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# Calculus III -- Fall 2017 -- midterm

**MATH 231 -- Fall 2017 -- Midterm exam**

Name:_________________________ $\qquad$ 6 problems, 10 points each

- Write the problems in the given order, each problem on a separate page.
- Show enough work to justify your answers.

$\bullet$ **1.** Suppose an object is dropped from a 100 feet building. You are standing 100 feet away from the building and tracing the object with a laser. Express the angle of the laser as a function of time.

$\bullet$ **2.** If the velocity of an object at time $t$ is given by $V(t)=<t^2+1,\cos t>$, what is its position $P$ at time $t=3$ if the object starts at the origin?

$\bullet$ **3.** Find the equation of the line tangent to the curve
$$x=t^2,\ y=t^3,\ z=t^4,$$
at the point $(1,1,1).$

$\bullet$ **4.** Suppose during the first $2$ seconds of its flight an object progressed from point $(0,0)$ to $(1,0)$ to $(1,1)$. What was its (a) average velocity, (b) average acceleration?

$\bullet$ **5.** Find the curvature of the curve $F(t)=<t-1,t^2+2>$.

$\bullet$ **6.** Plot a few points of the graph of the function $h(x,y)=3-x-2y$ to demonstrate that this is a plane.