f ', the first derivative of a differentiable function, f. Use the

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1 f, f ', and The graph given to the right is the graph of graph to answer the questions below. f '' Relationships and The Extreme Value Theorem 1. On the interval [0, 8], are there any values where f(x) is not differentiable? Give a reason for your answer. f ', the first derivative of a differentiable function, f. Use the 2. On what interval(s) is f '' > 0? < 0? Give reasons for your answers. 3. On what intervals is f increasing? Decreasing? Give reasons for your answers. 4. What is the value of f ''(4)? 5. What is the value of f ''(8)? Explain your reasoning. Explain your reasoning If g( x) x e f ( x) and f(2) = 3, what is the equation of the normal line to the graph of g at x = 2? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 390

2 Consider the function f ( x) 2x ax bx 5. Given the table of information below, answer the questions that follow. x < < x < < x < 2 2 > f ' Positive 0 Negative Negative Negative 0 Positive f '' Negative Negative Negative 0 Positive Positive Positive a. Determine intervals of increasing and decreasing values of f. Justify your answers. b. Determine and classify all x values of relative extrema of f. Justify your answers. c. Determine the intervals of concavity of f. Justify your answer. d. Determine the values of a and b in the equation of f. Show your work. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 391

3 The Extreme Value Theorem 3 2 Consider the cubic function f ( x) x 6x 9x 2 to answer the following questions. a. Determine the intervals where f is increasing and decreasing. Justify your answers. b. Determine the coordinates of the relative extrema of f. Justify your answers. c. Sketch an accurate graph of f on the interval [ 4, 1] d. Identify the absolute maximum value of f on the axes below. on the interval [ 4, 1]. e. Identify the absolute minimum value of f on the interval [ 4, 1]. f. Identify the absolute maximum value of f on the interval [ 4, 1]. g. Identify the absolute minimum value of f on the interval [ 4, 1]. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 392

4 The Extreme Value Theorem: Given the functions below, determine the absolute extreme values of the function on the given interval f ( x) x 2x 3x 2 on [ 1, 3] 2. g( x) sin x cos x on, f on [-3, 6] 4. ( x ) ln x ( x) x 2 3 h on [ 1, 3] Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 393

5 The Derivate as a Rate of Change Mean Value Theorem and Rolle s Theorem Consider the values of a differentiable function, f(x), in the table below to answer the questions that follow. Plot the points and connect them on the grid below. x f(x) In calculus, the derivative has many interpretations. One of the most important interpretations is that the derivative represents the Rate of Change of a Function. When speaking of rate of change, there are two rates of change that can be found that are associated with a function average rate of change and instantaneous rate of change. Average Rate of Change of f(x) on an Interval Instantaneous Rate of Change of f(x) at a Point Find the average rate of change of f(x) on the Is the instantaneous rate of change of f at x = 4 interval [2, 12]. greater than the rate of change at x = 6? Justify. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 394

6 Rolle s Theorem Consider the function, f(x), presented on the previous page. Does Rolle s Theorem apply on the following intervals? Explain why or why not? Interval [2, 14] Interval [2, 8] For each of the functions below, determine whether Rolle s Theorem is applicable or not. Then, apply the theorem to find the values of c guaranteed to exist x on the interval [ 3, 0] 2. g( x) x sin 2x g( x) on the interval [ 4, 1] x 2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 395

7 Rolle s Theorem guarantees that if a function is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there is guaranteed to exist a value of c where the instantaneous rate of change is equal to zero. The Mean Value Theorem is similar. In fact, Rolle s Theorem is a specific case of what is known in calculus as the Mean Value Theorem. Mean Value Theorem 5 Consider the function h( x) 3. The graph of h(x) is pictured below. Does the M.V.T. apply on the x interval [ 1, 5]? Explain why or why not. Does the M.V.T. apply on the interval [1, 5]? Why or why not? Graphically, what does the M.V.T. guarantee for the function on the interval [1, 5]? Draw this on the graph to the left. Apply the M.V.T. to find the value(s) of c guaranteed for h(x) on the interval [1, 5] Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 396

8 2 Explain why you cannot apply the Mean Value Theorem for f ( x ) x 3 2 on the interval [ 1, 1]. Find the equation of the tangent line to the graph of f ( x) 2x sin x 1on the interval (0, π) at the point which is guaranteed by the mean value theorem. The Mean Value Theorem guarantees that if a function is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there is guaranteed to exist a value of c where the instantaneous rate of change at x = c is equal to the average rate of change of f on the interval [a, b]. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 397

9 Applying Theorems in Calculus Intermediate Value Theorem, Extreme Value Theorem, Rolle s Theorem, and Mean Value Theorem Before we begin, let s remember what each of these theorems says about a function. Intermediate Value Theorem Extreme Value Theorem Rolle s Theorem Mean Value Theorem Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 398

10 The rate at which water flows out of a pipe, in gallons per hour, is given by a differentiable function R of time t. The table below shows the rate as measured every 3 hours for a 24-hour period. t (hours) R(t) (gallons per hour) a. Estimate the value of R '(5), indicating correct units of measure. Explain what this value means about R(t). b. Using correct units of measure, find the average rate of change of R(t) from t = 3 to t = 18. c. Is there some time t, 0 < t < 24, such that R '( t) = 0? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 399

11 The total order and transportation cost C(x), measured in dollars, of bottles of Pepsi Cola is approximated by the function 1 x C ( x) 10,000, x x 3 where x is the order size in number of bottles of Pepsi Cola in hundreds. Answer the following questions. a. Is there guaranteed a value of r on the interval 0 < r < 3 such that the average rate of change of cost is equal to C '( r)? Give a reason for your answer. b. Is there a value of r on the interval 3 < r < 6 such that C '( r) 0. Give a reason for your answer and if such a value of r exists, then find that value of r. c. For 3 < x < 9, what is the greatest cost for order and transportation? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 400

12 A car company introduces a new car for which the number of cars sold, S, is modeled by the function 9 S ( t) 300 5, t 2 where t is the time in months. a. Find the value of S '(2.5). Using correct units, explain what this value represents in the context of this problem. b. Find the average rate of change of cars sold over the first 12 months. Indicate correct units of measure and explain what this value represents in the context of this problem. c. Is it possible that a value of c for 0 < c < 12 exists such that S '( c) is equal to the average rate of change? Give a reason for your answer and if such a value of c exists, find the value. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 401

13 x f(x) f '( x) g(x) g '( x) The functions f and g are differentiable for all real numbers, and g is strictly increasing. The table above gives values of the functions and their first derivatives at selected values of x. The function h is given by the equation h ( x) f ( g( x)) 6. a. Find the equation of the tangent line drawn to the graph of h when x = 3. b. Find the rate of change of h for the interval 1 < x < 3. c. Explain why there must be a value of r for 1 < r < 3 such that h(r) = 2. d. Explain why there is a value of c for 1 < c < 3 such that h '( c) 5. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 402