Chapter 4-Applications of the Derivative Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John...

Chapter 4-Applications of the Derivative

Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

Chapter 4-Applications of the Derivative4.1 Related Rates


The Role of the Chain Rule in Related Rates Problems

If x depends on a third variable t, then so does y through the equation y = g(x).


4.1 Related Rates


The Role of the Chain Rule in Related Rates Problems

EXAMPLE: If a train that is approaching a platform with speed s (measured in m/s) sounds a horn with frequency 500 Hz, then, according to the Doppler effect, the frequency heard by a stationary observer on the platform is

If the train is decelerating at a constant rate of 4 m/s2, what is the rate of change of º when the train’s speed is 25 m/s?


4.1 Related Rates


The Role of Implicit Differentiation

EXAMPLE: Suppose that y3 + xy − 4x = 0. If x = 4 and dx/dt = 3 when t = 5, what is dy/dt when t = 5?


4.1 Related Rates


Basic Steps For Solving a Related Rates Problem

1. Identify the quantities that are varying, and identify the variable (this is often “time”) with respect to which the change in these quantities is taking place.2. Establish a relationship (an equation) between the quantities isolated in Step 1.3. Differentiate the equation from Step 2 with respect to the variable identified in Step 1. Be sure to apply the Chain Rule carefully.4. Substitute the numerical data into the equation from Step 3 and solve for the unknown rate of change.


4.1 Related Rates


Basic Steps For Solving a Related Rates Problem

EXAMPLE: A sample of a new polymer is in the shape of a cube. It is subjected to heat so that its expansion properties may be studied. The surface area of the cube is increasing at the rate of 6 square inches per minute. How fast is the volume increasing at the moment when the surface area is 150 square inches?


4.1 Related Rates


Quick Quiz

1. Suppose that y = x3 and that x = 2 and dy/dt = 60 when t = 10? What is dx/dt when t = 10?2. Suppose that x and y are positive variables, that 2x + y3 = 14, that x is a function of t, and that dx/dt = 4for all t. What is dy/dt when x = 3?

Chapter 4-Applications of the Derivative4.2 The Mean Value Theorem


Maxima and MinimaLet f be a function with domain S. We say that f has a local maximum at the point c in S if there is a > 0 such that f(x) ≤ f(c) for all x in S such that |x − c| < . We call f(c) a local maximum value for f. Similarly, We say that f has a local minimum at the point c in S if there is a > 0 such that f(x) ≥ f(c) for all x in S such that |x − c| < . We call f(c) a local minimum value for f.


4.2 The Mean Value Theorem


Maxima and Minima

EXAMPLE: Discuss local and absolute extreme values for the function f(x) = sec (x), −3 ≤ x ≤ 3.




Locating Maxima and Minima

THEOREM: (Fermat) Let f be defined on an open interval that contains the point c. Suppose that f is differentiable at c. If f has a local extreme value at c, then f’(c) = 0.




Locating Maxima and Minima

EXAMPLE: Use the first derivative to locate local and absolute extrema for the function f(x) = cos (x).




Rolle’s Theorem and the Mean Value Theorem

THEOREM: (Rolle’s Theorem) Let f be a function that is continuous on [a, b] and differentiable on (a, b). If f(a) = f(b), then there is a number c (a, b) such that f’(c) = 0.

THEOREM: (Mean Value Theorem) If f is a function that is continuous on [a, b] and differentiable on (a, b), then there is a number c (a, b) such that




Rolle’s Theorem and the Mean Value Theorem

EXAMPLE: An automobile travels 120 miles in three hours. Assuming that the position function p is continuous on the closed interval [0, 3] and differentiable on the open interval (0, 3), can we conclude that at some moment in time the car is going precisely 40 miles per hour?




An Application of the Mean Value Theorem

THEOREM: Let f be a differentiable function on an interval (). If f’(x) = 0 for each x in (), then f is a constant function.

THEOREM: If F and G are differentiable functions such that F’(x) = G’(x) for every x in (), then there is a constant C such that G(x) = F(x) + C for every x in the interval () .




An Application of the Mean Value Theorem

EXAMPLE: Suppose that F’(x) = 2x for all x and that F (1) = 10. Find F (4).




Quick Quiz1. Suppose that f is differentiable on an open interval containing the point c.

a) True or false: if f’ (c) = 0, then a local extremum of f (x) occurs at x = c.b) True or false: if a local extremum of f (x) occurs at x = c, then f’ (c) = 0.

2. The function f(x) = xe−x has a local maximum. Where does it occur?3. What upper bound on the number of local extrema of f (x) = x4 + ax3 + bx2 + cx + d can be deduced from Fermat’s Theorem?4. If f is continuous on [7, 13], differentiable on (7, 13), and if f (7) = 11 and f (13) = 41, then what value must f’ (x) assume for some x in (7, 13)?5. If F’ (x) = 2 cos (x) and F () = 3, then what is F (/6)?

Chapter 4-Applications of the Derivative4.3 Maxima and Minima of

Functions


Using the Derivative to Tell When a Function is Increasing or Decreasing

THEOREM: If f’(x)>0 for each x in an interval I, then f is increasing on I. If f’(x)<0 for each x in an interval I, then f is decreasing on I.


4.3 Maxima and Minima of Functions



EXAMPLE: Examine the function f(x) = x3 −6x2 +13x−7 to determine intervals on which it is increasing or decreasing.





EXAMPLE: On what intervals is the function f(x) = x3 −3x2 -9x+5 increasing? On which intervals is it decreasing?




Critical Points and The First Derivative Test for Local Extrema

DEFINITION: Let c be a point in an open interval on which f is continuous. We call c a critical point for f if one ofthe following two conditions holds:i) f is not differentiable at c; orii) f is differentiable at c and f’(c) = 0.





THEOREM: (The First Derivative Test) Suppose that I = () is an open interval contained in the domain ofa continuous function f, that a point c in I is a critical point for f, and that f is differentiable at every point of Iother than c.(i) If f’(x) < 0 for < x < c and f0(x) > 0 for c < x < , then f has a local minimum at c.(ii) If f’(x) > 0 for < x < c and f0(x) < 0 for c < x < then f has a local maximum at c.(iii) If f’(x) does not change sign at c, then f has neither a local minimum nor a local maximum at c.





EXAMPLE: Find and analyze the critical points for the function f(x) = x − sin (x).

EXAMPLE: Find and analyze the critical points for the function f(x) = x · (x − 1)1/3.




Quick Quiz

1. True or false: If f (x) is differentiable for every real number x and if f’ (c) = 0, then f has either a local minimum or a local maximum when x = c.2. True or false: If f (x) is differentiable for every real number x, if f’(x) < 0 for all x < c, and if f’ (x) > 0 for all x > c,then f (x) has a local minimum at x = c.3. True or false: If f (x) is defined for every real number x, if f’(x) > 0 for all x < c, and if f’(x) < 0 for all x > c, thenf (x) has a local maximum at x = c.4. Find and analyze the local extrema for the function f(x) = x3/3 + x2 − 15x + 5.

Chapter 4-Applications of the Derivative4.4 Applied Maximum-

Minimum Problems


EXAMPLE 1 The total mechanical power P (in watts) that an individual requires for walking a fixed distance at constant speed v (in kilometers per hour) with step length s (in meters) is

where L is the individual’s maximum step size. In this example we use L = 1 m. The positive constants and depend on the distance walked and the individual’s gait. In this example we use = 1.4 and = 45. For what s is P (s) minimized if v = 5? If v = 6?


4.4 Applied Maximum-Minimum Problems


BASIC RULE FOR FINDING EXTREMA OF A CONTINUOUS FUNCTION ON A CLOSED BOUNDED INTERVAL

To find the extrema of a continuous function f on a closed interval [a, b], we test

(i) The points in (a, b) where f is not differentiable;(ii) The points in (a, b) where f’ exists and equals 0;(iii) The endpoints a and b.

In brief, we should test the critical points and the endpoints.

Closed Intervals




EXAMPLE: A man builds a rectangular garden along the side of his house. He will put fencing on three sides. If he has 200 feet of fencing available, then what dimensions will yield the garden of greatest area?

Closed Intervals

EXAMPLE: A swimmer is 600 m straight out from a landmark on shore, as shown in Figure 4. She wants tomeet some friends who are 800 m down the beach from the landmark. The swimmer can swim at a rate of 100 m/min and run on the beach at the rate of 200 m/min. Toward what point on shore should she swim in order to minimize the time it takes for her to join her friends?




Examples with the Solution at an Endpoint

EXAMPLE: A 10 inch piece of wire can be bent into a circle or a square. Or, it can be cut into two pieces. In this case a circle will be formed from the first piece and a square from the other. What is the maximal area that can result?




An Example Involving a Transcendental FunctionEXAMPLE: Mr. Woodman is viewing a six foot long tapestry that is hung lengthwise on a wall of a 20 ft by 20 ft room. The bottom end of the tapestry is two feet above his eye level. At what distance from the tapestry should Mr. Woodman stand in order to obtain the most favorable view? That is, for what value of x is angle maximized?




Quick Quiz1. True or false: If f is continuous on [a, b], then its maximum and minimum must occur at critical points of thefunction.2. True or false: If f is defined and continuous on [1, 5], if f is differentiable on the intervals (1, 2) and (2, 5), and if f’(3) = 0, then the maximum value of f (x) must be among the numbers f (1), f (2), f (3), and f (5).3. True or false: If f is defined and continuous on [2, 4], if f (2) = 0 and f (4) = 3, and if f’(x) is not equal to 0 for any x in the interval (2, 4), then the maximum value of f (x) must be 3.4. Calculate the minimum values of f (x) = x + 100/x on the intervals [1, 5], [5, 12], and [12, 20].

Chapter 4-Applications of the Derivative4.5 Concavity


DEFINITION: Let the domain of a differentiable function f contain an open interval I. If f’ increases as x moves from left to right in I, then the graph of f is said to be concave up on I . If f’ decreases as x moves from left to right in I, then the graph of f is said to be concave down on I


4.5 Concavity


EXAMPLE: Discuss the concavity of f(x) = 1+x2 and g(x) = x4


4.5 Concavity


THEOREM: (The Second Derivative Test for Concavity) Suppose that the function f is twice differentiable on an open interval I.a) If f’’ (x) > 0 for every x in I then the graph of f is concave up on I.b) If f’’ (x) < 0 for every x in I then the graph of f is concave down on I.

Using the Second Derivative to Test for Concavity


4.5 Concavity


EXAMPLE: Apply the Second Derivative Test for Concavity to the function f(x) = 1/x.

Using the Second Derivative to Test for Concavity


4.5 Concavity


DEFINITION: Let f be a continuous function on an open interval I. If the graph of y = f (x) changes concavity as x passes from one side to the other of a point c in I, then the point (c, f (c)) on the graph of f is called a point of inflection (or an inflection point). We also say that f has a point of inflection at c.

Points of Inflection


4.5 Concavity


EXAMPLE: Examine the graph of f(x) = x-(x-1)3 for concavity and points of inflection.

Points of Inflection


4.5 Concavity


THEOREM: (The Second Derivative Test for Extrema) Let f be twice differentiable (both f0 and f00 exist) on an open interval containing a point c at which f’(c) = 0.

i. If f’’(c) > 0 then c is a local minimum;ii. If f’’(c) < 0 then c is a local maximum.iii. If f’’(c) = 0 then no conclusion is possible from this test.

The Second Derivative Test at a Critical Point


4.5 Concavity


EXAMPLE: Use Fermat’s Theorem and the Second Derivative Test for Extrema to determine the local extrema of f(x)= 2x3+15x2+24x+23

The Second Derivative Test at a Critical Point


4.5 Concavity


1. True or false: f’’ (4) = 0 tells us that (4, f (4)) is a point of inflection for y = f (x).2. True or false: f’’ (x) < 0 on (1, 5) and f’’ (x) > 0 on (5, 5.1) tells us that (5, f (5)) is a point of inflection for y = f (x).3. Suppose that f’’ (x) = (x + 2)4(x − 3). Does f(x) have a point of inflection at x = −2? At x = 3?4. True or false: f’’ (4) < 0 tells us that f (x) has a maximum at x = 4.5. True or false: f’ (4) = 0 and f’’ (4) > 0 tell us that f (x) has a maximum at x = 4.

Quick Quiz

Chapter 4-Applications of the Derivative4.6 Graphing Functions


Basic Strategy of Curve Sketching1. Determine the domain and the range.2. Find all horizontal and vertical asymptotes.3. Calculate the first derivative and find the critical points for the function.4. Find the intervals on which the function is increasing or decreasing.5. Calculate the second derivative and find the intervals on which the function is concave up or concave down.6. Identify all local maxima, local minima, and points of inflection.7. Plot these points, as well as the y-intercept (if applicable) and any x-intercepts (if they can be computed). Sketch the asymptotes.8. Connect the points plotted in step 7, keeping in mind concavity, local extrema, and asymptotes.


4.6 Graphing Functions


Basic Strategy of Curve Sketching

EXAMPLE: Graph the function f(x) = 5x/(x − 2)2

EXAMPLE: Sketch the graph of g(x) = 4x3 + x4.

EXAMPLE: Sketch the graph of




Periodic Functions

EXAMPLE: Sketch the graph of f(x) = sin(x)/(2+cos(x)).




Skew-Asymptotes

DEFINITION: The line y = mx + b is a skew-asymptote of f if

or




Skew-Asymptotes

EXAMPLE: Sketch the graph of f (x) = x3/(x2 − 4).




Quick Quiz1. What are the periods of sin (x/2), sec (x), and cot (x)?2. What is the skew-asymptote of f (x) =(3x2 − 2x + 5 cos (x))/x?3. If f (x) = 2x3/ (x − 1) 2, then f’ (x) = 2x2 (x − 3) / (x − 1) 3 and f’’ (x) = 12x/ (x − 1)4. Determine (i) the interval or intervals on which f increases, (ii) the interval or intervals on which f decreases, (iii) all local extrema, (iv) the interval or intervals on which the graph of f is concave up, (v) the interval or intervals on which the graph of f is concave down, (vi) all points of inflection, (vii) all horizontal and vertical asymptotes, and (viii) all skew-asymptotes.

Chapter 4-Applications of the Derivative4.7 l’Hôpital’s Rule


l’Hôpital’s Rule the Indeterminate form 0/0

THEOREM: (l’Hôpital’s Rule) Let f and g be differentiable functions on (a, c) U (c, b). If

then

provided that the limit on the right exists as a finite or infinite limit.


4.7 l’Hôpital’s Rule


l’Hôpital’s Rule the Indeterminate form 0/0

EXAMPLE: Evaluate

EXAMPLE: Calculate




l’Hôpital’s Rule the Indeterminate form ∞/ ∞

THEOREM: Let f(x) and g(x) be differentiable functions on (a, c)U(c, b). If lim xc f(x) and lim xc g(x) both exist and equal + ∞ or − ∞ (they may have the same sign or different signs) then

provided this last limit exists either as a finite or infinite limit.




l’Hôpital’s Rule the Indeterminate form ∞/ ∞

EXAMPLE: Evaluate




The Indeterminate form 0∞

EXAMPLE: Evaluate the limit




l’Hôpital’s Rule for limits at ∞

THEOREM: Let f and g be differentiable functions. If lim x ∞ f(x) = lim x ∞ g(x) = 0, or if lim x ∞ f(x) =± ∞ and lim x ∞ g(x) = ± ∞, then

Provided that this last limit exists either as a finite or infinite limit. The same result holds for the limit as x -∞.




l’Hôpital’s Rule for limits at ∞

EXAMPLE: Evaluate




The Indeterminate forms 00, 1∞, and ∞0

EXAMPLE: Evaluate

EXAMPLE: Evaluate

EXAMPLE: Evaluate




Quick Quiz1. If lim x c f(x) = lim x c g(x) = , then what must be in order to apply l’Hôpital’s Rule to the limit lim x c (f(x)/g (x))?2. Evaluate lim x 0 tan (3x) / tan (2x).3. For which indeterminate forms is it helpful to first apply the logarithm?4. Evaluate

Chapter 4-Applications of the Derivative4.8 The Newton-Raphson

Method



4.8 The Newton-Raphson Method


EXAMPLE: Let P denote the point at which the graphs of y = cos (x) and y = x cross. A very rough estimate of the x-coordinate of P is /3. Apply the fundamental idea behind the Newton-Raphson method to find a better approximation.

The Geometry of the Newton-Raphson Method




The Newton-Raphson Method

Calculating with the Newton-Raphson Method

If f is a differentiable function, then the (n + 1)st estimate xn+1 for a zero of f is obtained from the nth estimate xn by the formula

provided that f’(xn) ≠ 0.




Calculating with the Newton-Raphson Method

EXAMPLE: Use the Newton-Raphson Method to determine to within an accuracy of 10−7.




Quick Quiz

1. If we wish to apply the Newton-Raphson Method to calculate 41/5 by setting x1 = 1 and letting xn+1 = (xn) for n ≥ 1, then what function can we use?2. If we use the Newton-Raphson Method to find a root of f (x) = x5 +2x−40 and x = 2 is our first estimate of the root, then what is our second?

Chapter 4-Applications of the Derivative4.9 Antidifferentiation and

Applications


Antidifferentiation

DEFINITION: Let f be defined on an open interval I. If F is a differentiable function such that F’(x) = f(x) for all x in I, then F is said to be an antiderivative of f on I. The collection of all antiderivatives of a function f is denoted by

This expression is called the indefinite integral of f.


4.9 Antidifferentiation and Applications


Antidifferentiation

EXAMPLE: Calculate x3 dx.




Antidifferentiating Powers of x





EXAMPLE: Show that





THEOREM: Let f and g be functions whose domains contain an open interval I. Let F be an antiderivative for f and G an antiderivative for g. Then





EXAMPLE: Calculate

EXAMPLE: Calculate the indefinite integral




Antidifferentiation of other Functions

EXAMPLE: Calculate sin (5x) dx.




Velocity and Acceleration

Velocity is an antiderivative of acceleration. Position is an antiderivative of velocity.




Velocity and Acceleration

EXAMPLE: A police car accelerates from rest at a rate of 3 mi/min2. How far will it have traveled at the moment that it reaches a velocity of 65 mi/hr?

EXAMPLE: An object is dropped from a window on a calm day. It strikes the ground precisely 4 seconds later. From what height was the object dropped?




Quick Quiz1. Suppose that F (x) is an antiderivative of cos (x) such that F (/4) = . What is F (/3)?2. True or false: There is exactly one antiderivative F(x) of x−2 on (0,1) such that F (1) = 100.3. True or false: There is exactly one antiderivative F(x) of 1/x on {x : x ≠0} such that F (1) = 100.4. True or false: ln (|x|) dx = 1/x + C.

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Chapter 4-Applications of the Derivative Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John...

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