Unit C Overview

1. Unit C Overview Application of Derivatives

2. Strategy for Solving Related Rate Problems • Identify all given quantities and quantities to be determined. • Write an equation relating the variables • Use chain rule to implicitly differentiate both sides of the equation with respects to time t. • Substitute and Solve

3. Finding related rate equations • Assume the radius r of a sphere is a differentiable function of t and let V be the volume of the sphere. Find an equation that related dV/dt and dr/dt

4. Finding Related Rate Equations Assume that the width w and the length l of a rectangle are functions of t and let A be the area of the rectangle. Find an equation that relates these. A = l•w = • l + • w

5. Finding related rate equations • Assume that the radius and height h of a cone are differentiable functions of t and let V = volume of the cone. Find an equation that relates dV/dt, dr/dt and dh/dt.

6. Example 1 Ripples in a pond • A pebble is dropped in a pond, causing ripples in the form of circles. The radius r of the outer ripple is increasing at a constant rate of 1 ft per second. When the radius is 4ft, at what rate is the total area A of the circle of water changing? • 1. The variables are area and radius of a circle. • 2. Equation to use: A = πr² • Given info: = 1 and r = 4 • Find: • = 2πr • • Plug in: • = 2π(4) • • = 8π ft²/sec

7. Example 2 Given info: r = 2 Find: Equation relating them: V = πr³ Differentiate: = 4πr² • Substitute: 4.5 = 4π(2)² • = 0.09 ft per minute • Air is being pumped into a spherical balloon at a rate of 4.5 cubic ft per minute. Find the rate of change of the radius when the radius is 2 ft.

8. Example 3 A Rising Balloon A hot-air balloon rising straight up from a level field is tracked by a range finder 700 feet from the lift-off point. At the moment the range finder’s elevation angle is , the angle is increasing at the rate of 0.16 radians per minute. How fast is the balloon rising at that moment?

9. Example 3 A Rising Balloon A hot-air balloon rising straight up from a level field is tracked by a range finder 700 feet from the lift-off point. At the moment the range finder’s elevation angle is , the angle is increasing at the rate of 0.16 radians per minute. How fast is the balloon rising at that moment?

10. Example 3 A Rising Balloon

11. Example 4 A Ladder 0.25 m/s

12. Example 5 • The radius of the base of a cylinder is increasing at a rate of 10ft/sec. • The height of the cylinder is fixed at 6 ft. • At a certain instant, the radius is 1 ft. • What is the rate of change of the volume of the cylinder at that instant (in cubic feet per second)?

13. Equation that relates: V = πr²h What do we know: What do we want to know R = 1ft h = 6ft = 10 = 0 Differentiate Implicitly: = π(r² + 2rh) Substitute: = π(1²•0 + 2•1•6•10) Solve: = 120π • The radius of the base of a cylinder is increasing at a rate of 10ft/sec. • The height of the cylinder is fixed at 6 ft. • At a certain instant, the radius is 1 ft. • What is the rate of change of the volume of the cylinder at that instant (in cubic feet per second)?

14. HW • Pg. 154 # 15, 18a, 20, 22

15. Extreme Values of Functions

16. Definition: Absolute (Global) Extreme Values Given a function f with domain D, then f (c) is the absolute maximum value of the function if and only if f (x) <f (c) for all x in the domain. Given a function f with domain D, then f (c) is the absolute minimum value of the function if and only if f (x) >f (c) for all x in the domain.

17. Examples Absolute Maximum Only Absolute Minimum Only

18. Exploring Extreme Values • On [ -] • f(x) = sinx • Takes on maximum value of 1 and minimum value of -1

19. Exploring Extreme Values • On [ -] • f(x) = cosx • Takes on maximum value of 1(once) and minimum value of 0 (twice)

20. Exploring Extreme Values • y = x² Domain: (-∞, ∞) • No absolute max • Absolute Minimum at (0,0)

21. Exploring Extreme Values Absolute Maximum of 4 at x=2 Absolut Minimum of 0 at x = 0 • y = x² Domain: [0,2]

22. Theorem 1: The Extreme Value Theorem If f is a continuous function on a closed interval [a, b], then f has both a maximum value and a minimum value on the interval.

24. Example of E.V.T • The function is continuous over a closed interval [a,b] • Max @ x = b • Min @ x = c2 a c1 c2 b

25. Definition: Local (Relative) Extreme Values Let c be an interior point on the domain of function f. Then f (c) is a local maximum value at c if and only if f(x) <f(c) for all x in the open interval containing c. f (c) is a local minimum value at c if and only if f(x) >f(c) for all x in the open interval containing c.

27. Decide whether each point is an absolute max/min, relative max/min, or neither.

28. Theorem 2: Local Extreme Values Theorem If a function f has a local maximum value or a local minimum value at an interior point c of its domain, and if f ’ exists at c, then f ’(c) = 0.

29. Definition: Critical Point A critical point at x = c on the interior of the domain of the function f exists where f ’(c) = 0 or f ’(c) is undefined. (Your book refers to this as a critical number) If f has a relative minimum or maximum at x = c, then c is a critical point (number) of f.

30. Finding Absolute Extrema • Find derivative of function • Find all critical points of f(x) that are in the interval [a,b], you do this by setting derivative equal to 0. • Evaluate the original function at the critical points and at end points • Identify the absolute extrema

31. Example 1 • Determine the absolute extrema for g(t) = 2t³ + 3t² - 12 t + 4 on [-4,2] Get the derivative: g’(t) = 6t² + 6t – 12 = 0 So we have two critical points at t = -2 and t = 1 Both points lie in the interval so we will use both of them.

32. Original functiong(t) = 2t³ + 3t² - 12 t + 4 • g(-2) = 24 • g(1) = -3 • g(-4) = -28 • g(2) = 8 • Our absolute max of g(t) is 24 and occurs at t= -2 • Our absolute min of g(t) is -28 and occurs at t = -4

33. Example 2 • Find the extrema of f(x) = 3x⁴ - 4x³ on the interval [-1, 2]. • Differentiate first: • f’(x) = 12x³ - 12x² • 12x³ - 12x² = 0 at • CP: x = 0 and x = 1

34. Critical numbers x = 0 and 1 • Evaluate at the critical numbers and endpoints. f(x) = 3x⁴ - 4x³ • F(-1) = 7 • F(0) = 0 • F(1) = -1 • F(2) = 16 Minimum Maximum

35. Example 3 • Find the extrema of f(x) = 2sinx – cos2x on the interval [0, 2𝝿]. • F’(x) = 2cosx + 2sin2x • 2cosx + 2sin2x = 0 when x = , , ,

36. Critical Numbers and endpoints f(x) = 2sinx – cos2x • F(0) = -1 • F() = 3 • F() = - 3/2 • F() = -1 • F() = -3/2 • F(2𝝿) = -1 Maximum Minimum minimum

37. Example 4 • Locate the absolute extrema of the function on the closed interval [0,5]. • F(x) = 2x + 5 • F’(x) = 2 • No critical numbers • F(0) = 5 • F(5) = 15 Minimum Maximum

38. Example 5 Find the absolute maximum/minimum, local maximum/minimum, and all other critical points. No absolute max/min. (look at the graph!) Local Maximum at (–1, 15); Local Minimum at (2, –12).

39. HW • Pg. 169 # 2, 19, 23, 53

40. Mean Value Theorem

41. Mean Value Theorem for Derivatives Given a function f that is continuous at every point on a closed interval [a, b] and differentiable at every point on the interval (a, b), there exists at least one point c in the open interval where .

42. Mean Value Theorem for Derivatives • Line tangent to the graph at point c. • Notice it is parallel to the chord with endpoints a and b. • Therefore the slope of the two lines are parallel. • Which means the derivative of f(c) or f’(c) is equal to the slope of the chord containing a and b. Chord containing endpoints a and b c a b

43. Example 1 Exploring the Mean Value Theorem Show that the function f (x) = 9 – x2 satisfies the hypotheses of the Mean Value Theorem on the interval [0, 2]. Then find a solution c to the equation over the interval.

44. Example 2: Exploring the MVT • Explain why each of the following functions fails to satisfy the MVT on the interval [-1, 1] • Looking at the graph below, you can see that there is a corner at x = 0, which means the graph is NOT differentiable at that point, meaning the equation fails the MVT at 0, which is within the stated interval.

45. Example 2: Exploring the MVT • Explain why each of the following functions fails to satisfy the MVT on the interval [-1, 1] Looking at the graph, you can see that there is a point of discontinuity @ x = 1, which means it does not satisfy the MVT, given the interval of [-1, 1]

46. Example 3 Interpreting the Mean Value Theorem A truck driver handed a turnpike ticket to a toll booth attendant. The ticket indicated that the driver covered 159 miles in 2 hours. The speed limit on the turnpike is 65 mph. Should the truck driver be cited for speeding? Why or Why not? Yes, the driver should be cited for speeding; according to the MVT, at some point during the trip the driver’s speed was 79.5 mph.

47. HW • Pg. 176 # 39

48. First Derivative Test (increasing, decreasing)

49. Definition: Where a Function is Increasing or Decreasing Let f be a continuous function on the interval [a, b] and differentiable on (a, b). The function increases on [a, b] if f ’ > 0 at each point of (a, b). The function decreases on [a, b] if f ’ < 0 at each point of (a, b).

50. Example Determining Where Graphs Rise or Fall Use analytic methods to find the local extrema and the intervals on which the function is increasing or decreasing. The function is increasing on the interval (-∞, 2.5), decreasing (2.5, ∞), maximum at x = 2.5. Extrema: Decreasing: Increasing: