אְּדשמַּ יָּא ִ ָּבְּ ִסיַּעת 1969 AP Calculus Exam 1. Which of the following defines a function f for which f(-x)= - f(x) Analysis: 1. Looking for an odd function. 2. f(x)= log (x) is neither. x cannot be negative. 3. is neither odd or even. 2. ln(x-2) <0, if and only f 2<x<3 Analysis: Ln (x-a)<0, it is decreasing (a,a+1). 3. Calculations: and if f is continuous at x=2, then k = 4. Calculations: 5. If , then the value of Calculations: 6x+2x +2y+2y (2x+2y)= -6x-2y =0 at x=1 is 6. What is Analysis: 1. This is the limit definition of derivative. 2. 7. For what value of k will have a relative maximum at x=-2? Calculations: Analysis: 1. Find f’(x) 2. Set f’(x)=-2 3. 8. If P(x)=(x+2)(x+k) and if the remainder is 12 when p(x) is divided by (x-1), then k=3 Analysis: a) P(x) is the remainder at the given x-value. b) Substitute the x values with x=1 c) When a function is evaluated at k, f(k)=remainder, the result will be reminder when that function is divided by (x-k) d) Dividing the function by (x-1) implies that the function is being evaluated at x=1. Calculations: Example: 9. When the area ,in square units, of an expanding circle is increasing twice as fast as its radius in linear units , the radius is Analysis: 10. The set of all points , where t is a real number, is the graph of y Analysis: Which function will have an output of t for the input of Calculations: 11. The point on the curve 𝑥 2 +2y=0 is the nearest point (0, Analysis: 1. 2. Thus, the point is (x, 3. 4. 5. ) −1 ) 2 6. 12. If and g(x)=2x, then the solution set of f(g(x))= g(f(x)) Analysis: 1. Do the composition. 2. Equate and solve for x Calculations: 13. The region bounded by the x-axis and the part of the graph of y=cos x between separated into two regions by the line x=k. If the area of the region for area of the region for Calculations: then k= is is three times the 14. If the function f is defined by , then , the inverse of f, is defined by Calculations: 15. If f’(x) and g’(x) exist and f’(x)>g’(x) for all real x, then the graph of y=f(x) and the graph y=g(x) intersect no more than once. Analysis: a) f(x) increases faster than g(x) b) f(x)=-𝑒 −𝑥 and g(x)=𝑒 −𝑥 (graph shown) c) The graphs could intersect, i.e., f(x)=2x and g(x)=x. However, if they intersect, they will intersect no more than once because f(x) grows faster than g(x). 17. The graph of has a POI at? Calculations: 18. If for all x, then the value of the derivative f’(x) at x=3 is The deritiative is nonexistent . 19. A point moves on the x-axis in such way that its velocity at time t, t>0, is given by value of t does v attain its maximum? Calculations: . At what Analysis: 1. It is not asking where v(t) is increasing but rather where v(t) achieves its maximum. Thus, a(t)=0 𝑥 20. An equation for a tangent to graph of y=sin−1 2 at the origin is x-2y=0 Calculations: 21. At x=0, which of the following is true of the function f defined f(x)=𝑥 2 +𝑒 −2𝑥 : f is decreasing. Analysis a) Interested in knowing what is happening at x=0. Since f’ (0) =-2, f is decreasing at x=0. b) Thus, is a continuous function. c) Zero is not a CP. Therefore, it is not an extremum. d) Calculations: 22. Calculations: 23. The area of the region bounded by the curve Calculations: Graph of 24. , the x-axis, the y-axis and the line x=2 is equal what is in terms of x? Calculations: 25. A region in the plane bounded by 1 𝑥 , the line x=m, and the line x=2m, m>0. The region is independent of m. Analysis: a) It is independent of m since there is no m in the result. b) Note that for 2m the answer will always be ln 2, and for 3m the answer will be ln 3, Calculations: 1 26. ∫0 √𝑥 2 − 2𝑥 + 1 dx Analysis: Calculations: Need to use the negative branch, since the boundaries of integration (0,1). Analysis: 1. Taking the square root, need to use the AV 27. If Analysis: 28. A function defined by f(x)= √3 cosx+3sinx has an amplitude of 2√3 Analysis: a) Find f’(x)=0 b) Substitute the value of x= into f(x) c) This will yield the amplitude. Calculations: Calculations: Thus, the amplitude is 29. Analysis: Calculations: 30. If a function is continuous for all x and if f has a relative maximum at (-1,4) and a relative min at (2, -3), then f crosses both axes. Analysis a) Continuous, but not differentiable. b) There is no POI between x=-1 and x=3, but might occur at either point. c) |𝑥| does not have asymptotes. d) The graph of f intersects both axes. e) f'(-1)=0 is not always true since there could be a corner at x=-1 . Thus it would be undefined. 31. If f’(x)=-f(x) and f (1) =1 Analysis: 1. Need to evaluate each choice at x=1 2. Its derivative must equal its negative. Calculations: 32. If a, b,c,d and e are real numbers and a ≠ 0, then the polynomial equation a𝑥 7 +b𝑥 5 +c𝑥 3 +dX+e=has at least one real root Analysis: a) Depending on the Location of the curve and its end behavior, the curve will intersect the x-axis at least once. b) A polynomial is guaranteed to have at least one real root. c) 33. What is the average (mean) value of over the interval 34. Which of the following isa an equation of a curve that intersects at right angles every curve of the family Analysis: 1. Find f’(x) 2. Find its negative reciprocal 3. Find the derivative of all the choices. Calculations: 35. At t=o a particle starts AT REST and moves along a line in such a way that at time t its acceleration is 24𝑡 2 feet per second per second> Through how many feet does the particle move during the first 2 seconds. Analysis: a) Integrate twice and evaluate, b) Verbiage: “at rest” implies that v(0)=0 Calculations: 36. The approximate value of y=√4 + 𝑠𝑖𝑛𝑥 at x=.12, obtained from the tangent to the graph at x=0 Analysis: 1 a) m=4 and the point (0,2) b) The point (0,2) is used since the tangent is evaluated at x=0. Calculations: Point (0,2) y-2= 0.25x y=0.25(0.12)+2=2.03; 37. Which is the best of the following polynomial approximation to cos (2x) near x=0 Analysis: a) f’(x) and f’’(x) for f(x)=cos (2x) must equal f’(x) and f’’(x) of the choices given. Thus, must evaluate f’(x) and f’’(x) of each choice. 38.∫ Calculations: 𝑥2 3 𝑒𝑥 Analysis: a) Note that if ln was involved, the derivative of the denominator would have been 3 and just 3. 1 Calculations: 𝑑𝑦 39. If y=tan u, u=v-𝑣 and v=ln x, what is the value of 𝑑𝑥 at x=e Calculations: Use the quotient rule 1 1 40. If n is a non-negative integer, then ∫0 𝑥 𝑛 𝑑𝑥 = ∫0 (1 − 𝑥)𝑛 dx for all n. Analysis: The value for each answer will be where n is any power. 41. If Calculations: , , then is a number between Calculations 42. What are all values of k for which the graph of y=𝑥 3 -3𝑥 2 +k will have three distinct xintercepts. Analysis a) f'(x)= 3x(x-2) b) Max at x=0, (0,k) c) Min at f(2)=𝑥 3 -3𝑥 2 + 𝑘= 8-12+k d) Thus, Min at (2, k-4) e) Total vertical distance 4: from k to k-4 f) K cannot be more than 4 since k>4, the min. will be above the x-axis. g) If k=0, it will have only two distinct solutions. h) There will be 3 distinct x-intercepts if the maximum and the minimum are on opposite sides of the x-axis. Calculations: Graph 43. Calculations: 𝜋𝑥 44. The Fundamental period of the function defined by f(x)=3-2𝑐𝑜𝑠 2 ( 3 ) is 3. Analysis: Calculations a) Period= b) c) 𝑑 𝑑 𝑑2 45. if 𝑑𝑥 (f(x))=g(x) and 𝑑𝑥(g(x))=f(𝑥 2 ),then 𝑑𝑥 2(f(𝑥 3 ) = 9𝑥 4 f(𝑥 6 )+6x g(𝑥 3 ) Analysis: The input is Thus, f'(input)= g(input)( the derivative of the input) Similarly, g'(input)= f( input squared) (derivative of the input). Calculations: