1. Chapter 4-Applications of the Derivative Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

2. Chapter 4-Applications of the Derivative 4.1 Related Rates Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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).

3. Chapter 4-Applications of the Derivative 4.1 Related Rates Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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. Chapter 4-Applications of the Derivative 4.1 Related Rates Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Role of Implicit Differentiation EXAMPLE: Suppose that y 3 + xy − 4x = 0. If x = 4 and dx/dt = 3 when t = 5, what is dy/dt when t = 5?

5. Chapter 4-Applications of the Derivative 4.1 Related Rates Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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.

6. Chapter 4-Applications of the Derivative 4.1 Related Rates Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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?

7. Chapter 4-Applications of the Derivative 4.1 Related Rates Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. Suppose that y = x 3 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 + y 3 = 14, that x is a function of t, and that dx/dt = 4 for all t. What is dy/dt when x = 3?

8. Chapter 4-Applications of the Derivative 4.2 The Mean Value Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Maxima and Minima Let 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| 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.

9. Chapter 4-Applications of the Derivative 4.2 The Mean Value Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Maxima and Minima EXAMPLE: Discuss local and absolute extreme values for the function f(x) = sec (x), −3  ≤ x ≤ 3 .

10. Chapter 4-Applications of the Derivative 4.2 The Mean Value Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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.

11. Chapter 4-Applications of the Derivative 4.2 The Mean Value Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Locating Maxima and Minima EXAMPLE: Use the first derivative to locate local and absolute extrema for the function f(x) = cos (x).

12. Chapter 4-Applications of the Derivative 4.2 The Mean Value Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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

13. Chapter 4-Applications of the Derivative 4.2 The Mean Value Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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?

14. Chapter 4-Applications of the Derivative 4.2 The Mean Value Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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 (  ).

15. Chapter 4-Applications of the Derivative 4.2 The Mean Value Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved An Application of the Mean Value Theorem EXAMPLE: Suppose that F’(x) = 2x for all x and that F (1) = 10. Find F (4).

16. Chapter 4-Applications of the Derivative 4.2 The Mean Value Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. 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) = x 4 + ax 3 + bx 2 + 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)?

17. Chapter 4-Applications of the Derivative 4.3 Maxima and Minima of Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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.

18. Chapter 4-Applications of the Derivative 4.3 Maxima and Minima of Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Using the Derivative to Tell When a Function is Increasing or Decreasing EXAMPLE: Examine the function f(x) = x3 −6x2 +13x−7 to determine intervals on which it is increasing or decreasing.

19. Chapter 4-Applications of the Derivative 4.3 Maxima and Minima of Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Using the Derivative to Tell When a Function is Increasing or Decreasing EXAMPLE: On what intervals is the function f(x) = x 3 −3x 2 -9x+5 increasing? On which intervals is it decreasing?

20. Chapter 4-Applications of the Derivative 4.3 Maxima and Minima of Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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 of the following two conditions holds: i) f is not differentiable at c; or ii) f is differentiable at c and f’(c) = 0.

21. Chapter 4-Applications of the Derivative 4.3 Maxima and Minima of Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Critical Points and The First Derivative Test for Local Extrema THEOREM: (The First Derivative Test) Suppose that I = (  ) is an open interval contained in the domain of a continuous function f, that a point c in I is a critical point for f, and that f is differentiable at every point of I other than c. (i) If f’(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.

22. Chapter 4-Applications of the Derivative 4.3 Maxima and Minima of Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Critical Points and The First Derivative Test for Local Extrema 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.

23. Chapter 4-Applications of the Derivative 4.3 Maxima and Minima of Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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, 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, then f (x) has a local maximum at x = c. 4. Find and analyze the local extrema for the function f(x) = x 3 /3 + x 2 − 15x + 5.

24. Chapter 4-Applications of the Derivative 4.4 Applied Maximum- Minimum Problems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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?

25. Chapter 4-Applications of the Derivative 4.4 Applied Maximum-Minimum Problems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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

26. Chapter 4-Applications of the Derivative 4.4 Applied Maximum-Minimum Problems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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 to meet 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?

27. Chapter 4-Applications of the Derivative 4.4 Applied Maximum-Minimum Problems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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?

28. Chapter 4-Applications of the Derivative 4.4 Applied Maximum-Minimum Problems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved An Example Involving a Transcendental Function EXAMPLE: 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?

29. Chapter 4-Applications of the Derivative 4.4 Applied Maximum-Minimum Problems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. True or false: If f is continuous on [a, b], then its maximum and minimum must occur at critical points of the function. 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].

30. Chapter 4-Applications of the Derivative 4.5 Concavity Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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

31. Chapter 4-Applications of the Derivative 4.5 Concavity Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Discuss the concavity of f(x) = 1+x 2 and g(x) = x 4

32. Chapter 4-Applications of the Derivative 4.5 Concavity Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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

33. Chapter 4-Applications of the Derivative 4.5 Concavity Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Apply the Second Derivative Test for Concavity to the function f(x) = 1/x. Using the Second Derivative to Test for Concavity

34. Chapter 4-Applications of the Derivative 4.5 Concavity Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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

35. Chapter 4-Applications of the Derivative 4.5 Concavity Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Examine the graph of f(x) = x-(x-1) 3 for concavity and points of inflection. Points of Inflection

36. Chapter 4-Applications of the Derivative 4.5 Concavity Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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

37. Chapter 4-Applications of the Derivative 4.5 Concavity Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Use Fermat’s Theorem and the Second Derivative Test for Extrema to determine the local extrema of f(x)= 2x 3 +15x 2 +24x+23 The Second Derivative Test at a Critical Point

38. Chapter 4-Applications of the Derivative 4.5 Concavity Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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 (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

39. Chapter 4-Applications of the Derivative 4.6 Graphing Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Basic Strategy of Curve Sketching 1. 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.

40. Chapter 4-Applications of the Derivative 4.6 Graphing Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Basic Strategy of Curve Sketching EXAMPLE: Graph the function f(x) = 5x/(x − 2) 2 EXAMPLE: Sketch the graph of g(x) = 4x 3 + x 4. EXAMPLE: Sketch the graph of

41. Chapter 4-Applications of the Derivative 4.6 Graphing Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Periodic Functions EXAMPLE: Sketch the graph of f(x) = sin(x)/(2+cos(x)).

42. Chapter 4-Applications of the Derivative 4.6 Graphing Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Skew-Asymptotes DEFINITION: The line y = mx + b is a skew- asymptote of f if or

43. Chapter 4-Applications of the Derivative 4.6 Graphing Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Skew-Asymptotes EXAMPLE: Sketch the graph of f (x) = x 3 /(x 2 − 4).

44. Chapter 4-Applications of the Derivative 4.6 Graphing Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. What are the periods of sin (x/2), sec (x), and cot (x)? 2. What is the skew-asymptote of f (x) =(3x 2 − 2x + 5 cos (x))/x? 3. If f (x) = 2x 3 / (x − 1) 2, then f’ (x) = 2x 2 (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.

45. Chapter 4-Applications of the Derivative 4.7 l’Hôpital’s Rule Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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.

46. Chapter 4-Applications of the Derivative 4.7 l’Hôpital’s Rule Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved l’Hôpital’s Rule the Indeterminate form 0/0 EXAMPLE: Evaluate EXAMPLE: Calculate

47. Chapter 4-Applications of the Derivative 4.7 l’Hôpital’s Rule Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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 x  c f(x) and lim x  c 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.

48. Chapter 4-Applications of the Derivative 4.7 l’Hôpital’s Rule Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved l’Hôpital’s Rule the Indeterminate form ∞/ ∞ EXAMPLE: Evaluate

49. Chapter 4-Applications of the Derivative 4.7 l’Hôpital’s Rule Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Indeterminate form 0  ∞ EXAMPLE: Evaluate the limit

50. Chapter 4-Applications of the Derivative 4.7 l’Hôpital’s Rule Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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  -∞.

51. Chapter 4-Applications of the Derivative 4.7 l’Hôpital’s Rule Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved l’Hôpital’s Rule for limits at ∞ EXAMPLE: Evaluate

52. Chapter 4-Applications of the Derivative 4.7 l’Hôpital’s Rule Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Indeterminate forms 0 0, 1 ∞, and ∞ 0 EXAMPLE: Evaluate

53. Chapter 4-Applications of the Derivative 4.7 l’Hôpital’s Rule Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. 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

54. Chapter 4-Applications of the Derivative 4.8 The Newton-Raphson Method Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

55. Chapter 4-Applications of the Derivative 4.8 The Newton-Raphson Method Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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

56. Chapter 4-Applications of the Derivative 4.8 The Newton-Raphson Method Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Newton-Raphson Method Calculating with the Newton-Raphson Method If f is a differentiable function, then the (n + 1) st estimate x n+1 for a zero of f is obtained from the n th estimate x n by the formula provided that f’(x n ) ≠ 0.

57. Chapter 4-Applications of the Derivative 4.8 The Newton-Raphson Method Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Calculating with the Newton-Raphson Method EXAMPLE: Use the Newton-Raphson Method to determine to within an accuracy of 10 −7.

58. Chapter 4-Applications of the Derivative 4.8 The Newton-Raphson Method Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. If we wish to apply the Newton-Raphson Method to calculate 4 1/5 by setting x 1 = 1 and letting x n+1 =  (x n ) for n ≥ 1, then what function  can we use? 2. If we use the Newton-Raphson Method to find a root of f (x) = x 5 +2x−40 and x = 2 is our first estimate of the root, then what is our second?

59. Chapter 4-Applications of the Derivative 4.9 Antidifferentiation and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 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.

60. Chapter 4-Applications of the Derivative 4.9 Antidifferentiation and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Antidifferentiation EXAMPLE: Calculate  x 3 dx.

61. Chapter 4-Applications of the Derivative 4.9 Antidifferentiation and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Antidifferentiating Powers of x

62. Chapter 4-Applications of the Derivative 4.9 Antidifferentiation and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Antidifferentiating Powers of x EXAMPLE: Show that

63. Chapter 4-Applications of the Derivative 4.9 Antidifferentiation and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Antidifferentiating Powers of x 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

64. Chapter 4-Applications of the Derivative 4.9 Antidifferentiation and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Antidifferentiating Powers of x EXAMPLE: Calculate EXAMPLE: Calculate the indefinite integral

65. Chapter 4-Applications of the Derivative 4.9 Antidifferentiation and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Antidifferentiation of other Functions EXAMPLE: Calculate  sin (5x) dx.

66. Chapter 4-Applications of the Derivative 4.9 Antidifferentiation and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Velocity and Acceleration Velocity is an antiderivative of acceleration. Position is an antiderivative of velocity.

67. Chapter 4-Applications of the Derivative 4.9 Antidifferentiation and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Velocity and Acceleration EXAMPLE: A police car accelerates from rest at a rate of 3 mi/min 2. 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?

68. Chapter 4-Applications of the Derivative 4.9 Antidifferentiation and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. 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.