Calculus Quiz

If y = sin(3x) then dy/dx =

• -3cos(3x)

• -cos(3x)

• -1/3cos(3x)

• cos(3x)

• 3cos(3x)

lim(x -> 0) (e^x - cos(x) - 2x) / (x^2 - 2x) is

• -1/2

• 0

• 1/2

• 1

• non-existent

int (3x+1)^5 dx =

For (0 ≤ t ≤ 13), an object travels along an elliptical path given by the parametric equations x = 3cos t and y = 4sin t. At the point where t = 13, the object leaves the path and travels along the line tangent to the path at that point. What is the slope of the line on which the object travels?

• -4/3

• -3/4

• -4 tan 13/3

• -3 tan 13/4

• -3/4 tan 13

Let y = f(x) be the solution to the differential equation dy/dx = x + y with the initial condition f(1) = 2. What is the approximation for f(2) if Euler's method is used, starting at x = 1 with a step size of 0.5?

• 3

• 5

• 6

• 10

• 12

What are all values of p for which the integral ∫₁ᴵⁿᶠⁱⁿⁱᵗʸ 1/x^(2p) dx converges?

The position of a particle moving in the xy-plane is given by the parametric equations x = t^3 - 3t^2 and y = 2t^3 - 3t^2 - 12t. For what values of t is the particle at rest?

• (A) -1 only

• (B) 0 only

• (C) 2 only

• (D) -1 and 2 only

• (E) -1, 0, and 2

Calculate the integral ∫ x^2 cos(x^3) dx.

• (A) -1/3 sin(x^3) + C

• (B) 1/3 sin(x^3) + C

• (C) -x^3/3 sin(x^3) + C

• (D) x^3/3 sin(x^3) + C

• (E) x^3/3 sin(x^4/4) + C

If f(x) = ln(x+4) + e^(3x), then f'(0) is

• (A) -2/5

• (B) 1/5

• (C) 1/4

• (D) 2/5

• (E) nonexistent

What is the value of the series \( \sum_{n=1}^{\infty} \frac{2^{n+1}}{3^n} \)?

• 1

• 2

• 4

• 6

• The series diverges.