x 3x 1 if x 3 On problems 8 9, use the definition of continuity to find the values of k and/or m that will make the function continuous everywhere.

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1 CALCULUS AB WORKSHEET ON CONTINUITY AND INTERMEDIATE VALUE THEOREM Work the following on notebook paper. On problems 1 4, sketch the graph of a function f that satisfies the stated conditions. 1. f has a limit at x = 3, but it is not continuous at x = 3.. f is not continuous at x = 3, but if its value at x = 3 is changed from to, f becomes continuous at x = 3. f f has a removable discontinuity at x = c for which 4. f has a removable discontinuity at x = c for which f c f c f is undefined. is defined. 3 1 On problems 5 7, use the definition of continuity to prove that the function is discontinuous at the given value of a. Sketch the graph of the function. x 3x x x 5x4 if x 3 e if x 0 5. f x, a 1 6. g x x 9 a 3 7. h x a 0 x 1 x if x 0 1 if x 3 On problems 8 9, use the definition of continuity to find the values of k and/or m that will make the function continuous everywhere. x 5, x kx, x 8. f x 9. g x m( x 3) k, 1 x x k, x 3 x x 7, x 1 On problems 10 1, a function f and a closed interval [a, b] are given. Show whether the conditions of the Intermediate Value Theorem hold for the given value of k. If the conditions hold, find a number c such that f c k. If the theorem does not hold, give the reason. Whether the theorem holds or not, sketch the curve and the line y = k f x x x 11. f x 5 x 1. f x x a, b 0, 3 a, b 4.5, 3 a, b 3, 5 5 k 1 k 3 k 6 f x x x Given the function (a) Does f x 7 somewhere on the interval 1, 3 show why or why not. (b) Must f x 1 somewhere on the interval 1, 3 show why or why not.? Use the Intermediate Value Theorem to? Use the Intermediate Value Theorem to TURN->>>

2 14. Use the Intermediate Value Theorem to show that x 3 x0 1, has a root in the interval. 15. One night in January, the outside temperature at midnight was 4 F. At 10 AM the next morning, the temperature was 57 F. (a) Must there have been a time between midnight and 10 AM when the temperature was 50 F? Explain how you know. (b) Must there have been a time between midnight and 10 AM when the temperature was 40 F? Explain how you know. (c) Could there have been a time between midnight and 10 AM when the temperature was 40 F? Explain how you know. 16. One afternoon you were driving on Hwy 90, headed for College Station. At PM, you were driving 60 miles per hour. At 3 PM, you were driving 55 miles per hour. (a) Must there have been a time between PM and 3PM when you were driving 57 miles per hour? Explain how you know. (b) Must there have been a time between PM and 3PM when you were driving 45 miles per hour? Explain how you know. (c) Could there have been a time between PM and 3PM when you were driving 45 miles per hour? Explain how you know.

3 CALCULUS AB WORKSHEET 1 ON.1 Work the following on notebook paper. Give decimal answers correct to three decimal places. 1. With an initial deposit of $1000, the balance in a bank account after t years is f t t dollars. (a) What are the units of the rate of change of (b) Compute the average rate of change of f t f t? on the following time intervals. Time interval Average rate of change 1.9, 1.99, 1.999,,.1,.01,.001 (c) Use your answers to (b) to estimate the instantaneous rate of change of f t at t = years.. The figure on the right shows the estimated number N of internet users in Chile, based on data from the United Nations Statistics Division. (a) Estimate the rate of change of N at t = (b) Does the rate of change increase or decrease as t increases? Explain. (c) Compute the average rate of change over [001, 005]. (d) Is the rate of change at t = 00 greater than or less than the average rate of change you found in (c)? 3. Given f xx 5. x (a) Sketch the function and the tangent line to the graph at the point where x = 1. (b) Based on your sketch, do you expect f 1 to be positive or negative? f 1 three different ways: 1) by using the definition of the derivative ) by using the alternative form of the definition of the derivative (c) Compute 3) by using the shortcut that you learned last year. (d) Write the equation of the tangent line to the graph of point-slope form, y y m x x 1 1. y f x at x = 1. Leave your equation in TURN->>>

4 4. Use the figure on the right to answer the following. (a) Find the slope of the secant line through, f and.5, f.5. Is it larger or smaller than Explain. f h f (b) Estimate for h this quantity represent? Is it larger or smaller than (c) Estimate f 1 and f. f h 0.5. What does? f? f h f (d) Find a value of h for which 0. h 5. Use the figure on the right to answer the following. (a) Determine for a = 1,, 4, 7. f a (b) For which values of x is f x 0? (c) Which is larger, f5.5 or f 6.5? (d) Do you think that f 3 exists? Explain. 6. Sketch the graph of f x sin x on 0,, and guess the value of f. Then calculate the difference quotient at x for two small positive and negative values of h. Are these calculations consistent with your guess? 7. Use the graph of the function on the right to determine the intervals along the x-axis on which the derivative is positive. f x h f x Use the definition of the derivative, f x lim, to find f x on problems h0 h 3 8. f xx 5x 9. f x 1 x f x x 4

5 Answers to Worksheet 1 on.1 1. (a) dollars year (b) , , , , , (c) About dollars per year. (a) About 0.3 million internet users per year (b) As t increases, the slope of the tangent line decreases so the rate of change decreases. (c) About 0.35 million internet uses per year (d) Greater. The tangent line at t = 00 has a great slope than the secant line that passes through (001, N(001) and (005, N(005). 3. The derivative is positive for 1 x.5 and 3.5 x , 0, 4 1, 7 0 (b) 7 9 (c) is larger because the slope of the tangent line at x = 5.5 is greater than the slope of 4. (a) f f f f x f 5.5 the tangent line at x = 6.5 (d) No. As x approaches 3 from the left, the slope is 0, but as x approaches 3 from the right, the slope is 1. Since the slopes are not approaching the same value, 5. (a) The secant line through, and.5,.5 f 3 does not exist. f f is steeper than the tangent line at x = so the secant line has a larger slope than the tangent line. (b) The secant line through (, ) and (.5,.5) has a slope of 1. f.5 f (c) f 0.8 or (a) Sketch (b) negative (c) 1) 3 ) 3 3) 3 y 4 3 x 1 (d) x 10x 1 x 3 1 x 4

6 CALCULUS WORKSHEET ON DERIVATIVES Work problems 6, 7, and 8 on notebook paper. 1. For each of the graphs shown below, list the x-values for which the function appears to be: (i) not continuous (ii) not differentiable. Match the points labeled on the curve with the given slopes. Slope Point y f x shown in the figure, arrange the following numbers in ascending order: 3. For the graph The slope of the graph at A The slope of the graph at B The slope of the graph at C The slope of the line AB The number 0 The number 1 4. For the function shown below, at what labeled points is the slope of the graph: positive? negative? At which labeled point does the graph have the: greatest (i.e., most positive) slope? the least slope (i.e., negative and with the largest magnitude)? TURN>>>

7 x y x y. Estimate the slope of the graph at the points, and, 5. (a) 1 1 Work problems 6, 7, and 8 on notebook paper. 6. Let f be a function which satisfies the property f x y f x f y xy for all real f h numbers x and y, and suppose that to find f x. lim h0 h 7. Use the definition of the derivative 3 7. Let f be a function which satisfies f 1 h f 1 3h 4h 5h for all real numbers h. Find f 1. f x f 8. Suppose f is a function for which lim 0. Which of the following questions x x MUST be true, MIGHT be true, or can NEVER be true? Explain your answers. (a) (b) (c) f x is continuous at x. f x is continuous at x 0. f 0 (d) lim f x f x. (e) f

8 Answers to Worksheet on Derivatives 1. (i) 1 (ii) 1,, 3 (i) None (ii), 4. F, C, E, A, B, D 3. 0 < slope at C < slope at B < slope of AB < 1 < slope at A 4. positive? A and D negative? C and F greatest? A least? F 5. (a) 0,.5 (b).5, 6. 7 x (a) Must (b) Might (c) Might (d) Must (e) Never

9 CALCULUS AB WORKSHEET ON DERIVATIVES USING DATA Work the following on notebook paper. Give decimal answers correct to three decimal places. 1. A roast turkey is taken from an oven and placed on a counter to cool. The table shows the temperature of the turkey at various times over a three hour period. t (minutes) ( F) T t (a) Use the data in the table to find T 90 (b) Use the data in the table to find T Using appropriate units, explain the meaning of your answer.. Using appropriate units, explain the meaning of your answer. (c) For 0 t 180, must there be a time t when the temperature of the turkey is 15 F? Justify your answer. (d) Use the data in the table to find an approximation for. Show the computations that lead to your answer. Using appropriate units, explain the meaning of your answer. T 75. The number of locations of a popular coffeehouse chain is modeled by a differentiable function N for where t is the number of years since 1996 and is the number of locations. Values of 0 t 8, at selected values of t are shown in the table below. N t t (years) Nt locations N Using appropriate units, explain the meaning of your answer. (b) Use the data in the table to find N (c) Use the data in the table to find an approximation for N5. Show the computations that lead to your answer. Using appropriate units, explain the meaning of your answer. (a) Use the data in the table to find Nt. Using appropriate units, explain the meaning of your answer. 3. A hot cup of coffee is taken into a classroom and set on a desk to cool. The temperature of the coffee is modeled by a differentiable function T for 0 t 1, is measured in degrees Fahrenheit. Values of T t where t is measure in minutes and at selected values of t are shown in the table below. t (minutes) Tt ( F) (a) For 0 t 1, must there be a time t when the temperature of the coffee is 99 F? Justify your answer. (b) Use the data in the table to find an approximation for T 3. Show the computations that lead to your answer. Using appropriate units, explain the meaning of your answer. (c) Use the data in the table to find the average rate of change of T t on the time period 5 t 1 minutes. Show the computations that lead to your answer. 1 y t Fahrenheit and t is measured in minutes. Find y. 0.3t (d) A model for the temperature is given by e, where (e) Use the model given in part (d) to find the average rate of change of 3 T t y t is measured in degrees yt on the time period

10 minutes. TURN->>> 4. The temperature of the water in a pond is modeled by a differentiable function T for 0 t 15, 5 t 1 where t is measure in days and values of t are shown in the table below. Wt is measured in degrees Celsius. Values of t (days) Wt ( C) (a) Use data from the table to find W (b) Use data from the table to find 9 W 1 31 (c) Use data from the table to find an approximation for Wt at selected. Using appropriate units, explain the meaning of your answer.. Using appropriate units, explain the meaning of your answer. W 7. Show the computations that lead to your answer. (d) Based on values in the table, what is the smallest number of instances at which the temperature of the pond could equal 3 C on the open interval 0 < t < 15? Justify your answer. 3 (e) A student proposes the function P, given by P t 0 10te t, as a model for the temperature of the water in the pond at time t, where t is measured in days and P 7 Pt. Using appropriate units, explain the meaning of your answer. is measured in degrees Celsius. Find 5. A car travels on a straight road. During the time interval 0 t 60 the car s velocity vt, measured in feet per second, is a twice-differentiable function. The table below shows selected values of this function. t (sec) (ft/sec) vt (a) Use data from the table to find an approximation for v17. Show the computations that lead to your answer. Using appropriate units, explain the meaning of your answer. (b) Based on the values in the table, what is the smallest number of instances at which the velocity of the car could be 0 feet per second on the open interval 0 t 60? Justify your answer. 6. A test plane flies in a straight line with positive velocity v is a differentiable function of t. Selected values of v t, in miles per minute at time t minutes, where v t for 0 t 40 are shown in the table below. t (min) vt (mpm) (a) Use data from the table to find an approximation for v18. Show the computations that lead to your answer. Using appropriate units, explain the meaning of your answer. (b) Use data from the table to find the average acceleration of the particle over the time interval 5 t 5 minutes. (Hint: The average acceleration is the average rate of change of the velocity.) (c) Based on the values in the table, what is the smallest number of instances at which the velocity of the plane could be 8 miles per minute on the open interval 0 t 40? Justify your answer. t 7t (d) The function f, defined by f t 6 cos 3sin, is used to model the velocity of the plane, in miles per minute, for 0 t 40. According to this model, what is the acceleration of

11 the plane at t = 3? Indicate units of measure. Answers to Derivatives using Data 1. (a) = 110 F. When t = 90 minutes, the temperature of the turkey is 110 F.. (a) T 90 (b) T = 30 minutes. When the temperature of the turkey is 150 F, the time is 30 minutes. (c) Yes. Since T is a continuous function and T T , , and ,, the Intermediate Value Theorem guarantees a t in (60, 90) so that Tt 15. (d) T F / min. When t = 75 minutes, the temperature of the turkey is dropping at a rate of F / min. 3 = 1886 locations. When t = years (in 1998), the number of locations is N (b) (c) N = 6 years. When the number of locations is 4617, the time is 6 years (in 00) N locations / year. When t = 5 years, the number of locations is 6 4 increasing at a rate of locations / year. 3. (a) Yes. Since T is a continuous function and T T (b) 103, 5 95, and ,, the Intermediate Value Theorem guarantees a t in (, 5) so that Tt T3 F / min. When t = 3 minutes, the temperature of the coffee is 5 3 dropping at a rate of 8 F / min. 3 (c) Average rate of change = F / min (d) (e) y3.846 F/ min F / min 4. (a) W C / day. When t = 7 days, the temperature of the pond is dropping at a rate of 4 C / day. 3 (b) Two times. Since W is a continuous function and W W 0 0, 3 31, and , the Intermediate Value Theorem guarantees a t in (0, 3) where Tt 3. Since W9 4, W1, and 3 4, the Intermediate Value Theorem guarantees a t in (9, 3) where Tt 3 (c) One time. Since W is a continuous and differentiable function, the Mean Value Theorem guarantees a value of t somewhere in the interval (0, 15) where Wt 0. (d) P C/day. When t = 7 days, the temperature of the pond is dropping at a rate of 1.93 C/day.

12 When t = 18 minutes, the acceleration of the plane is mi / min (or the velocity of the plane is decreasing at a rate of 0.5mi / min ) (b) Average acceleration = 0.34mi / min 5 5 v 0 7.0, v 5 9., and , the 5. (a) v mi / min (c) Two times. Since v is a continuous function and Intermediate Value Theorem guarantees a t in (0, 5) where v10 9.5, v15 7.0, and in (10, 15) where. vt 8. Since the Intermediate Value Theorem guarantees a t vt 8 (d) Two times. Since v is a continuous and differentiable function, the Mean Value Theorem guarantees a value of t somewhere in the interval (0, 15) and somewhere in the interval. v t (5, 30) where 0 (e) f mi / min f 5 f 5 (f) Average acceleration = mi / min When t = 17 minutes, the velocity of the car is ft / sec 5 (or the velocity of the car is decreasing at a rate of ft / sec ). 8 8 (b) Two times. Since v is a continuous function and v v 14 and v v 33, the Intermediate Value 6. (a) v 17 ft / sec Theorem guarantees a t in (0, 14) where vt 0 and a t in (33, 47) where vt 0. (c) Two times. Since v is differentiable on [0, 60], v must also be continuous there. v 14 v 4 and v 4 v 33, v attains a maximum value on (14, 33). Since Similarly, since v v v v and 47 60, v attains a minimum on (33, 60). v is differentiable so vt 0 at each relative extreme point on (0, 60). Therefore vt 0 for at least two values of t in [0, 60].

13 CALCULUS AB REVIEW WORKSHEET ON 1.4 &.1 Work the following on notebook paper except for problems 3. Use your calculator only on problem 1, and give decimal answers correct to three decimal places. 1. A hot cup of coffee is poured and set on a table to cool. The table shows the temperature of the coffee at various times over a fifteen minute period. t (minutes) Tt ( F) (a) Use data from the table to find an approximation for T 5. Show the computations that lead to your answer. Using appropriate units, explain the meaning of your answer. (b) For 0 15, must there be a time t when the temperature of the coffee is 15 F? Justify your answer (c) A model for the temperature is given by t y t e, where is measured in 3 degrees Fahrenheit and t is measured in minutes. Use the model to find Using appropriate units, t explain the meaning of your answer. (d) Use the model given in (c) to find y 5 yt y 5.. Using appropriate units, explain the meaning of your answer. No calculator. Draw tangent lines on the graph of f at five points, and record the slopes of the lines in the table below. Then sketch the graph of the derivative on the grid on the right.. Graph of f x Graph of f x x f x 3. x f x

14 Use the alternative form of the derivative to find the derivative at x = c if it exists. If the derivative does not exist, use the alternative form to show why. All work must be shown. x 1, x 4. f x c 4x5, x Does f exist? If yes, then f = What happens on the graph of f at x =? Choices are: smooth curve, sharp turn, cusp, vertical tangent 4 x, x 5. f x c x 6x 10, x Does f exist? If yes, then f = What happens on the graph of f at x =? Choices are: smooth curve, sharp turn, cusp, vertical tangent (a) For what value(s) of x is f not continuous? (b) Classify the discontinuities as point, jump, or asymptotic. (c) Classify the discontinuities as removable or nonremovable. x 1, x0 x x6 6. f x 7. f x x1, x0 x 9 8. Use the definition of continuity to find k so that f will be continuous everywhere given kx, x 1 x f k 1 x, x 1 9. Use the definition of continuity to find a and b so that f will be continuous everywhere given 17 if x 3 f x ax b if 3 x 1 3 if x Given f x x x 6, [ a, b] [ 4,1], k 1. (a) Does the IVT hold? Show why or why not. (b) If the theorem holds, find the c which the theorem guarantees. (c) Sketch y f x, the line y k, and the point ( c, k) that you found Given f x, [ a, b] [0,], k. x 1 4 (a) Does the IVT hold? Show why or why not. (b) If the theorem holds, find the c which the theorem guarantees. (c) Sketch y f x, the line y k, and the point ( c, k) that you found. Use the definition of the derivative to find the derivative f x x f x x

AP Calculus AB Chapter 1 Limits

AP Calculus AB Chapter 1 Limits AP Calculus AB Chapter Limits SY: 206 207 Mr. Kunihiro . Limits Numerical & Graphical Show all of your work on ANOTHER SHEET of FOLDER PAPER. In Exercises and 2, a stone is tossed vertically into the air