Math 161, Calculus 1
Sample Final Exam

Name:

Answer all questions.

1. Give definitions for
   1. $${\lim_{x\to a} f(x) =L}$$
   2. $f$ is continuous on $[a,b]$
2. What three major problems give rise to the concept of the derivative?
3. Show how to find the derivative of $f(x)=\sqrt{x}$ using the definition.

4. Give a $\delta$-$\epsilon$ proof that $$\lim_{x\to 1} x^3=1$$.

5. Find the following limits:
   1. $${\lim_{x\to 0} \frac{(x^2-1)\sin(x)}{x}}$$
   2. $${\lim_{x\to 3} \frac{3x^2+2}{\sqrt{x+6}}}$$
   3. $${\lim_{x\to \infty} \frac{\sqrt{x^2+1}}{2x-1}}$$
6. Find the following derivatives:
   1. $f'(x)$ if $${f(x)=\frac{\tan(x)}{x^2+\frac{1}{x^2}}}$$
   2. $\frac{dy}{dx}$ if $y= \sin(x+\sqrt[3]{x^2+3x})$
   3. $D_x(f)(x)$ if $f(x)=(x^5+3)(\cos(x))\sqrt{x^3+1}$

7. State carefully the following theorems:
   1. The Intermediate Value Theorem
   2. The Mean Value Theorem

8. Give a proof from the definitions of derivative and limit of the following part of the Candidate Theorem:

   If $f'(a)>0$ then $f$ assumes neither a local maximum nor a local minimum at a.

9. Prove the squeeze theorem.
10. Prove from the definition that $(kf)'(x)=k(f'(x))$ if $k$ is a constant.

11. A jewelry box is to be made with a square top. The box is to have a volume of 1000 cm$^3$. What are the most economical dimentions if the material for the top costs three times as much as the material for the sides and bottom?

12. Consider the function $${h(x)= \frac{2x^2}{x^2-4}}$$
    1. Where is $h$ increasing?
    2. Where is $h$ concave down?
    3. What are the asymptotes of $h$?