Exam 2

1. Find derivatives for the following functions. (50%)

   1. $$f(x)=2 x e^{x} - \frac{1}{\sqrt{x}}$$

   2. $$f(x) = \sin^{3} x$$

   3. $$f(x) = \frac{\sin(5-x)}{x^2}$$

   4. $$f(x) = x e^{\tan x}$$

   5. $$f(x) = 2^{\sin x} \cos x$$

   6. $$f(x) = \frac{a^{2} - x^{2}}{a^{2} + x^{2}}$$   assume$$\ a \$$   is a constant$$$$

   7. $$f(x) = \sqrt{a^{2} - \sin^{2} x}$$   assume$$\ a \$$   is a constant$$$$

   8. $$f(x) = \frac{e^{ax} - e^{-ax}}{e^{ax} + e^{-ax}}$$   assume$$\ a \$$   is a constant$$$$

   9. $$f(x) = e^{kx} (\sin ax + \cos bx)$$   assume$$\ a , \ b \$$   and$$\ k \$$   are constants$$$$

   10. $$f(x) = \left(\frac{1}{x} - \frac{1}{x^{2}} \right) \left(2 x^{3} + 4 \right)$$

2. Find $y^{\prime}$ if $x^{6}+y^{6}=1$ . (10%)
3. Find the equation of the tangent line for the equation $y^3-xy=-6$ at $(7,2)$ . (10%)
4. Find the local linearization of $f(x)=\sqrt[3]{1+3x}$ near $x=0$ . (10%)
5. Find the following limits. (20%)

   1. $$\lim_{x \rightarrow 1} \frac{\ln x}{x - 1}$$

   2. $$\lim_{x \rightarrow -1} \frac{x^{2} -1 }{x+1}$$

   3. $$\lim_{x \rightarrow 0} \frac{e^{x} - 1 - x}{x^{2}}$$

   4. $$\lim_{x \rightarrow 0^{+}} x \ln x$$