This quiz works best with JavaScript enabled. Home > Cbse > Class 12 > Science > Mathematics > Class 12 Mathematics Chapter 6 Applications Of Derivatives – Quiz 1 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 6 Applications Of Derivatives Quiz 1 (60 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. If C(x) = 500 + 20x, the marginal cost MC(x) is: A) 500. B) 20x. C) 20. D) 520. Show Answer Correct Answer: C) 20. 2. If the position of a particle is represented by x(t) =-t$^{2}$ + 1, what is its instantaneous velocity at t = 1? A) V =-2. B) V =-1. C) V = 1. D) V = 0. Show Answer Correct Answer: A) V =-2. 3. Find the third derivative of the function:f (x) = 2x-5x$^{6}$ A) F '" (x) =-150x$^{4}$. B) F '" (x)= 2-30x. C) F '" (x) =-600x$^{3}$. D) F '" (x) =-30x$^{5}$. E) F '" (x) = 2-30x$^{5}$. Show Answer Correct Answer: C) F '" (x) =-600x$^{3}$. 4. Find $k^{" }\left(x\right)$ $k\left(x\right)=8x^{-3}-24x^{-2}+12x^{-1}$ A) $-480x^{-6}+576x^{-5}-72x^{-4}$. B) $96x^{-5}-144x^{-4}+24x^{-3}$. C) $-24x^{-4}+48x^{-3}-12x^{-2}$. D) $0$. Show Answer Correct Answer: B) $96x^{-5}-144x^{-4}+24x^{-3}$. 5. The function f has a first derivative $f'\left(x\right)=x\left(x-3\right)^2\left(x+1\right)$ A) -1 only. B) 0 only. C) -1 and 0. D) -1 and 3. E) -1, 0 and 3. Show Answer Correct Answer: A) -1 only. 6. We need to enclose a field with a fence. We have 500 feet of fencing material and a building is on one side of the field so we won't need any fencing on that side. Determine the dimensions of the field that will enclose the largest area. A) X=125 ft, y=100 ft. B) X=150 ft, y=200 ft. C) X=200 ft, y=150 ft. D) X=250 ft, y=125 ft. Show Answer Correct Answer: D) X=250 ft, y=125 ft. 7. If a function switches from increasing to decreasing, what occurs between? A) Absolute maximum. B) Relative minimum. C) Relative maximum. D) Absolute minimum. Show Answer Correct Answer: C) Relative maximum. 8. Identify the conclusion of Rolle's Theorem. A) $f'\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$. B) $f'\left(c\right)=0$. C) All the above. D) None of the above. Show Answer Correct Answer: B) $f'\left(c\right)=0$. 9. An objects distance from its starting point at time t is given by the equation s(t) = t$^{3}$-6t$^{2}$-4. What is the speed of the object when its acceleration is 0? A) 44. B) -24. C) 2. D) 22. Show Answer Correct Answer: C) 2. 10. The function f(x) = x + cos x is A) Always increasing. B) Always decreasing. C) Ncreasing for a certain range of x. D) None of these. Show Answer Correct Answer: A) Always increasing. 11. If f' > 0 on an interval, then f is increasing on that interval. A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: A) True. 12. If the first derivative of a function changes from positive to negative at a certain point, then that point is a known as a relative minimum. A) True (Facts). B) False (Not Facts). C) All the above. D) None of the above. Show Answer Correct Answer: B) False (Not Facts). 13. Given x(t)=t$^{3}$-4t$^{2}$+7, what is the initial position? A) 0. B) -10. C) 11. D) 7. Show Answer Correct Answer: D) 7. 14. The demand function D(p)=120-3p. At price p=10, the elasticity of demand is: A) -1/3. B) -0.1. C) -10/120. D) -3. Show Answer Correct Answer: A) -1/3. 15. If f '(x) >0, what is true about f(x)? A) It is concave down there. B) It is concave up there. C) It has an inflection point. D) It's increasing. Show Answer Correct Answer: D) It's increasing. 16. If $f'\left(x\right)=\sqrt{2x^2-1}$ $y=f\left(x^2\right), $ $\frac{dy}{dx}$ A) 0. B) 1. C) 2. D) 7. Show Answer Correct Answer: C) 2. 17. Find the x-coordinate of the inflection point of f(x)=6x$^{2}$-x$^{3}$. A) -2. B) 0. C) 2. D) None. Show Answer Correct Answer: C) 2. 18. When f ''(x) changes from negative to positive, there is A) A maximum. B) A minimum. C) A point of inflection. D) None of the above. Show Answer Correct Answer: C) A point of inflection. 19. Using the tangent line approximation for $f\left(x\right)=\sqrt{x}$ $\sqrt{8.2}$ A) 2.867. B) 2.733. C) 2.712. D) 2.863. E) 3.133. Show Answer Correct Answer: A) 2.867. 20. To find the velocity function, you need the A) Third derivative. B) Second derivative. C) First derivative. D) Original function. Show Answer Correct Answer: C) First derivative. 21. If fixed costs increase but marginal cost function MC(x) is unchanged, the marginal cost curve shifts upward. A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: B) False. 22. What is the differential equation, $dy=f'(x)dx$ $\sqrt{52}$ A) $dy=3$. B) $dy=\frac{1}{7}$. C) $dy=\frac{1}{14}$. D) $dy=\frac{3}{14}$. Show Answer Correct Answer: D) $dy=\frac{3}{14}$. 23. Does y =-4x$^{2}$ have a maximum or minimum value? A) Maximum. B) Minimum. C) Both. D) Neither. Show Answer Correct Answer: A) Maximum. 24. A rectangular page is to contain 144 sq. inches of print. the margins on each side are 1 inch. Find the dimensions of the page that would use the least paper A) 13, 13. B) 14, 14. C) 16, 16. D) 15, 15. Show Answer Correct Answer: B) 14, 14. 25. The function $f\left(x\right)=e^x$ $f" \left(x\right)>0$ A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: A) True. 26. Refer to # 19.Determine the Displacement when the acceleration is 16 m/s$^{2}$ A) 16m. B) 32m. C) 12m. D) 6m. Show Answer Correct Answer: D) 6m. 27. F(t) =2t$^{2}$-t+1 is the equation of Velocity. Determine the Acceleration Equation A) Y'-t-4. B) Y'=4. C) Y'=4t. D) Y'=4t-1. Show Answer Correct Answer: D) Y'=4t-1. 28. Find the differential for the function y = cos(x) A) Dy =-sin(x) dx. B) Dx = cos(x) dy. C) Dy = cos (x) dx. D) Dx =-sin(x) dy. Show Answer Correct Answer: A) Dy =-sin(x) dx. 29. A manager notices marginal cost rising sharply while marginal revenue is flat. Best short-run action? A) Increase production aggressively. B) Reduce production or investigate cost drivers. C) Ignore and keep producing the same. D) Immediately close the plant. Show Answer Correct Answer: B) Reduce production or investigate cost drivers. 30. Given revenue R(x)=100x-2x$^{2}$. The marginal revenue MR(x) is: A) 100-2x. B) 100-4x. C) 100x-4x$^{2}$. D) 200x-2x$^{2}$. Show Answer Correct Answer: B) 100-4x. 31. If f '(3) = 0 and f"(3) < 0, then which of the following must be true? A) There is an inflection point at x = 3. B) There is a local min at x = 3. C) There is an x-intercept at x = 3. D) There is a local max at x=3. Show Answer Correct Answer: D) There is a local max at x=3. 32. Marginal revenue (MR) is the derivative of total revenue with respect to quantity. A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: A) True. 33. If f'(a) = 0 and f" (a) < 0, then x = a is the location of a relative A) Saddle point. B) Maximum. C) Minimum. D) Inflection point. Show Answer Correct Answer: B) Maximum. 34. A car is moving away from a point at a rate of 60 km/h. If the car is 100 m away from the point, how fast is the distance from the car to a point 200 m away changing? A) 5 m/s. B) 30 m/s. C) 16.67 m/s. D) 50 km/h. Show Answer Correct Answer: C) 16.67 m/s. 35. Apply optimization techniques to find the maximum area of a rectangle with a perimeter of 20. A) 15. B) 25. C) 30. D) 20. Show Answer Correct Answer: B) 25. 36. What is the highest point on a function called? A) Absolute minimum. B) Relative minimum. C) Absolute maximum. D) Relative maximum. Show Answer Correct Answer: C) Absolute maximum. 37. The function h(x) is differentiable and decreasing for all real numbers. On what intervals is the function $y=h\left(x^3-4x^2\right)$ A) (-infinity, 0) only. B) (-infinity, 0) and (8/3, infinity). C) (0, 8/3). D) (4, infinity). Show Answer Correct Answer: C) (0, 8/3). 38. How many relative extrema does f have if $f'\left(x\right)=(x-2)^2(x+5)$ A) 0. B) 1. C) 2. D) 3. Show Answer Correct Answer: B) 1. 39. Find y' if y = cos$^{5}$(4x$^{3}$) A) -5cos$^{4}$(4x$^{3}$)sin(4x$^{3}$). B) -60x$^{2}$sin$^{4}$(4x$^{3}$). C) -5cos$^{4}$(4x$^{3}$)sin(12x$^{2}$). D) -60x$^{2}$cos$^{4}$(4x$^{3}$)sin(4x$^{3}$). Show Answer Correct Answer: D) -60x$^{2}$cos$^{4}$(4x$^{3}$)sin(4x$^{3}$). 40. $\frac{\text{d}}{\text{d}x}\left[x^3\right]$ A) 6. B) 3. C) $6x$. D) $3x^2$. Show Answer Correct Answer: D) $3x^2$. 41. What is the decimal approximation of $\sqrt{52}$ $dy=f'(x)dx$ A) $\frac{3}{14}$. B) 10.5. C) 7.2111. D) 7.214. Show Answer Correct Answer: D) 7.214. 42. What is the "bottom of a valley" on a function (even if it is a cusp) called? A) Absolute maximum. B) Relative minumum. C) Absolute minimum. D) Relative maximum. Show Answer Correct Answer: B) Relative minumum. 43. Given $g\left(x\right)=\frac{3}{x^2}-\frac{5}{x^4}+\frac{2}{x}$ $g^{" '}\left(x\right)$ A) $360x^{-6}-4200x^{-8}+48x^{-5}$. B) $-72x^{-5}+600x^{-7}-12x^{-4}$. C) $18x^{-4}-100x^{-6}+4x^{-3}$. D) $-6x^{-3}+20x^{-5}-2x^{-2}$. Show Answer Correct Answer: B) $-72x^{-5}+600x^{-7}-12x^{-4}$. 44. Let f be a continuous function for which f(-2)=1 and f(5)=-3. The Intermediate Value Theorem guarantees that A) F(c)=2 for at least one c between-3 and 1. B) F(c)=0 for at least one c between-2 and 5. C) F(c)=0 for at least one c between-3 and 1. D) F(c)=2 for at least one c between-2 and 5. Show Answer Correct Answer: B) F(c)=0 for at least one c between-2 and 5. 45. Which statement is true about the slope of the tangent to a revenue curve at a point? A) It gives the average revenue at that output. B) It gives the marginal revenue (instantaneous change in revenue). C) It gives the total revenue up to that point. D) It gives the fixed cost. Show Answer Correct Answer: B) It gives the marginal revenue (instantaneous change in revenue). 46. For a function f(x), f ' (-3) = 5 indicates f(x) is ..... at x=-3. A) Increasing. B) Concave up. C) Decreasing. D) Concave down. Show Answer Correct Answer: A) Increasing. 47. Discuss the concavity and the point(s) of inflection (if any) of the function:$f\left(x\right)=x^3-3x^2+10$ A) Concave up in $\left(1, \infty\right)$. B) Concave down in $\left(-\infty, 1\right)$. C) Concave down in $\left(-\infty, \infty\right)$. D) Concave up in $\left(-\infty, 1\right)$. Show Answer Correct Answer: B) Concave down in $\left(-\infty, 1\right)$. 48. A particle moves along the x axis with its acceleration given by a(t) = t$^{2}$-2t for 0 < t < 6. At time t = 0, the velocity is 10. What is the maximum velocity of this particle on the specified interval? A) 10. B) 26. C) 40. D) 46. Show Answer Correct Answer: D) 46. 49. What are identified by finding where the second derivative equals zero or is undefined? A) Relative extrema. B) Absolute extrema. C) Possible points of inflection. D) Points of inflection. Show Answer Correct Answer: C) Possible points of inflection. 50. If f"(x) >0, what is true about f(x)? A) It is concave up there. B) It's zero. C) It has an inflection point. D) It is concave down there. Show Answer Correct Answer: A) It is concave up there. 51. Given a function g(x), if g'(x)=0 at a certain value of x, then g(x) has ..... at x. A) A critical point. B) An inflection point. C) A maximum. D) A minimum. Show Answer Correct Answer: A) A critical point. 52. If R(w) = w$^{3}$ then R'(-2) is negative. A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: B) False. 53. If demand becomes more elastic after a price change, a firm is more likely to: A) Keep price unchanged. B) Remove the product from market. C) Increase price to raise revenue. D) Decrease price to raise revenue. Show Answer Correct Answer: D) Decrease price to raise revenue. 54. If f(x) = $3x^2 + 12x-4$ A) 9. B) 12. C) 6. D) 3. Show Answer Correct Answer: A) 9. 55. It is a part of the problem that needs to be maximized or minimized. A) Concept equation. B) Problem equation. C) Constraint equation. D) Optimization equation. Show Answer Correct Answer: D) Optimization equation. 56. If a function is concave down at a critical number, what must occur at that critical number? A) Relative maximum. B) Absolute minimum. C) Absolute maximum. D) Relative minimum. Show Answer Correct Answer: A) Relative maximum. 57. A conical tank is being filled with water at a rate of 3 cubic meters per minute. If the tank has a height of 10 meters and a radius of 5 meters, how fast is the water level rising when the water is 6 meters deep? A) 1/20 m/min. B) 1/30 m/min. C) 1/25 m/min. D) 1/35 m/min. Show Answer Correct Answer: C) 1/25 m/min. 58. If $y=x^x, $ $\frac{dy}{dx}$ A) $2e^e$. B) $e^e$. C) $e$. D) $2e$. Show Answer Correct Answer: A) $2e^e$. 59. The concavity of a function is described by its ..... A) Second derivative. B) First derivative. C) Third derivative. D) Any derivative determines concavity. Show Answer Correct Answer: A) Second derivative. 60. How many possible points of inflection does f have if $f'\left(x\right)=(x-2)^2(x+5)$ A) 3. B) 2. C) 0. D) 1. Show Answer Correct Answer: B) 2. 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