AP Calc BC Differential Equations and Integral

Question 1

If dy/dx = xy and y(0) = 2, what is y(2)?

Accepted Answer

2e²

Answer

Separating gives dy/y = x dx, so ln|y| = x²/2 + C. Since y(0) = 2, y = 2e^(x²/2), and y(2) = 2e².

Question 2

A population satisfies dP/dt = 0.2P(1 − P/500). For what value of P is the population growing most rapidly?

Accepted Answer

250

Answer

The growth rate 0.2P(1 − P/500) is a downward-opening parabola in P. Its maximum is halfway between the zeros at P = 0 and P = 500, which is P = 250.

Question 3

Euler's method with two steps of size 0.5 is used to approximate y(1) for the differential equation dy/dx = x + y with y(0) = 1. What is the approximation?

Accepted Answer

2.5

Answer

At (0, 1) the slope is 1, so y(0.5) ≈ 1 + 0.5(1) = 1.5. At (0.5, 1.5) the slope is 2, so y(1) ≈ 1.5 + 0.5(2) = 2.5.

Question 4

What is the area of the region enclosed by the graphs of y = x² and y = 2x?

Accepted Answer

4/3

Answer

The curves meet where x² = 2x, at x = 0 and x = 2, and 2x is the upper curve on that interval. The area is ∫₀² (2x − x²) dx = 4 − 8/3 = 4/3.

Question 5

The region bounded by y = x², the x-axis, and the line x = 2 is revolved about the y-axis. What is the volume of the solid generated?

Accepted Answer

8π

Answer

Using washers in y, the outer radius is 2 and the inner radius is √y, for 0 ≤ y ≤ 4. The volume is π∫₀⁴ (4 − y) dy = π(16 − 8) = 8π.

AP Calc BC Practice Test

AP Calc BC Practice Test

AP Calc BC Differential Equations and Integral