40. Find dy in terms of x and y for the function x + y = 6. dx y a. . X b. 6 dy 2 X N - + y c . a dy y dy 2 12 d. dry dx 2 d'y y e. dx 2 41. The function f(x) = x has two tangents drawn to it at the points on its graph where x = -1 and x = 1. What are the coordinates of the point where the two tangent lines intersect? a. (0, 1) b. (0, 0) C. (0, -1) d. (0, -2) e. none of these 42. The function f(x) = x has two normal lines drawn to it at the points on its graph where x = -1 and x = 1. What are the coordinates of the point where the two normal lines intersect each other? a. (0, 1) b. (0, 0) C. (0, -1) d. (0, -2) e. none of these Unit 2 Evaluation 186 MTHH 07143. A tangent line and a normal line are drawn to the function f(x) = " + 5x + 6 at the x2 - 4 point where x = 3. What is the product of the slopes of the tangent line and the normal line when the function is passing through the point where x = 3 will be? a. -1 b. 0 C. 1 d. 2 e. none of these 44. The equation for the tangent line to a function y = f(x) at (a, f(a)) is a. y = f '( x )y b. y = f '(a) c. y - a = f(x) . (x - a) d. y - fla) = f'(a) . (x - a) JIN 45. Find the equation for the tangent line of f(x) = (x3 + 8 )(2x - 4) at a = 8. a. y - 144 = 28(x - 8) b. y = 28x - 224 c. y - 144 = -16(x - 8) d. y - 144 = 2 (x -8) 46. If f(x) = (3x - 1) "s, then f '(3) = 1 a. True b. False 47. If f(x) = x + x- 2 then f(2) would equal 8. What would the slope of the inverse function to f(x) be when x = 8? a. 8 b. 13 1 d 1 13 e. none of these Unit 2 Evaluation 187 MTHH 07148. Find f '(x) if f(x) = x ( x + 2)?. a. 2x . (x + 2) b. (x + 2)(x + 2x+ 1) c. 4x(x + 2)(x + 1) d. none of these 49. If f(x) is continuous on [a, b], which of the following must be true? 1. f(x) is differentiable on (a, b) Il. f(x) is defined for all x in [a, b] Ill. f(x) has a maximum on (a, b) a. I b. II C. III d. All of these are true. 50. If f(x) is differentiable on (a, b), which of the following must be true? I. f(x) is continuous on (a, b) Il. f(x) is defined for all x in [a, b] Ill. There exists a point c in [a, b] such that f(c) = 0 a. I b. II C. III d. All of these are true. Carefully check your answers on this evaluation and make any corrections you feel are necessary. When you are satisfied that you have answered the questions to the best of your ability, transfer your answers to an answer sheet. Please refer to the information sheet that came with your course materials. Unit 2 Evaluation 188 MTHH 0714. Which of the following statements is correct? a. The derivative of a product is the product of the derivatives. b. The derivative of a quotient is the quotient of the derivatives. c. The derivative of a constant k is k. d. The derivative of a constant times a function is the same as the constant times the derivative of the function. e. The derivative of a power is the power of the derivative. 5. What is the derivative of the function f(x) = x + 2x _ _ + 5? X a. f '(x) = x +2x -1 b. f ( x ) = 3x + 4x -_ + 5 * 2 c. f '(x) = 3x + 4x- * 2 d. f '( x) = 3x + 4x + 6. What is the derivative of f(x) = cot x? (Hint: cot x = Cos x sin x a. f '(x) = cos x . sin x b. f ' (x) = sec x c. f '(x) = - csc x d. f '(x) = tan x e. f '(x) = cot x 7. In using the chain rule to calculate the derivative of f(x) = sin(4x + 1), you would let u stand for: a. u = sin x b. u = 4x + 1 C. U = 4x d. u= x 8. In using the chain rule to calculate the derivative of f(x) = (x + 2x + 1) , would equal du U a. True b. False Unit 2 Evaluation 178 MTHH 0719. Given the function f(x) = (x + 2) , calculate f "(-1). a. 2.000 b. 6.000 c. 3.000 d. -1.000 10. Which of the following functions are differentiable for all real numbers x? a. f(x) = sin x b. f(x) = tan x c. f(x) = 1 x| 11. If a function is differentiable at a point then it must also be continuous at that point. a. True b. False 12. If a function is continuous at a point then it must be differentiable at that point. a. True b. False 13. The equation of a circle centered at (0, 0) with a radius of 2 is x + y = 2". What is the slope of the tangent to the circle at the point (1, v/3 )? a. -1 b. - V3 3 C. - V2 2 d. - V3 e. none of these 14. What is the slope of the tangent line to x cos y - sin y = 0 at the point (0, 7 )? a. 0 -2 1 b. 272 C. -1 d. undefined Unit 2 Evaluation 179 MTHH 07115. The equation vy - cos x - 1 = 0 crosses the y-axis at one location. What is the slope of the tangent to the graph as it is crossing the y-axis? a. -1 b. 0 c. V2 d. 2 e. none of these 16. Find y ' if 2x - 3y = 4xy. a. 6x2 - 4y 4x +9y2 6 x2 b. 4x +9y2 6x2 - 92 - 4y 4 x d. 4x +9y2 6x2 - 4y 17. Find y 'if 8 xy + 6x = 3y. a. 4y + 6\\xy +4x 3 xy b . 3 xy - 4y 4 x 4x - 3 xy C. 6 xy + 4y 4y + 6 xy 3 xy - 4x WIN 18. Where is f(x) = (x - 2)3 differentiable? a. (-00, 00) b. (2, 00) C. (-00, 2) U (2, 00 ) d. (-00, 0) U (0, 00) Unit 2 Evaluation 180 MTHH 071d. e do .860an u dx . none of these a. {00, no) a .True b. False Unit 2 Evaluation 20. Where is x) = )1 3x differentiable? 22. Find the equation of the line tangent to x) = 181 3 x4 19. If you combine the chain rule with the tangent rule, d1 [tan u] = x atx= -2. 23. The derivative of the sum of two functions is the sum of their derivatives. MTHH 071 24. The derivative of the difference of two functions is the difference of their derivatives. a. True b. False 25. An object moves according to the position equation s(t) = t + sin t, where s is measured in meters. What is the average velocity of the object during the time interval from t = 0 to t = 27 seconds? a. 0.318 m/s b. 1 m/s c. 2 m/s d. 3. 14 m/s e. none of these 26. An object moves according to the position equation s(t) = t + sin t, where s is measured in meters. What is the average acceleration of the object during the time interval from t = 0 to t = 2 7 seconds? a. 0 m/s b. 1 m/s2 c. 2 m/s d. 3.14 m/s e. none of these 27. An object moves according to the position equation s(t) = sin t, where s is measured in meters. What is the instantaneous velocity of the object at time t = 2 7 seconds? a. 0 m/s b. 1 m/s c. 2 m/s d. 3.14 m/s e. none of these 28. An object moves according to the position equation s(t) = sin t, where s is measured in meters. What is the instantaneous acceleration of the object at time t = 2 x seconds? a. 0 m/s2 b. 1 m/s2 c. 2 m/s2 d. 3.14 m/s e. none of these Unit 2 Evaluation 182 MTHH 07129. You can buy rectangular-shaped sponges that have been vacuum packed to take up less volume for shipment and storage. When they are soaked in water, the volume expands in such a way that the linear dimensions of the soaked sponges all increase by the same ratio. A particular sponge starts out with dimensions of 1 inch high, 4 inches wide, and 6 inches long. When the sponge is put in water, at the end of the rst hour its height has doubled and is increasing at the linear rate of 1 inch per hour. So, at the end of the first hour a. the volume of the sponge is 48 cubic inches, and its volume is increasing at the rate of 3 cubic inches per hour. b. the volume of the sponge is 48 cubic inches. and its volume is increasing at the rate of 9 cubic inches per hour. 0. the volume of the sponge is 48 cubic inches. and its volume is increasing at the rate of 27 cubic inches per hour. d. the volume of the sponge is 192 cubic inches, and its volume is increasing at the rate of 288 cubic inches per hour. 30. A 10 cm thick grindstone is initially 200 cm in diameter, and it is wearing away at a rate of 50 cmalhr. At what rate is its diameter decreasing? a. cm/hr 31. A water heater in the shape of a circular cylinder is standing vertically. resting on one of the round ends. The tank is 60 inches tall and has a volume of 30 gallons. If water is flowing into the tank at the rate of 5 gallons per minute, at what rate is the level of the water in the tank increasing? a. 10 inches per minute b. 12 inches per minute c. 15 inches per minute d. 20 inches per minute Unit 2 Evaluation 183 MTHH 071 32. A 40 foot long ladder is resting against a wall. The base of the ladder at point B is being pulled away from the wall at the rate of 1 foot per second. If the ladder remains in contact with the ground and with the wall as the base of the ladder is pulled away from the wall, how fast is the top of the ladder (point A) moving down the wall at same the time that the bottom of the ladder (point B) is 39 feet from the base of the wall? a. 1 ft/sec b. 2.2 ft/sec c. 4.4 ft/sec d. 8.8 ft/sec e. 10.4 ft/sec 33. Use the alternative form of the derivative to find the derivative of the function f(x) = x - 9 at x = 5. a. f '(5) = 1 b. f '(5) = 250 c. f '(5) = 2 d. f '(5) = 125 e. f '(5) = 10 34. Describe the x-values at which f(x) = x2 - is differentiable. - 9 a. f(x) is differentiable at x = 13. b. f(x) is differentiable everywhere except at x = 13. c. f(x) is differentiable everywhere except at x = 0. d. f(x) is differentiable on the interval (-2, 2). e. f(x) is differentiable on the interval (2, co). 35. Describe all values of x, if any, at which the graph of the function has a horizontal tangent. y(x) = x + 12x- + 8. a. x = 0 b. x = -8 c. x = 0 and x = -8 d. x = 0 and x = 8 e. The graph has no horizontal tangents. Unit 2 Evaluation 184 MTHH 07136. Find the second derivative of the function f(x) = 8x9 . 4 a. f "(x) = -160 81 -13 b. f " ( x ) = 5 x 9 81 -13 c. f "(x) =. - -160 x 9 81 -13 d. f "(x) = 160 9 81 -13 e. f "(x) = 8x 9 37. Find the second derivative of the function f(x) = x sin x. a. f "(x) = 2 sin x + 2x cos x - x sin x b. f "(x) = 2 sin x + 4x cos x - x sin x c. f "(x) = sin x + 4x cos x - x sin x d. f "(x) = 2 sin x + 4x cos x - x sin x 38. The velocity of an object in meters per second is v(t) = 25 - f, 0 s ts 5. Find the acceleration of the object when t = 3. a. 3 meters per second squared b. -6 meters per second squared c. 6 meters per second squared d. -2 meters per second squared 39. Find the second derivative of the function f(x) = (3x5 + 7)'. a. f "(x) = 63x(7 + 3x) (14 + 63x) b. f " (x) = 63x(7 + 3x ) (14 + 60x) c. f "(x) = 63x(7 + 3x ) (14 + 60x ) d. f "(x) = 63x(7 + 3x ) (14 + 63x) e. f "(x) = 63x(7 + 3x ) (14 - 60x ) Unit 2 Evaluation 185 MTHH 071Name ID. Number Unit 2 Evaluation 0 Evaluation 02 AP Calculus AB 1 (MTHH 071 057) This evaluation will cover the lessons in this unit. It is open book, meaning you can use your course materials. You will need to understand, analyze, and apply the information you have learned in order to answer the questions correctly. To Submit the evaluation by mail. follow the directions on your Enrollment Information Sheet. To take the evaluation online. access the online version of your course; use the navigation panel to access the prep page for this evaluation and follow the directions provided. Note: A graphing calculator may be used on unit evaluations. You may also use scratch paper to work out the solutions. J Multiple-Choice Select the response that best completes the statement or answers the question. 'I. The slope of the tangent line to a function x) at a point where x = a is measured by which of the following? a. Iim f(x+h)f(x) xph h f(x + h) x) b. Iim h>0 m = f(x+h) f(x) (x+h)x d. f'(h) f(x + h) x} 2. If x) = x2, then Iim would be worth hpO a. 2 b. x2 + 2m + n2 C. X d. none of these 3. If x) = 3x + 5, then the value off 17:) is a. 3.500 b. 5.000 c. 3.000 d. 5.300 Unit 2 Evaluation 177 MTHH 071