Calculus I -- Fall 2018 -- final

MATH 229 -- Fall 2018 -- Final exam

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

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

$\bullet$ 1. Sketch the graph of a differentiable function $f$ that is continuous on $(-\infty,6]$ and has an absolute maximum at $-2$, a local maximum at $0$, a local minimum at $5$, an inflection point at $4$, and a horizontal asymptote $y=-1$.

$\bullet$ 2. The velocity of the object at time t is given by $v(t)=1+3t^2$. If at time $t=1$ the object is at position $x=0$, where is it at time $t=3$?

$\bullet$ 3. Find all antiderivatives of the function $f(x)=x^{\pi}+e^{x}-1/x+e$.

$\bullet$ 4. (a) Analyze the first and second derivatives of the function $f(x)=x^4-2x^2$. (b) Use part (a) to sketch the graph of $f$.

$\bullet$ 5. (a) Finish the statement: "If $h'(x)=0$ for all $x$ within $(a,b)$ then...". (b) Finish the statement: "If $f'(x)=g'(x)$ for all $x$ within $(a,b)$ then...".

$\bullet$ 6. Find two numbers $x,y$ whose sum is $2$ and whose product is a maximum.

$\bullet$ 7. Set up but do not solve the optimization problem for the following situation: "Among all rectangles with perimeter $1$, find the one with the largest area".

$\bullet$ 8. The graph of a function $f(x)$ is given below. Estimate the values of the derivative $f'(x)$ for $x=0,3$, and $5$.

$\bullet$ 9. Solve the following equation: $\sqrt{x^2-7}-3=0$.

$\bullet$ 10. 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.