2023 AP CALCULUS BC EXPLANATION SECTION 1 PART B NO.81 - 85

81. A thin rectangular sheet of metal expands in a hot oven. At the instant when the width of the rectangular sheet of metal is 200 millimeters and the length is 300 millimeters, the width is increasing at a rate of 2 millimeters per minute, and the length is increasing at a rate of 3 millimeters per minute. What is the rate of change of the area, in square millimeters per minute, of the rectangular sheet of metal at that instant ?

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82. A particle travels along x-axis so that is velocity at time t ≥ 0 is given by dy/dx = sin (t^2) . What is the total distance the particle travels from t = 0 to t = 2.7 ?

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83. The function f is continuous on the closed interval [a,b] where a and b are constants and a < b . If f (a) < 0 < f (b) , which of the following statements must be true ?

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84. The alternating series

n = 1Σ ∞

(-1)^n+1 / (2n + 2)

converges to S. If the sum of the first six terms of the series is used to approximates S. What is the alternating series error bound ?

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85. The graph of a twice- differentiable function f is shown for 0 < x < 4 . The graph of f has a horizontal tangent at x = 2 and points of inflection at x = 1 and x = 3. On what intervals do f ' (x) and f '' (x) have the same sign ?

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2023 AP CALCULUS BC EXPLANATION SECTION 1 PART B NO.76 - 80

 76. If f ' (x) = 1 + 2x - x^3 - √(x^4 + 1) , then f has a local minimum at x = ...

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77. 

The preceding figure shows the graph of g. If g(x) = ₁∫ ^x f (t) dt, which of the following could be the graph of y = f (x) ?

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78. The function f has derivatives of all orders for all real numbers. The table shown gives values of f and its first three derivatives at selected values of x. What is the value of _-1∫ ¹ f '' (x) dx ?

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79. On certain day, an industrial plant manufactures steel at a rate modeled by R (t) = 10 + 30e^-0,6t tons per hour, where t is measured in hours since the plant opened that day. What is the average rate at which the plant manufactures steel, in tons per hour, over the time interval 0 ≤ t ≤ 5 ?

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80. The power series

converges at x = 3. Which of the following must be true ?

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2023 AP CALCULUS BC EXPLANATION SECTION 1 PART A NO.26 - 30

 26. Let f be function defined by f(x) = x^3 + x. If g is the reverse function of f, what is the slope of the line tangent to the graph of g at the point (2,1) ?

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27. 

The function f is continuous on the closed interval [0,5] and differentiable on the open interval (0,5). Selected values of f are given in the table shown. The value c = 4 satisfies the conclusion of the Mean Value Theorem for f on the closed interval [0,5]. What is the slope of the line tangent to graph of f at x = 4 ?

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28. Let f be a function with f(0) = 0 and derivative given by f'(x) = cos (x^2) . Which of the following is the Maclaurin series for f ?

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29. 

Let R be the region bounded by the graph of y = ln x , the vertical line x = e^2 , and the x-axis , as shown in the figure. Which of tthe following integrals are equal to the area of R ?

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30.

lim n →∞

k=1Σ n 

(3/n k)^2 (3/n) is ...

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2023 AP CALCULUS BC EXPLANATION SECTION 1 PART A NO.21-25

 21. The function g has a continuous second derivative. If g (0) = 2 , g ' (0) = 4, g (3) = 5, and g ' (3) = 6, what is the value of ₀∫ ³ x . g''(x) dx ?

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22. The Maclaurin series for the function f is given below. What is the value of f⁽⁴⁾ (0) ?

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23. Let f be a differentiable function such that f (π) = 2 and f ' (π) = - 3. What is the slope of the line tangent to the polar curve r = f(π) at θ = π ?

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24. _{- ∞}∫ ¹ x^3 . 2^(x^3) dx is ...

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25.  ∫  dx / (x^2 + 4x + 5) = ...

2023 AP CALCULUS BC EXPLANATION SECTION 1 PART A NO.16 - 20

16. 

f(x) = x^2 - 2x for x≤ 4

f(x) = g(x) for x>4

let f be the function defined as shown. If f is differentiable, which of the following could define g(x) ?

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17. The function y = f(x) satisfies the differential equation dy/dx = 2xy with the initial condition f(0) = 5. What is the value of f(2) ?

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18. Which of the following gives the length of the curve y = tan x from x = - (π/4) to x = π/4 ?

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19. If y = f(x) is a function such that dy/dx = (2x - y) / x , then d^2y / dx^2 = ...

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20. Which of the following series converges to 1/e ?

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2023 AP CALCULUS BC EXPLANATION SECTION 1 PART A NO.11-15

11. A particle moves in the xy plane so that its position at any time is given by the parametric equations x(t) = t^2 - t^3 and y(t) = 9 - 6t. What is the speed of the particle at time t = 2 ?

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12. Which of the following is the solution to the differential equation dy/dx = -3y with the initial condition y(0) = 7 ?

a. y = 7e^-3x

b. y = e^-3x + 6

c. y = -3/2x^2 + 7

d. y = √(49-6x)

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13. Consider the series. Which of the following statements is true ?

a. The series converge absolutely

b. The series converge conditionally

c. The series diverges

d. It cannot be determined whether the series converges or diverges from the information given

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14. If f is a function such that lim x→∞ = 0 , which of the following could be an expressive for f(x)?

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15. ∫(x+1)/(x(x+3)) dx = …

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2023 AP CALCULUS BC EXPLANATION SECTION 1 PART A NO.6-10

6. Let y = f(x) be the solution to the differential equation dy/dx = x + y^2 with initial condition f(1) = 2. What is the approximatation for f(1,4) , if Euler’s method is used, starting at x=1 with two steps of equal size ?

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7.

A portion of the slope field for the differential equation dy/dx = x - y is shown in the figure, if the linear function y = f(x) is the solution to the differential equation such that f(-1) = -2 then f(3) = …

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8. If f(x) = sin (e^3x) then f ‘ (x) = …

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9. If the series converges, where A ≠ 0 and B ≠ 0 , what is the sum of the series ?

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10. Of the following series, which can be used with the comparison test to show that diverges?

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2023 AP CALCULUS BC EXPLANATION SECTION 1 PART A NO.1-5

1. If g(x) = x^2 √(x+1) , then g’ (3) = …

2. ₀∫⁴ 100 e^(x/4) dx = ...

3. 

An airplane’s height above sea level is modeled by the differentiable function y=h(t) , where t is the number of minutes take off and h(t) is measured in feet. The graph of h is shown in the figure for 0≤t ≤15. Of the following, which is the best estimate of the rate at which the plane is rising above sea level, in feet per minute, at time t=9?

4. Let f be a vector valued function with f(0) = (-2,3). If the instantaneousrate of change of f with respect to t is given by (3t^2 +4t+1, 2t^3 + t -2 ) what if f(1)?

5. Lim x→4

(2-√x) / (x-4)