Calculus I -- Spring 2017 -- final

MATH 229 -- Spring 2017 -- final exam

Name:_________________________ $\qquad$ 8 problems, 80 points total

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

$\bullet$ 1. Solve the following equation: $\sqrt{x^2-7}-3=0$.

$\bullet$ 2. Compute the one-sided limits of the function given below at $x=-1$ and $x=3$: \[ f(x)=\left\{ \begin{align} {}% -x+1 &\quad \text{ if } x<-1;\\ x^{2}+1 &\quad \text{ if } -1\leq x<3;\\ e^x &\quad \text{ if }x>3. \end{align} \ \right. \]

$\bullet$ 3. Use implicit differentiation to find an equation of the line tangent to the curve $x^{1/2}+xy=2$ passing through the point $(1,1)$.

$\bullet$ 4. Find the point on the line $y=1-2x$ that is closest to the origin.

$\bullet$ 5. Describe the behavior of the function plotted below without referring to the picture:

$\bullet$ 6. Suppose $s(t)$ represents the position of a particle at time $t$ and $v(t)$ its velocity. If $v(t)=\sin t-\cos t$ and the initial position is $s(0)=0,$ find the position $s(1).$

$\bullet$ 7. The Fundamental Theorem of Calculus includes the formula $\int _a^bf(x)\, dx=F(b)-F(a)$. (a) State the whole theorem. (b) Provide definitions of the items appearing in the formula.

$\bullet$ 8. Evaluate $$\int_{-1}^1 \sin \frac{x}{3} \, dx.$$