Math3A Exam #02 Solution Fall 2017

Transcription

1 Math3A Exam #02 Solution Fall Use the limit definition of the derivative to find f (x) given f ( x) x Use the local linear approximation for f x x at x0 8 to approximate and write your approximation as a single fraction. Note: y f x x x f x 0 0 0

2 2 dy 3. Find the exact values of x that make 0 dx given x 1 y 4 x 1.

3 4. Give a graph of the function below. Find the intervals of increasing and decreasing and the intervals of concavity. Show any x or y intercepts and label only the x-values of any relative extrema or inflection points. Note: This graph does not have any vertical or horizontal asymptotes f x 4x 2x

4 5. Give a graph of the rational function. Find the intervals of increasing and decreasing and the intervals of concavity. Show any x or y intercepts and label only the x-values of any relative extrema or inflection points. Show any vertical or horizontal asymptotes on your graph. f x 3 x 3 1 x

5 Find the equation of the tangent line on the curve x y 3xy at the point equation of the tangent line in slope intercept form. 3 3, 2 2. Write the 7. The diagram below details the rectilinear motion of a robot with time in seconds. Answer the following questions. t 5 t 20 t 0 t 3 t m a) What is the distance traveled on the time interval 3,20? b) What is the displacement on the time interval 5,20? c) What is the average speed on the time interval 0,12? d) What is the average velocity on the time interval 3,20?

6 8. An entry window consisting of a rectangle topped with a semi-circle is to have a perimeter of 20 feet. Find the radius of the semicircle if the area of the window is to be maximized. r h 2r

7 9. Find the radius and the height of the right circular cylinder of largest volume that can be inscribed in a right circular cone with radius 6 inches and height 16 inches? r 16 inches h 6 inches

8 10. The position function of a particle moving on a horizontal axis is shown below. Use the graph to answer the following questions. cm Inf. Point Inf. Point Inf. Point Inf. Point sec a) Is the acceleration positive, negative or zero at t 1 sec? b) Is the particle speeding up or slowing down through t 6 sec? c) What is the total distance traveled from 2 and 14 seconds? d) Is the acceleration positive, negative or zero at t 9 sec? e) What is the particle s displacement during the first 9 seconds? f) Is the acceleration positive, negative or zero at t 4 sec? g) Is the particle speeding up or slowing down through t 13 sec? h) Is the velocity positive, negative or zero at t 4 sec? i) Is the acceleration positive, negative or zero at t 5 sec? j) Is the velocity positive, negative or zero at t 7 sec?

9 11. Below is the graph of a velocity function of a particle moving on a horizontal axis. Use this graph to answer the questions below. v(t) t a) Through t 9 seconds is the particle speeding up or slowing down? b) At t 5 seconds is the particle s acceleration positive, negative, or zero? c) At t 11 seconds is the particle s velocity positive, negative, or zero? d) Through t 13 is the particle speeding up or slowing down? e) Through t 3 seconds is the particle s position function increasing, decreasing, or neither? f) Through t 8 seconds is the particle s position function increasing, decreasing, or neither? g) At t 1 seconds is the particle s acceleration positive, negative, or zero? h) At t 13 seconds is the particle s acceleration positive, negative, or zero?

10 12. Given the graph of the first derivative, sketch a graph of f (x) and f ''(x) on the indicated set of axis. f (x) a b 0 c d f '(x) a b 0 c d f ''(x) a b 0 c d

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21 Calculus I Sample Exam #02 1. Use the limit definition of the derivative to find f (x) given 1 f( x) 1 x. 2. Find the local linear approximation for f x 3 x at x0 4. Note: y f x x x f x 0 0 0

22 3. First find dy dx given 2 sin xy x y. Next find dy dx x y 1 4. Find dy dx x 2 16 if y x tan x

23 5. Give a graph of the function below. Find the intervals of increasing and decreasing and the intervals of concavity. Show any x or y intercepts and label only the x-values of any relative extrema or inflection points.. Note: This graph does not have any vertical or horizontal asymptotes f x 3x 3x

24 6. Give a graph of the rational function. Find the intervals of increasing and decreasing and the intervals of concavity. Show any x or y intercepts and label only the x-values of any relative extrema or inflection points. Show any vertical or horizontal asymptotes on your graph. f x x 1 2 2x 2

25 7. Define the derivatives below as being either positive, negative, zero or undefined Inf. Point Inf. Point a) f 5 b) f 2 c) f 6 d) f 2 i) f 1 j) f 3 k) f 5 l) f 3

26 8. Find the radius and the height of the right circular cylinder of largest volume that can be inscribed in a right circular cone with radius 4 inches and height 10 inches. Vcylinder 2 r h r 10 inches h 4 inches

27 9. A conical water tank with vertex down has a radius of 4 ft at the top and is 10 ft high. If water flows into the tank at a rate of 12 ft 3 / min, how fast is the depth of the water increasing when the water is 5 feet deep? Inlet 4 ft Vcone r h r 10 ft h

28 10. The position function of a particle moving on a horizontal axis is shown below. Use the graph to answer the following questions. meters Inf. Point Inf. Point Inf. Point sec a) Is the acceleration positive, negative or zero at t 1 sec? b) Is the particle speeding up or slowing down at t 3 sec? c) What is the total distance travel during the first 8 seconds? d) Is the acceleration positive, negative or zero at t 6 sec? e) What is the total distance travel between 9 and 14 seconds? f) Is the acceleration positive, negative or zero at t 4 sec? g) Is the particle speeding up or slowing down at t 13 sec? h) Is the velocity positive, negative or zero at t 13 sec? i) Is the acceleration positive, negative or zero at t 13 sec? j) Is the velocity positive, negative or zero at t 9 sec?

29 11. (16 Points) Given the graph of the first derivative, sketch a graph of f (x) and f ''(x) on the indicated set of axis. f (x) a b 0 c d f '(x) a b 0 c d f ''(x) a b 0 c d

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39 Math3A Exam #02 Spring Use the limit definition of the derivative to find f (x) given f( x) = 2x Find the local linear approximation for f ( x) = 3 x at x 0 = 4. Note: y = f ( x )( x x ) + f ( x ) 0 0 0

40 3. Find dy x 2 y 3 + sin xy = 3x 2. dx given ( ) 4. Find the equation of the tangent line to the curve y 2 = tan x at π x =. 4

41 5. Give a graph of the function below. Find the intervals of increasing and decreasing and the intervals of concavity. Show any x or y intercepts and label only the x-values of any relative extrema or inflection points. Note: This graph does not have any vertical or horizontal asymptotes. ( ) f x = 2x 4x

42 6. Give a graph of the rational function. Find the intervals of increasing and decreasing and the intervals of concavity. Show any x or y intercepts and label only the x-values of any relative extrema or inflection points. Show any vertical or horizontal asymptotes on your graph. 3( x 2) 2 f ( x) = 2 x

43 7. Find the following. a) 2 d y dx 2 x = 3 if x 1 y = + x 1 b) dy dx 4 x= π if y = x tan x 2 1

44 8. Suppose we want to construct a cylindrical tin can with a closed top and bottom. If the volume of the can is to be 3 16π in, what are the dimensions of the can to minimize the surface area? Reminder: You will not receive credit for this problem if you do not show how you reached your solution. Surface Area of a Cylinder = 2π rh

45 9. A rocket, rising vertically, is tracked by a radar station that is on the ground 5 miles from the launch pad. How fast is the rocket rising when it is 4 miles high and its distance from the radar station is increasing at a rate of 2000 miles per hour? D h 5 miles

46 10. The position function of a particle moving on a horizontal axis is shown below. Use cm the graph to answer the following questions Inf. Point Inf. Point Inf. Point Inf. Point sec a) Is the acceleration positive, negative or zero at t = 1 sec? b) Is the particle speeding up or slowing down at t = 6 sec? c) What is the total distance traveled between 5 and 14 seconds? d) Is the acceleration positive, negative or zero at t = 9 sec? e) What is the total distance traveled for the first 7 seconds? f) Is the acceleration positive, negative or zero at t = 4 sec? g) Is the particle speeding up or slowing down at t = 13 sec? h) Is the velocity positive, negative or zero at t = 4 sec? i) Is the acceleration positive, negative or zero at t = 5 sec? j) Is the velocity positive, negative or zero at t = 7 sec?

47 11. Given the graph of the first derivative, sketch a graph of f (x) and f ''(x) on the indicated set of axis. f (x) a b 0 c d f '(x) a b 0 c d f ''(x) a b 0 c d

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57 Math3A Exam #02 Spring 2016 Solutions 1. Use the limit definition of the derivative to find f (x) given f( x) 1 x 1. Be sure to clearly show how you reached your solution using the limit definition Find the local linear approximation for f x tan x Note: y f x x x f x at x0. 4

58 3. (10 Points) Suppose that the radius of a sphere is measured to be 2 inches with a measurement error of at most sphere. 1 inch. Use differentials to estimate the error in the computed volume of the 16 Note: 4 V r (10 Points) Find the equation of the tangent line in slope-intercept form to the curve 2 sin xy x y at the point,.

59 5. Sketch a graph of the function below. Find the intervals of increasing and decreasing and the intervals of concavity. Show any x or y intercepts and label only the x-values of any relative extrema or inflection points. Note: This graph does not have any vertical or horizontal asymptotes f x 3x 6x

60 6. Sketch a graph of the rational function. Find the intervals of increasing and decreasing and the intervals of concavity. Show any x or y intercepts and label only the x-values of any relative extrema or inflection points. Show any vertical or horizontal asymptotes on your graph. 2 x 1 f x 3 x

61 7. Find the following. a) 2 d y dx 2 x if y cos x 2 b) dy dx 4 x 2 if y x tan 1 x

62 8. A box with a square base is taller than it is wide. In order to send the box through the U.S. mail, the height of the box and the perimeter of the base can be no more than 108 inches. What is the maximum volume of such a box? Reminder: You will not receive credit for this problem if you do not show how you reached your solution. h s s

63 9. Sand pouring from a chute forms a conical pile whose height is always equal to the diameter. If the Chute height increases at a rate of 5 ft/min, at what rate is the sand pouring from the chute when the pile is 10 feet high? Reminder: You will not receive credit for this problem if you do not show how you reached your solution. Vcone r h h d

64 10. The position function of a particle moving on a horizontal axis is shown below. Use the graph to cm answer the following questions Inf. Point Inf. Point Inf. Point Inf. Point sec a) Is the acceleration positive, negative or zero at t 1 sec? b) Is the particle speeding up or slowing down at t 6 sec? c) What is the total distance traveled between 5 and 14 seconds? d) Is the acceleration positive, negative or zero at t 9 sec? e) What is the total distance traveled for the first 7 seconds? f) Is the acceleration positive, negative or zero at t 4 sec? g) Is the particle speeding up or slowing down at t 13 sec? h) Is the velocity positive, negative or zero at t 4 sec? i) Is the acceleration positive, negative or zero at t 5 sec? j) Is the velocity positive, negative or zero at t 7 sec?

65 11. Given the graph of the first derivative, sketch a graph of f (x) and f ''(x) on the indicated set of axis. f (x) a b 0 c d f '(x) a b 0 c d f ''(x) a b 0 c d

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