CARNEGIE MELLON UNIVERSITY
DEPARTMENT OF MATHEMATICAL SCIENCES
21-120 Review Exam 1, Summer , 2005

Your exam shall consist of 10-12 problems similar to the homework and quiz problems. All homework problems are posted on the course web site at http://www.math.cmu.edu/$\sim$tim/ap05/ap05.html . Thursday, July 7th will be a review day. Please ask any questions you may have from the sections 2.1-2.9,3.1,3.2,3.4. The exam will be held in class, from 10:30-11:50, on July 8th. Calculators, notes and texts will not be permitted during the exam. What follows is a practice exam. This is designed for you to take in 80 minutes. The actual exam will consist of different questions. To fully prepare please practice many homework problems.

1. Let $f(x)=x^2+1$. Express the slope of the tangent line, $m_{\rm tan}$ at the point $(-2,5)$ in terms of a limit. You do not need to evaluate the limit. Also, provide a detailed sketch which illustrates this situation.

2. Find the finite or infinite limit, if it exists.
   
   $$\lim_{x \rightarrow 2^-} \frac{x^2-x-2}{(x-2)^2}$$
   
   

3. Find the finite or infinite limit, if it exists.
   
   $$\lim_{x \rightarrow 1} \frac{ x-1 +\vert x-1\vert}{x^2-1}$$
   
   

4. Find the finite or infinite limit, if it exists.
   
   $$\lim_{x \rightarrow 0} \sqrt{\vert x\vert} \sin \frac{1}{x^2}$$
   
   

5. Find the finite or infinite limit, if it exists.
   
   $$\lim_{x \rightarrow 0} \sqrt{\vert x\vert} \sin \frac{1}{x^2}$$
   
   

6. Find the finite or infinite limit, if it exists.
   
   $$\lim_{x \rightarrow -\infty} \frac{2x+\sqrt{1+x^2}}{\sqrt[3]{1+x^3} }$$
   
   

7. Using the precise definition of limit, prove
   
   $$\lim_{x \rightarrow -3} \frac{1+5x}{7}=-2$$
   
   

8. Using the definition of the derivative, find $f^{\prime }(-2)$, where $f(x)=\frac{3-x}{4+2x}$.

9. Using the definition of the derivative, find $f^{\prime }(x)$, where $f(x)=\frac{1}{\sqrt{1+x^2}}$.

10. Prove that the equation $e^x=1-x$ has a solution. Please clearly state any theorem you use and justify your steps.

11. Using any method, find $f^{\prime }(x)$, where $f(x)=\frac{3x^2+e^x-\tan x}{4x^3+\cos x -6\sec x}$.

12. Using any method, find $f^{\prime}(0)$, where $f(x)=(a\cos x+b\sin x)e^x$, and $a$ and $b$ are constants.