.y-intercept is in the form of (0,b) if it is a solution point of the equation.
Example 1: Find the x- and y- intercepts of the graph of the equation .
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Example 2: Find the center and radius of the circle .
A point of intersection of two graphs is an ordered pair that is a solution point of both graphs. To find the points analytically, set the two y-values or two x-values equal to each other and solve the equation.
Example 3: Find the points of intersection (if any) of the graphs of the equations.
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a) ;
b)
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Graphs of six basic algebraic equations:
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Lecture Sections Objective Assignment
2 1.3slope-intercept, point-slope form of a linear equation
1.3: 7, 9, 19, 23, 39, 45, 49, 53, 65, 69,
equations of parallel and perpendicular lines
Understanding Goals:1. To understand the slope-intercept form of a linear equation and how to use it to
sketch graphs and to write equations of a line.2. To understand how to find slopes of lines passing through two points.3. To understand how to find equations of parallel and perpendicular lines.4. To understand how to use linear equations to model and solve real-life problems.
Slope of a line:
Example 1: Sketch the graph of .
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Example 2: Find the slope of the line passing through each pair of points.
a) and .
b) and .
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Example 3: Find the equation of the line that has a slope of 2 and passes through the point .
Summary:
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Example 4: Write the equations of the lines through the point (a) parallel to the line .(b) perpendicular to the line .
Example 5: Write the equations of the lines through the point
Understanding Goals:1. To understand what limit is.2. To be able to find limits of functions graphically and numerically.3. To use the properties of limits to evaluate limits of functions.4. To use different analytic techniques to evaluate limits of functions.5. To understand one-sided limits to be able to evaluate it.6. To be able to recognized unbounded behavior of functions.
Example 1: Find the limits of the following functions both numerically and graphically as .
a)
Question: We know that , is it true that ?
What is the behavior of the graph of this function near
1. To understand the meaning of “a function is continuous at a point or on a closed interval” and know how to determine it.
2. To understand how the greatest integer function is defined and how to use it to model and solve real-life problem.
3. To understand how to use compound interest model to solve real-life problems.
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Example 1: Discuss the continuity of each function:
(a)
(b)
(c)
If is continuous then, .
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Example 2: Discuss the continuity of each function:
(a)
(b)
(c)
(d) The greatest integer function at .
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Example 3: Determine where the function is discontinuous.
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Lecture Sections Objectives Assignment
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2.1
Tangent line, limit definition of derivativeDifferentiability and continuity
2.1: 5, 9, 15-47 eoo. 49-55 odds, 68-71
Understanding Goals:
1. To understand how to identify tangent lines to a graph at a point.2. To understand how to use the limit definition to find the slopes of graphs at
points.3. To understand the limit definition of the derivative and use it to find derivatives of
functions.4. To understand the relationship between differentiability and continuity.
For a line, the slope (rate of change) is the same at every point on the line. How about for graphs other than lines?
Question: How can we determine the rate at which a graph rises or falls at a single point?
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The tangent line to the graph of a function at a point is the line that best approximates the graph at that point. How do we define a tangent line then?
Example 1: Find the equation of the tangent line to the graph of at the point .
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Example 2: Find the slope of the graph of .
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Notation:
Example 3: Find the derivative of .
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Example 4: Find the derivative of .
Example 5: Find the derivative of y with respect to t for the
function .
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Lecture Sections Objectives Assignment
7 2.2Some Rules for Differentiation
2.2: 3, 5, 9, 13-53 eoo. 55, 57
Understanding Goals:
1. To understand how to use the Constant, Power, Constant Multiple, Sum and Difference Rules to find the derivatives of functions.
2. To understand how to use derivative to solve real-life problems.
Example 1: Find the derivate of the following functions:
(a) (b) ,
(c) , (d) ,
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Example 2: Find the derivate of the following functions:
(a) , (b) ,
(c) , (d) ,
Example 3: If possible, find the slopes of the graph of when
Example 4: Differentiate each function.
(a)
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(b)
(c)
(d)
Example 5: Find the slope of the graph of at .
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Example 6: Find an equation of the tangent line to the graph of
at the point .
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Lecture Sections Objective Assignment
8 2.3 Rates of Change2.3: 3-9 odds, 13, 15, 19, 21, 27, 33(c. d. e)
Understanding Goals:1. To understand the difference between average rates of change and instantaneous
rates of change.2. To understand how to find the average rates of change of functions over intervals.3. To understand how to find the instantaneous rates of change of functions at
points.4. To understand how to find the marginal revenues, costs and profits for products.
Two applications of derivatives: slope and rate of change.Real-life applications of rate of change: velocity, acceleration, population growth rates, production rate…
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Example 1: The concentration C (in milligrams per milliliter) of a drug in a patient’s blood stream is monitored over 10-minute intervals, where t is measured in minutes, as shown in the table. Find the average rate of change over each interval.(a) [0, 10] (b) [20, 50] (c) [60, 80]
The average velocity is the average change in a function over a given time interval. If we let represent the position of a particle at time t, then the average velocity over the
time interval is calculated as:
Average velocity =
The instantaneous velocity of is found by determining the limiting value of the
average velocity as the time interval, approaches 0.
Instantaneous velocity =
Example 2: A man standing on a platform that is 200 meters above the ground drops a baseball. Let be the Law of Motion for the path of the baseball.
(a) Sketch the function by plotting the points at , and 6 seconds.
t 0 2 3 4 6s
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(b) Find the average velocity between 2 and 3 seconds, and between 3 and 4 seconds.
(c) Estimate the instantaneous velocity at 3 seconds.
(d) Find the average velocity between t and seconds.
(e) Determine the instantaneous velocity as at 3 seconds as h goes to zero.
The general position function for a free-falling object, neglecting air resistance is
where
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Example 3: At time , a diver jumps from a diving board that is 32 feet high. Because the diver’s initial velocity is 16 feet per seconds, his position function is .(a) When is the diver hit the water?(b) What is the diver’s velocity at impact?
Example 4: Find the instantaneous rate of change of
Rate of Change in Economics: Marginals
Total profit = Total revenue – total cost
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Example 5: Find the marginal cost for producing x units.
Example 6: The revenue (in dollars) from renting x apartments can be modeled by
(a) Find the additional revenue when the number of rentals is increased from 14 to 15.
(b) Find the marginal revenue when .
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Example 7: The position of an object at any time t (in hours) is given by,
Determine when the object is moving to the right and when the object is moving to the left.
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Lecture Sections Objectives Assignment9 2.4 The Product and Quotient Rule 2.4: 1-37 eoo. 39, 43, 45,47
Understanding Goals:
1. To understand how to use the Product Rule and Quotient Rule to find the derivatives of functions.
2. To be able to simplify derivatives.3. To understand the meaning of derivative and use it to answer questions about
real-life situations.
Note:
Example 1: Differentiate the following functions:
(a)
(b)
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(c)
Note:
Example 2: Differentiate the following functions:
(a)
(b)
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(c)
Example 3: Find the equation of the tangent line to the graph of
when .
Example 4: As blood moves from the heart through the major arteries out to the capillaries and back through the veins, the systolic pressure continuously drops. Consider a person whose blood pressure P (in millimeters of mercury) is given by
where t is measured in seconds. At what rate is the blood pressure changing 5 seconds after blood leaves the heart?
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Example 5: It is possible for one medication to reduce the effectiveness of other medicines taken simultaneously. If a patient taking an antibiotic also takes a kaolin-pectin medication for diarrhea, the antibiotic adsorbs to the surface of the kaolin particles and passes out of the body. The amount of antibiotic available to the body is reduced. Suppose the adsorption coefficient for the amount of antibiotic adsorbed to the kaolin
particles is given by: where C represents the resulting concentration of
antibiotic in the blood plasma (mg/dL), A represents the adsorption coefficient of antibiotic (mg of antibiotic/g of kaolin-pectin mixture), and k is a constant. Find an equation that represents the rate of change of the adsorption coefficient.
z
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Lecture Sections Objectives Assignment
10 2.5 The Chain Rule2.5: 1-39 eoo. 45-63 eoo. 67,
69, 73
Understanding Goals:
1. To understand where the Chain Rule is used and to find derivatives using it.2. To understand how to use the General Power Rule to find derivatives.3. To be able to write derivatives in simplified form.4. To understand the application of derivatives in real-life situations.5. To understand how to use the differentiation rules to differentiate algebraic
functions.
Example 1: Write each function as the composition of two functions:
(a) (b)
Example 2: Find the derivative of y respect to x and simplify your result, where.
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Example 3: Find the derivative of .
Example 4: Find the derivative of .
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Example 5: Find the derivative of .
Example 6: Find the derivative of using
(a) the Quotient Rule and (b) the Chain Rule respectively.
Example 7: Find the tangent line to the graph of when .
1. To understand the definition of higher-order derivatives.2. To understand how to use the position functions to determine the velocity and
acceleration of moving objects.
Notation for Higher-Order Derivatives
First derivative: , ,
Second derivative: , ,
Third derivative: , ,
Fourth derivative: , ,
. . .
nth derivative: , ,
The nth-order derivative of an n th-degree polynomial function
is the constant function . Each derivative of order higher than n is the zero function.
Example 1: Find the fourth derivative of .
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Example 2: Find the value of if .
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Example 3: A ball is thrown into the air from the top of a 160-foot cliff. The initial velocity of the ball is 48 feet per second, which implies that the position function is
where the time t is measured in seconds. Find the height, the velocity, and the acceleration of the ball when
1. Understand the difference between explicit form and implicit form of functions.2. Understand how to find derivatives explicitly.3. Understand how to find derivatives implicitly.4. Understand how to use implicit differentiation to answer questions about real-life
situation
Example 1: Use two ways to find .a) Solve for yb) Without Solving for y
The process that we used in the second solution to the previous example is called implicit differentiation. In the previous example we were able to just solve for y and avoid implicit differentiation. However, that won’t always be the case. For example it is difficult to express y as a function of x explicitly if .
Example 2: Differentiating with respect to x:
(a)
(b)
(c)
(d)
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Example 3: Use implicit differentiation to find :
(a)
(b)
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(c)
(d)
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Example 4: Find the tangent line to the graph given by at the point
1. Understand relationship between variables in a function.2. Understand how to use derivatives to solve related-rate problems.
Related rates: the rates of two of more related variables that are changing with respect to time.
Related rates problems: Find and solve a relation between different rates of changes.
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Example 1: A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate of 1 ft/second. When the radius is 4 feet, at what rate is the total area A of the disturbed water changing?
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Example 2: Air is being pumped into a spherical balloon at a rate of 4.5 . Find the rate of change of radius when the radius is 2 feet.
Example 3: A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 , find the rate of change of the depth of water when the water is 8 feet deep.
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Example 4: A tumor is modeled as being roughly spherical, with radius R. If the radius of the tumor is currently R = 0.54 cm and is increasing at the rate of 0.13 cm per month,
what is the corresponding rate of change of the volume ?
Example 5: A boat is pulled by a winch on a dock, and the winch is 12 feet above the deck of the boat. The winch pulls the rope at a rate of 4 feet per second. Find the speed of the boat when 13 feet of rope is out. What happens to the speed of the boat as it gets closer and closer to the dock?
1. Understand the definition of increasing and decreasing functions and know how to test it.
2. Understand what critical number(s) of a function is and know how to find it.3. Understand how to use critical number(s) to find the open intervals on which a
function is increasing or decreasing.4. Understand how to use increasing and decreasing functions to model and solve
real-life problems.
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Example 1: Show that the function is decreasing on the open interval
and increasing on the open interval .
At what x-values can change signs?
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Example 2: Find the open intervals on which is increasing or decreasing.
Interval
Test Value
Sign of f'(x)
Conclusion
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Example 3: Find the open intervals on which the function is increasing or
decreasing.
Example 4: Find the critical numbers and the open intervals on which the function
is increasing or decreasing.
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Example 5: Find the critical numbers and the open intervals on which the function
is increasing or decreasing.
Interval
Test Value
Sign of f'(x)
Conclusion
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Lecture Section Objectives Assignment15 3.2 Extrema and the First-Derivative Test 3.2: 1, 5-11 odd, 19-29 odd, 35, 39
Understanding Goals:
1. Understand the difference between relative extrema and absolute extrema.2. Understand how to use the First-Derivative Test to find the relative extrema of
functions.3. Understand how to find absolute extrema of continuous functions on a closed
interval.4. Understand how to find minimum and maximum values of real-life models and
interpret the results in context.
A function has a relative extremum at points where the function changes from increasing to decreasing or vice versa.
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For a continuous function, the relative extrema must occur at critical numbers of the function.
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Example 1: Find all relative extrema of the function .
Interval
Test Value
Sign of f'(x)
Conclusion
Example 2: Find all relative extrema of the function .
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Example 3: Find the minimum and maximum values of on the interval
.
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Example 4: Coughing force the trachea (windpipe) to contract, which affects the velocity v of the air passing through the trachea. The velocity of the air during coughing is
where k is constant, R is the normal radius of the trachea, and r is the radius during coughing. What radius will produce the maximum air velocity?
Example 5: Poiseuille’s Law asserts that the speed of blood that is r centimeters from the
central axis of an artery of radius R is , where c is a positive constant.
Where is the speed of the blood greatest?
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Lecture Section Objectives Assignment16 3.3 Concavity and The Second-derivative Test 3.3: 1, 7, 9-17 odd, 23-33 odd, 47, 65
Understanding Goals:
1. Understand the definition of concavity and points of inflection.2. Understand how to determine the intervals on which the graphs of functions are
concave upward or downward.3. Understand how to find the points of inflection of the graphs of the function and
how to use the second-derivative test to find the relative extrema of functions.4. Understand the concept of diminishing returns in economics and know how to
find it in an input-output model.
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Example 1: Determine the open intervals on which the graph of is concave
upward or downward.
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Interval
Test Value
Sign of f''(x)
Conclusion
Example 2: Determine the open intervals on which the graph of is concave
upward or downward.
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Example 3: Determine the points of inflection and discuss the concavity of the graph of .
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Question: Are all points where the second derivative is zero points of inflection?
Example 4 :The graph below represents . On what intervals is positive?
Negative? Are there any points where ?
Example 5 : The graph below represents . On what intervals is increasing?
1. Understand how to write a primary and secondary equation.2. Understand how to use the first- and second-derivative test to solve real-life
optimization problems.
Primary equation: an equation gives a formula for the quantity to be optimized.Secondary equation: an equation relates the independent variables of the primary equation, which is used to reduce the primary equation to one with a single independent variable.
Example 1: Find two numbers whose difference is 100 and whose product is a minimum.
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Example 2: We need to enclose a field with a fence. We have 500 feet of fencingmaterial and a building is on one side of the field and so won’t need any fencing.Determine the dimensions of the field that will enclose the largest area.
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Example 3: We are going to construct a box whose base length is 3 times the base width.The material used to build the top and bottom cost $10/ and the material used to buildthe sides cost $6/ . If the box must have a volume of 50 , determine the dimensionsthat will minimize the cost to build the box.
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Example 4: A sheet of cardboard 3 ft. by 4 ft. will be made into a box by cutting equal-sized squares from each corner and folding up the four edges. What will be the dimensions of the box with largest volume?
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Example 5: Determine the point(s) on that are closest to (0, 2).
1. To be able to analyze the graph of functions using previous knowledge of function.
2. To be able to recognize the graphs of simple polynomial functions.
So far, we have studied several concepts that are useful in analyzing the graph of a function.
x-intercept and y-intercept Symmetry Domain and range Continuity Vertical asymptotes Differentiability Relative extrema Concavity Points of inflection Horizontal asymptotes
1. Understand the definition of differentials of functions and when to use it.2. Understand how to use differentials to approximate changes and how to use it to
solve real-life problems.
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Example 1: Let . Find when and . Compare this value with for and .
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Example 2: A state game commission introduces 50 deer into newly acquired state
game lands. The population N of the herd can be modeled by where t is
the time in years. Use differentials to approximate the change in the herd size from to .
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Example 3: Find the differential of the given functions.
a)
b)
c)
d)
e)
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Example 4: The radius of a ball bearing is measured to be 0.7 inch. If the measurement is correct to within 0.01 inch, estimate the propagated error in the volume V of the ball bearing.
The relative error in the volume is defined to be the ratio of to .
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Example 5: Use differentials to approximate .
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Lecture Section Objectives Assignment
21 4.1-4.3
Exponential FunctionsNatural Exponential FunctionsDerivatives of Exponential Functions
1. Understand the properties of exponents and know how to use it to evaluate and simplify exponential expressions.rl
2. To be able to sketch the graphs of exponential functions.3. Understand how to evaluate and graph functions involving the natural exponential
function.4. To be able to solve compound interest and present value problems.5. To be able to differentiate natural exponential functions.6. Understand how to use calculus to analyze the graphs of functions that involve the
natural exponential function.7. Explore the normal probability density function.
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Example 1: After t years, the initial mass of 16 grams of a radioactive element whose
half-life is 30 years is given by , .
a) How much of the initial mass remains after 50 years?
b) After how many years 5 grams of the element will be remaining?
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Example 2: A bacterial culture is growing according to the logistic growth model
,
Where y is the culture weight (in grams) and t is the time (in hours). Find the weight of the culture after 0 hours, 1 hour, and 10 hours. What is the limit of the model as t increases without bound?
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Example 3: How much should be deposited in an account paying 7.8% interest compounded monthly in order to have a balance of $21,154.03 fours years from now?
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Example 4: The demand function for a product is modeled by
Find the price of product if the quantity demanded is (a) units and (b) units. What is the limit of the price as x increases without bound?
Example 5: Find the slope of the tangent lines to at the point and .
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Example 6: Differentiate each function.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
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Example 7: Show that the graph of the normal probability density function
has points of inflection at .
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Lecture Section Objectives Assignment
22-23 4.4-4.6
Logarithmic FunctionDerivatives of Logarithmic FunctionsExponential Growth and Decay
1. To be able to sketch the graphs of natural logarithmic function.2. To be able to use properties of logarithms to simplify, expand and condense
logarithmic expressions.3. Understand how to use inverse properties of exponential and logarithmic
functions to solve exponential and logarithmic equations.4. Use properties of natural logarithms to solve real-life problems.5. Understand how to differentiate natural logarithmic functions. 6. Understand how to use calculus to analyze the graphs of functions that involve the
natural logarithmic functions.7. To be able to evaluate logarithmic expressions involving other bases.8. Understand how to differentiate exponential and logarithmic functions involving
other bases.9. Understand how to use exponential growth and decay to model real-life situations.
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Example 1: Simplify each expression:
a)
b)
c)
d)
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Example 2: Use the properties of logarithms to rewrite each expression as a sum, difference or multiple of logarithms. (Assume x > 0 and y > 0)
a)
b)
c)
d)
Example 3: Use the properties of logarithms to rewrite each expression as the logarithm of a single quantity. (Assume x > 0 and y > 0)
a)
b)
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Example 4: Solve each equation.
a)
b)
c)
d)
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Example 5: Find the derivative of each function:
a)
b)
c)
d)
d)
e)
Example 6: Determine the relative extrema of the function .
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Change-of-base formula:
Example 7: Find the derivative of each function:
a)
b)
c)
Example 8: The temperatures at which water boils at selected pressures p (pounds per square inch) can be modeled by
Find the rate of change of the temperature when the pressure is 60 pounds per square inch.
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Example 9: A sample contains 1 gram of radium. How much radium will remain after 1000 years? (The half-life of Radium is 1599 years)
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Example 10: Find the half-life of a radioactive material if after 1 year of the initial amount remains.
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Example 11: The effective yield is the annual rate I that will produce the same interest per year as the nominal rate r compounded n times per year.a) For a rate r that is compounded continuously, show that the effective yield is .
b) Find the effective yield for a nominal rate of 6%, compounded continuously.
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Example 12: On the Richter scale, the magnitude R of an earthquake of intensity I is
given by , where is the minimum intensity used for comparison.
Assume .
a) Find the intensity of the 1906 San Francisco earthquake in which .
b) Find the factor by which the intensity is increased when the value of R is doubled.
1. Understand the definition of antiderivative.2. Understand how to use integral sign for antiderivatives.3. Understand how to use basic integration rules to find antiderivatives of functions.4. Understand how to fine particular solutions of indefinite integrals using initial
conditions.5. Understand how to use antiderivatives to solve real-life problems.
Question to think about:We know has derivative . Is the only function whose derivative is ?
Integration is the “inverse” of differentiation
Differentiation is the “inverse” of integration
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Example 1: Find
a)
b)
c)
d)
e)
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f)
g)
h)
i)
j)
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Differential equation: an equation involves x, y and derivatives of y.
Particular solution of differential equation:
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Example 2: Find the general solution of , and find the particular
solution that satisfies the initial condition .
Example 3: Solve the differential equation
a)
b)
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Example 4: A ball is thrown upward with an initial velocity of 64 ft/s from an initial height of 80 feet.a. Find the position function giving the height s as a function of the time t.b. When does the ball hit the ground?
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Lecture Section Objectives Assignment25 5.2 The General Power Rules 5.2: 1-27 odd, 35-47 odd
Understand Goals:
1. Understand how to use the General Power Rule to find indefinite integrals.2. Understand how to use substitution to find indefinite integrals.3. Understand how to use the General Power Rule to solve real-life problems.
Example 1: Find each indefinite integral.
a)
b)
c)
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d)
e)
Multiply and divide integrand by a necessary constant multiple based on the Constant
Multiple Rule:
Example 2: Find
a)
b)
c)
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d)
e)
Change of variables:
Example 3: Find
a)
b)
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Example 4: Find the indefinite integral in two ways. Explain any
1. Understand how to use the Exponential and Log Rule to find indefinite integrals.
Example 1: Find each indefinite integral:
a)
b)
c)
d)
e)
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f)
g)
Example 2: Find each indefinite integral:
a)
b)
c)
138
d)
e)
f)
h)
i)
j)
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Example 3: Because of an insufficient oxygen supply, the trout population in a lake is dying. The population’s rate of change can be modeled by
where t is the time in days. When , the population is 2500.
a) Write an equation that models the population P in terms of the time t.
b) What is the population after 15 days?
c) According to this model, how long will it take for the entire trout population to die?
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Lecture Section Objectives Assignment
27 5.4 Area and the Fundamental Theorem of Calculus5.4: 1-45 eoo, 53, 57, 61-73 odd, 75, 81, 91
Learning Objectives:
1. Understand the definition of definite integral and the difference between indefinite integral and definite integral.
2. Use the Fundamental Theorem of Calculus to evaluate definite integrals.3. Find the average values of functions over closed intervals.4. Use properties of even and odd functions to help evaluate definite integrals.5. Use definite integrals to solve marginal analysis and annuities problems.
We know how to use geometric formulas to find area of simple regions such as rectangles, triangles and circles. How can we find the area of nonstandard regions such as the region R shown in the Figure shown below?
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Example 1: Evaluate .
\
Proof of the Fundamental Theorem of Calculus
Let be a partition of [a, b] with a = x0 < x1 < x2 < ... < xn-1 < xn = b
Using this partition, can be rewritten as
By the Mean Value Theorem, there exists a number in each subinterval
(call it ci) such that
Because F' is f, . We let xi = xi - xi-1, which means we can
rewrite the sum, above, as
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Taking the limit as ,
Note: The definite integral do not necessarily represent areas and can be negative, zero, or positive.
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Example 2: Find the area of the region bounded by the x-axis and the graph of , .
Example 3: Evaluate each definite integral
a)
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b)
c)
d)
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Example 4: The velocity v of blood at a distance r from the center of an artery of radius R can be modeled by where k is a constant. Find the average velocity along a radius of the artery. (Use 0 and R as the limits of integration.)
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Example 5: Evaluate each definite integral
a)
b)
Example 6: You deposit $2000 each year for 15 years in an individual retirement account (IRA) paying 10% interest. How much will you have in your IRA after 15 years?
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Lecture Section Objectives Assignment
28 5.5 The Area of a region Bounded by Two Graphs5.5: 1-7 odd, 15-29 odd, 35, 37 , 51
Learning Objectives:
1. Use integral to find the areas of regions bounded by two graphs.2. Solve real-life problems using the areas of regions bounded by two graphs.
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The Area between Curves
Case I : The area between two continuous functions on the interval [a, b] with
or
Case II: The area between two continuous functions on the interval [c, d] with
or
* Note that if the area between f and g is , regardless of the
signs of f and g.
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Guidelines:
1. Sketch the curves to determine which function is the upper function.2. Set the functions equal to find their intersections if necessary.3. Set up the definite integral and evaluate.
Example 1: Find the area bounded by
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Example 2: Determine the area of the region enclosed by
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Example 3: Determine the area of the region bounded by
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Sometimes it is easier to integrate with respect to y than with respect to x to find the area between two curves.
Example 4: Determine the area of the region bounded by
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Example 5: An epidemic was spreading such that t weeks after its outbreak it had infected , people. Twenty-five weeks after the outbreak, a vaccine was developed and administered to the public. At that point, the number of people infected was governed by the model . Approximate the number of people that the vaccine prevented from becoming ill during the epidemic.
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Lecture Section Objectives Assignment
29 5.6The Definite Integral as the Limit of a Sum
– the Midpoint Rule 5.6: 1-12 odd, 17-25 odd
Learning Objective:1. Use the Midpoint Rule to approximate definite integrals.
Sometimes you cannot use the Fundamental Theorem of Calculus to evaluate a definite integral unless you can find an antiderivative of the integrand. For example,
, . In those cases, you can approximate the value of the
integral using some approximation techniques, like Midpoint Rule, Trapezoidal Rule and Simpson’s Rule.
Example 1: Use five rectangles to approximate the area of the region bounded by the graph of , the x-axis, and the lines and .
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Note: as n increases, the approximation tends to improve.
Example 2: Use the Midpoint Rule with to approximate the area of the region bounded by the graph of f and the x-axis over the interval.
,
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Example 3: Use the Midpoint Rule with to approximate the area of the region. Compare your result with the exact area obtained with a definite integral.
,
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The midpoint approximation becomes better and better as n increases. In fact, the limit of this sum as n approaches infinity is exactly equal to the definite integral on the closed
interval .
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Lecture Section Objectives Assignment30 6.5 The Trapezoidal Rule and Simpson’s Rule 6.5: 1-12 odd, 17-25 odd
Learning Objectives:
1. Use the Trapezoidal Rule and Simpson’s Rule to approximate definite integrals.2. Understand how to analyze the sizes of the errors when approximating definite
integrals with the Trapezoidal Rule and Simpson’s Rule.
In this section, we will develop two methods for estimating by thinking of the
integral as an area problem and using known shapes to estimate the area under the curve.
We will break up the interval [a, b] into n subintervals of width , where .
Then on each subinterval we will approximate the function with a straight line that is equal to the function values at either endpoint of the interval. Here is a sketch of this case for . Each of these objects is a trapezoid and the area of the trapezoid in the interval
is given by
So, if we use n subintervals the integral is approximately,
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Upon doing a little simplification we arrive at the general Trapezoid Rule.
where and
Note that all the function evaluations, with the exception of the first and last, aremultiplied by 2.
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In the Trapezoid Rule we approximated the curve with a straight line. For Simpson’sRule we are going to approximate the function with a quadratic and we’re going torequire that the quadratic agree with three of the points from our subintervals. Below is asketch of this using . Each of the approximations is colored differently so we can see how they actually work.
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Notice that each approximation actually covers two of the subintervals. This is thereason for requiring n to be even. It can be shown that the area under the approximationon the intervals and is,
If we use n subintervals the integral is then approximately,
Upon simplifying we arrive at the general Simpson’s Rule.
where n is even and
Note that all the function evaluations at points with odd subscripts are multiplied by 4 and all the function evaluations at points with even subscripts (except for the first and last) are multiplied by 2.
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Example 1: Using and all three rules to approximate the value of the following integral.
a. Maple gives the following value for this integral
=16.45262776
b. Midpoint Rule:
c. Trapezoid Rule:
d. Simpson’s Rule:
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None of the estimations in the previous example are all that good. The best approximation in this case is from the Simpson’s Rule and yet it’s still had an error ofalmost 1. To get a better estimation we would need to use a larger n. So, for completeness sake here are the estimates for some larger value of n.
Example 2: Cardiac output, the volume of blood pumped by a person’s heart over an interval of time can be measured by injecting dye into a vein near the heart. Cardiac
output can be measured by the quantity . F(L/min) represents the cardiac
output(flow rate), R(mg/s) is the constant rate at which the dye is injected into the vein, and C(t) (mg/dL) is the concentration of the dye in the bloodstream at time t(sec).
a. Use the following data and the trapezoidal rule to estimate area under the concentration curve of the 30 second measurement period.
b. Use Simpson’s rule to estimate area under the concentration curve of the 30 second measurement period.
c. If the flow at 30 seconds is 5L/min, find R. Use the estimate for area under the concentration curve determined by Simpson’s Rule.
Example 3: Clearance measures the volume of blood that flows through an organ of elimination per unit time from which all drug is extracted. Clearance can be determined by dividing the dosage of a drug by the area under the concentration curve (IV).
a. Find the clearance for a 50 mg dosage of a drug using the following times and respective concentrations.
1. Use the basic integration formulas and substitution to find indefinite integrals and evaluate definite integrals.
2. Understand how to use integration to solve real-life problems.
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Example 1: Use the substitution to find the indefinite integral .
Example 2: Find .
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Example 3: Find the indefinite integral .
Example 4: Evaluate .
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Example 5: The probability of recall in an experiment is modeled by
, where x is the percent of recall. What is the probability of
recalling between 40% and 80%?
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Lecture Section Objectives Assignment
32 6.3 Partial Fractions and Logistic Growth
6.3: 1-43 eoo, 55, 56, 57,
59
Learning Objectives:
1. Use partial fractions to find indefinite integrals.2. Use logistic growth functions to model real-life situations.
Compare the following two integrals:
Case 1: The numerator is the derivative of the denominator or a constant multiple of the derivative of the denominator,
Case 2: The numerator is not the derivative of the denominator and it is not a constant multiple of the derivative of the denominator. How do we evaluate it?
If we notice that
Then the integral can be evaluated.
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Example 1: Write the partial fraction decomposition for and evaluate it.
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Example 2: Find .
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Example 3: Find .
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Deriving the Logistic Growth Function
Example 4: Solve the equation . Assume and .
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Example 5: On a college campus, 50 students return from semester break with a
contagious flu virus. The virus has a history of spreading at a rate of
where N is the number of students infected after t days.
(a) Find the model giving the number of students infected with the virus in terms of the number of days since returning from semester break.
(b) If nothing is done to stop the virus from spreading, will the virus spread to infect half the student population of 1000 students? Explain your answer.
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Lecture Section Objectives Assignment
35-36 7.3-7.4Functions of Several Variables
Partial Derivatives
7.3: 1-13 eoo, 157.4: 1-27 eoo, 29, 35, 39,
43, 45, 55, 57, 63
Understanding Goals:
3. Understand how to evaluate functions of several variables.4. Understand how to find the domains and ranges of functions of several variables.5. Understand how to find the first partial derivatives and higher-order derivatives of
functions of two or more variables.6. Understand how to find the slopes of surfaces in the x- and y-directions and use
partial derivatives to answer questions about real-life situations.
Many quantities in science, business, and technology are functions of several variables.
Example 1: Evaluate Functions of Several VariablesFind the function values of .
a) For , find .
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b) For , find .
Example 2: Find the domain and range of the function .
Example 3: Find and for .
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Example 4: Find the first partial derivative of and evaluate each at the
point .
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Graphical Interpretation of Partial Derivatives:
Example 5: Find the slope of the surface given by at the point in
(a) the x-direction and (b) the y-direction.
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Example 6: Find the three partial derivatives of the function .
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Example 7: Find the second partial derivatives , , , and of
.
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Lecture Section Objective Assignment34 None Zero Order Differential Equations Attached
Zero-order Process
Modeled by a linear function , where k is the rate constant.
Rate of change is constant:
Example 1. Solve the decreasing rate equation for A.
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Example 2. Sketch the linear function A and the rate equation.
Example 3. Determine the half-life formula of a zero-order process.
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Example 4. The eliminating rate of alcohol by humans is constant at 0.02/hr.a. If the initial percent blood alcohol is 0.162, write the equation that can be used
to determine the amount of alcohol at any hour after ingestion (solve the rate equation for A).
b. The legal blood alcohol limit in many states is 0.08. At what time is the percent alcohol 0.08?
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ASSIGNMENT:1. Suppose the absorption rate by some organism of a chemical is constant at 0.12/h.
a. If the initial concentration in the blood is 0.02 mg/dL, find the equation that can
be used to determine the amount of chemical present at any time.b. If the amount of chemical exceeds 0.10 mg/dL, the animal will die. When will this happen?c. When is the concentration double the initial concentration?
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Lecture Section Objective Assignment34 None First Order Differential
Equations Finish the problems on this
worksheet
Quick-check: How do you know if a process is zero-order?
First-order Process
Modeled by a exponential function , where k is the rate constant.
Rate of change for a decreasing 1st order process is :
Rate of change for a increasing 1st order process is :
1. Solve the decreasing rate equation for A.
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2. Sketch the exponential function and the rate equation.
3. Determine the half-life formula for a decreasing 1st order process.
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4. The rate of a chemical reaction at any time (s) during the reaction is given by
a. If the initial concentration of the reactant was 1.2 M, find the function A.
b. Find the half life of the chemical.
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5. The rate of change of the concentration of cyclosporine in the blood plasma t minutes after an intravenous dose is given by
a. Determine the concentration function if the initial concentration is 0.1 mg/dL. What is the concentration after 1 hour?
b. What is the half-life of cyclosporine?
c. Sketch the concentration function on the appropriate domain.
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6. A certain industrial machine depreciates so that its value after t years becomes dollars. (Courtesy of Applied Calculus, Hoffman, L.)
a. What is the rate equation for the value of the machine?
b. When will the machine be worth half of its initial value?
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Lecture Section Objective Assignment34 None Second Order Differential Equations
Quick-check: How do you know if a process is first-order?
Second-order Process
Modeled by a rectangular hyperbolic function , where k is the rate
constant.
Rate of change for a decreasing 2nd order process is :
Rate of change for a increasing 2nd order process is :
1. Solve the rate equation of a decreasing 2nd order process for A.
1/[A] = 1/[A]0 + k t
2. Graph the rectangular hyperbolic function and the rate equation.
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3. Determine the half-life formula of A.
1. The decomposition of HI is a second-order process with rate constant at .
a. Determine the equation that models the chemical decomposition.
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b. Determine the half-life of HI.
2. The rate of change of the number of bacteria remaining following treatment with an antibiotic is proportional to the number of bacteria squared.
a. If the initial number of bacteria is 1200, find the equation that can be used to determine the number of bacteria at any time (hours).
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b. When will the number of bacteria be half of the initial amount?
