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Sample Midterm Exam 2\\
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Question 1  (10).
Find the derivative of the following functions: \\
a.  $ f(x) = x^8 \sin 5x $

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b.   $ f(x) = (x + \sin x)^{23} $

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c.   $ f(x) = \frac{x^2 - x}{ \tan x} $

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d.   $ f(x) = \frac{1 + (1/x)}{1 - (1/x)} $

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e.   $ f(x) = (2x+1)^7 (3x+1)^5 $


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Question 2  (15).  
a. State the definition of the derivative of a function $f$ at a point $x$.

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b.  Use the definition of the derivative to compute $f'(x)$ for
$  f(x) = \frac{2}{x}$

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Question 3  (10).
Find all the vertical and horizontal asymptotes of the graph of
$  f(x) = \frac{x^2 - 2x + 1}{x^2 - 1}$

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Question 4  (20). 
For each of the following, either find the limit or state that 
"no limit exists" and briefly explain why.
Show work used to get your answer.

a. $ \lim_{x \to 0} \frac{2 + 3\sin x} {x^3 + 1} $

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b. $ \lim_{x \to \infty} \cos x$
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c. $ \lim_{x \to \infty} \frac{\cos x}{x^2 - \sin x} $

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d. $ \lim_{x \to 2} \frac {x^2 - 4} { x-2}$. 

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Question 5  (10).
For the function $f(x) = x^2 + 2 \tan x - 2 $

a. Find the equation of the tangent line to the graph of $f(x)$ at
the point (0,-2).

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b. Show that $f(x) = 0$ at some point.

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Question 6  (10)
a. State the precise definition of what is meant by $ \lim_{x \to a} f(x) = L$.
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Use the precise definition of the limit to prove that $ \lim_{x \to 0} 5 x^2 - 4  = -4$. 

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Question 7  (5)
Give an example of a function $f(x)$ which is continuous at $x=1$ but
not differentiable at $x=1$.

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Question 8  (5)
Suppose $f$ and $g$ are functions and $f(3) = 2,\  f'(3) = 4,\  g(5) = 3,\  g'(5) = 7.$
Where can you calculate the derivative of $f \circ g$? What is it equal to?

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Question 9  (5)
Let $f(x) = \sqrt[4] {x^5} $. Find $f'(16)$.

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Question 10  (10).
Find an anti-derivative of the following functions: \\
a.  $ f(x) = 5/x^2  $


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b.  $ f(x) = 3  \sin x $


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c.  $ f(x) =  \csc x \cot x $

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