AP Calc BC Parametric, Polar, Vector and Series

Question 1

A curve is defined parametrically by x = t² and y = t³. What is dy/dx at t = 2?

Accepted Answer

3

Answer

dy/dx = (dy/dt)/(dx/dt) = 3t²/(2t) = 3t/2, which equals 3 at t = 2.

Question 2

What is the area enclosed by the polar curve r = 1 + cos θ for 0 ≤ θ ≤ 2π?

Accepted Answer

3π/2

Answer

Area = ½∫₀^{2π}(1 + cos θ)² dθ = ½(2π + 0 + π) = 3π/2.

Question 3

Which statement describes the series Σ_{n=1}^∞ (−1)ⁿ/√n?

Accepted Answer

It converges conditionally.

Answer

The terms 1/√n decrease to 0, so the alternating series test shows convergence. The series of absolute values, Σ1/√n, is a p-series with p = 1/2 ≤ 1 and diverges.

Question 4

What is the interval of convergence of Σ_{n=1}^∞ (x − 3)ⁿ / (n·2ⁿ)?

Accepted Answer

[1, 5)

Answer

The ratio test gives radius 2, so the series converges on (1, 5). At x = 1 it is the alternating harmonic series, which converges. At x = 5 it is the harmonic series, which diverges.

Question 5

In the Maclaurin series for f(x) = e^(−x²), what is the coefficient of x⁴?

Accepted Answer

1/2

Answer

Substitute u = −x² into eᵘ = 1 + u + u²/2 + …. The u²/2 term is (−x²)²/2 = x⁴/2, so the coefficient is 1/2.

AP Calc BC Practice Test

AP Calc BC Practice Test

AP Calc BC Parametric, Polar, Vector and Series