Rev2.pdf PDF Calculus Past Paper

Question number 1. Your answer was D. Correct. Lateral surface area of a right cylinder is given by A = 2πrh where r is the radius of the base and h is the height. If the radius and height are both increasing at a rate of 5 inches per second; at what rate is the lateral surface area changing whe...

Question number 1. Your answer was D. Correct. Lateral surface area of a right cylinder is given by A = 2πrh where r is the radius of the base and h is the height. If the radius and height are both increasing at a rate of 5 inches per second; at what rate is the lateral surface area changing when r = 8 and h = 10 inches? dA A = 36π dt dA B = 90π dt dA C = 800π dt dA D = 180π dt dA E = 4000π dt F None of the above. Question number 2. Your answer was E. Correct. 1 The volume of a right circular cone with radius r and height h is given by V = 2 πr h. The radius is 3 increasing at a rate of 2inches/sec, while the height is staying constant at h = 10 inches. At what rate is the volume changing when r = 5 inches? dV A = 150π dt  dV 100 B = π dt 3 dV 500 C = π dt 3 dV 40 D = π dt 3 dV 200 E = π dt 3 F None of the above. Question number 3. Your answer was A. Correct. 1 The volume of a right circular cone with radius r and height h is given by V. The radius is 2 = πr h 3 increasing at a rate of 10 inches/sec, while the height is decreasing at a rate of 2 inches/sec. At what rate is the volume changing when r = 12 and h = 3 inches? dV A = 144π dt dV B = 336π dt dV C = 24π dt dV D = −480π dt dV 718 E = π dt 3  F None of the above. Question number 4. Your answer was A. Correct. The volume of a right cylinder with radius r and height h is given by V = πr2 h. The radius is increasing at a rate of 2 inches/sec, while the volume is staying constant. At what rate is the height changing when r = 8 and h = 6 inches? dh A = −3 dt dh 4 B = − dt 3 dh C = 3 dt dh 3 D = dt 2 dh 3 E = − dt 2 F None of the above. Question number 5. Your answer was A. Correct. −− −−−− dP Given: P , x, y are all changing with time. If P 2 = 3√x + y 2 , find an expression for. dt  dP 9 dx dy A = (x + y ) dt P dt dt dP 9 dx dy B = ( + ) dt P dt dt dP 9 dx dy C = (2x + 2y ) 2 dt P dt dt dP dx dy D = 18 (x + y ) dt dt dt dP 18 dx dy E = (x + y ) dt P dt dt F None of the above. Question number 6. Your answer was F. Correct. 1 7 3 An object moves along the x-axis and its position is given by the function s(t) = t 4 − t 3 + t 2 − 1 6 6 2. Find the time(s), t , at which the acceleration is 0. A 0 1 B 3 1 C 0 , , and 2 3 1 D and 2 3  E 3 1 F 3 and 2 Question number 7. Your answer was C. Correct. Determine if f (x) = 4 √− − x − 3 x satisfies the Mean Value Theorem on [ 4, 49 ]. If so, find all numbers c on the interval that satisfy the theorem. 81 A 2 81 B 8 81 C 4 81 D − 4 E The Mean Value Theorem does not apply to this function on the given interval. Question number 8. Your answer was D. Correct. The domain of f (x) is all real numbers. Given the first derivative: ′ 3 2 3 f (x) = −18x (x − 5) (x − 9) (x + 5), state the intervals over which the function f (x) is increasing.  A (−5, 0) and (5, 9) B (−5, 0) and (9, ∞) C (−∞, −5) and (9, ∞) D (−∞, −5) and (0, 9) E (−∞, −5) and (5, 9) F None of the above. Question number 9. Your answer was E. Correct. Consider the function: g(x) 5 = −3x 4 + 2x − 6x + 2. State the x-coordinates of all inflection points for this function (if there is any). 2 A x = 0 and x = 5 1 B x = 5 2 C x = − 5 1 D x = 0 and x = 5 2 E x = 5 F None of the above. Question number 10. Your answer was C. Correct. Suppose that c = 3 is a critical number for a function f. Determine if f (c) is a local maximum, local minimum or neither if f ′ (x) =x − 9 x + 26 x − 24. 3 2 A Neither B Local Minimum C Local Maximum Question number 11. Your answer was E. Correct. Describe the concavity of the graph of f (x) = x 3 − x + 4 and find the points of inflection (if any). 1 1 1 A Concave down on (−∞, ) ; concave up on ( , ∞) ; point of inflection ( , 0) 6 6 6 B Concave up on (−∞, 0); concave down on (0, ∞); point of inflection (0, 4) C Concave down on (−∞, ∞) ; no points of inflection D Concave up on (−∞, ∞) ; no points of inflection E Concave down on (−∞, 0); concave up on (0, ∞); point of inflection (0, 4) Question number 12.  Your answer was A. Correct. Find the critical numbers of f (x) 4 = 2x − 4x 2 − 5 and classify all local extreme values. A Critical nos.: 0 and ± 1 ; local min: f (−1) = −7 and f (1) = −7 ; local max: f (0) = −5 B Critical no.: 0 ; local max: f (0) = −5 C No critical numbers, no extreme values. D Critical nos.: ± 1 ; local min: f (−1) = −7 ; local max: f (1) = −7 E Critical nos.: ± 1 ; local min: f (1) = −7 ; local max: f (−1) = −7 Question number 13. Your answer was B. Correct. Find the absolute extrema for f (x) = −x 3 + 6x − 1 over the interval [0, 6] – A Absolute maximum value is f (√2); absolute minimum value is f (0) – B Absolute maximum value is f (√2); absolute minimum value is f (6) C Absolute maximum value is f (6); absolute minimum value is f (0) D Absolute maximum value is f (0); absolute minimum value is f (6) – E Absolute maximum value is f (6); absolute minimum value is f (√2) F None of the above. Question number 14. Your answer was E. Correct. 1 Consider the function f (x) = 10(5x − 10x) 2 5 List all critical numbers and classify them as vertical tangent, cusp or neither. A x = 0(cusp); x = 1(vertical tangent); x = 2(cusp) B x = 0(vertical tangent); x = 1(vertical tangent); x = 2(neither) C x = 0(vertical tangent); x = 1(cusp); x = 2(vertical tangent) D x = 1(neither) only E x = 0(vertical tangent); x = 1(neither); x = 2(vertical tangent) F None of the above. Question number 15. Your answer was A. Correct. 4 Consider the function f (x) = 10(x2 − 11x + 24) 5 List all critical numbers and classify them as vertical tangent, cusp or neither. 11 A x = 3(cusp); x = 8(cusp); x = (neither) 2 11 B x = (neither) only 2 11 C x = 3(vertical tangent); x = 8(vertical tangent); x = (cusp) 2  11 D x = 3(vertical tangent); x = 8(vertical tangent); x = (neither) 2 11 E x = 3(cusp); x = 8(cusp); x = (vertical tangent) 2 F None of the above. Question number 16. Your answer was A. Correct. The graph of f ′′ (x) , the second derivative of f , is shown below. If the point (1, f (1)) is a critical number for f (x) , is it a local min, local max or neither? A Local minimum B Neither  C Local maximum Question number 17. Your answer was E. Correct. The graph of ′ f (x) is shown below. Where is f (x) increasing? A f (x) is increasing on the interval (-5, 3). B f (x) is increasing on the intervals (-5, 1) and (3, ∞). C f (x) is increasing on the interval (-5, ∞). D f (x) is increasing on the interval (-∞, 3). E f (x) is increasing on the intervals (-∞, -5) and (1, 3). Question number 18. Your answer was A. Correct. A graph of y = f (x) is shown below. Based on this graph, at which labeled point does f (x) satisfy the following conditions? ′ ′′ f < 0, f > 0, and f > 0 A None of these B Point Q C Point P D Point R Question number 19. Your answer was A. Correct.  2 x + 5 Determine the interval(s) where f (x) = is concave down. x A (−∞, 0) B (0, ∞) C (−∞, ∞) D (−∞, −10) ∪ (0, ∞) E (−10, 0) ∪ (10, ∞) F (−∞, 0) ∪ (10, ∞) G none of these. Question number 20. Your answer was E. Correct. 2x Find the vertical and horizontal asymptotes of f (x) =. 5x + 3 2 3 A Vertical asymptote at x = ; horizontal asymptote at y = −. 5 5 3 B Vertical asymptote at x = − ; horizontal asymptote at. y = 0 5 2 C No vertical asymptote; horizontal asymptote at y =. 5 3 D Vertical asymptote at x = − ; no horizontal asymptote. 5  3 2 E Vertical asymptote at x = − ; horizontal asymptote at y =. 5 5 Question number 21. Your answer was D. Correct. Given f (x) is a rational function with domain: R − {x = 8, x = 10} (all real numbers except for 8 and 10). −8(x − 3) Its first derivative is: f ′ (x) = 2 2 (x − 8) (x − 10) List all critical numbers and classify them as local min, local max or neither. A x = 3(local max); x = 8(neither); x = 10(neither). B x = 3(local min) only. C x = 3(local min); x = 8(neither); x = 10(neither). D x = 3(local max) only. E x = −3(local max); x = 8(neither); x = 10(neither). F None of the above. Question number 22. Your answer was C. Correct. Given f (x) is a rational function with domain: R − {x = −9, x = 9} (all real numbers except for -9 and 9). −9(x − 2)(x − 3) Its second derivative is: f ′′ (x) = 3 3 (x − 9) (x + 9) List x-coordinates of all points of inflection for f (x) (if any). A x = −2; x = −3; x = 9  B x = 3 only C x = 2 and x = 3 D x = 2; x = 3; x = −9; x = 9 E x = −9; x = 9 F None of the above. Question number 23. Your answer was D. Correct. Given f (x) is a rational function with domain: R − {x = 11} (all real numbers except for 11). 2 2(x − 7)(x − 2) Its second derivative is: f ′′ (x) = 4 (x − 11) List the intervals over which the function f (x) is concave UP. A (7, ∞) B (−∞, 7) C (2, 7) and (11, ∞) D (7, 11) and (11, ∞) E (−∞, 2) and (11, ∞) F (−∞, 2) and (7, 11) G None of the above. Question number 24. Your answer was D. Correct. Consider the function f (x) = 10 + sin2 (x) over the interval (0, π). Find all critical numbers in this interval and classify them as local min, local max or neither. π A x = (local min) 2 π B x = (local max) 6 π C x = (local min) 6 π D x = (local max) 2 π E x = (local min) 3 F None of the above. Question number 25. Your answer was A. Correct. Consider the function f (x) = 10 sin(x) cos(x) over the interval [0, π]. Find all critical numbers in this interval and classify them as local min, local max or neither. π 3π A x = (local max); x = (local min). 4 4 π π B x = (local max); x = (local min). 2 3  π π C x = (local max); x = (local min). 6 3 3π π D x = (local max); x = (local min). 4 4 π 5π E x = (local max); x = (local min). 6 6 F None of the above.