Mini Exam 10 — Timed Mini Exams

Questions in this quiz (6)

1. Let $$x>0$$. Suppose $$\frac{d}{dx}f(x)=g(x)$$ and $$\frac{d}{dx}g(x)=f(\sqrt{x})$$; then $$\frac{d^2}{dx^2}f(x^2)=$$

   • $$2xg(x^2)$$
   • $$2g(x^2)+4x^2f(x)$$
   • $$2g(x^2)+4x^2f(\sqrt{x^2})$$
   • $$2g(x^2)+4x^2f^2(x)$$

2. A particle moves counterclockwise on the circle $$x^2+y^2=16$$ with constant speed $$3$$. Its velocity vector when the particle is at $$(0,4)$$ equals

   • $$\langle-3,0\rangle$$
   • $$\langle3,0\rangle$$
   • $$\langle0,-3\rangle$$
   • $$\langle0,3\rangle$$

3. Let $$\mathbf{R}=\langle a\cos kt,\ a\sin kt\rangle$$ be the position vector of a moving particle. The acceleration vector equals

   • $$-k^2\mathbf{R}$$
   • $$-ak^2\mathbf{R}$$
   • $$-a^2k^2\mathbf{R}$$
   • $$-k\mathbf{R}$$

4. The length of the curve $$y=e^x$$ between $$(0,1)$$ and $$(1,e)$$ is approximately

   • $$1.718$$
   • $$2.003$$
   • $$2.587$$
   • $$3.194$$

5. The position of a moving object is given by $$P(t)=\langle2t,\ e^{2t}\rangle$$ Its acceleration is

   • constant in both magnitude and direction
   • constant in magnitude only
   • constant in direction only
   • constant in neither magnitude nor direction

6. Suppose we plot a particular solution of $$\frac{dy}{dx}=3y$$ from the initial point $$(0,1)$$ using Euler's method. After one step of size $$\Delta x=0.1$$, how big is the error?

   • $$0.06$$
   • $$0.09$$
   • $$0.14$$
   • $$0.19$$