Review for 2nd Exam

1. Exercises 1-52, page 159.
2. Find $g^{\prime\prime\prime}(-\pi/8)$ if $g(t)=\csc 2 t$ .
3. Find an equation of the tangent line to the curve at the given point.

   1. $$y=\frac{x}{x^{2}-2}, \quad (2, 1)$$

   2. $$y=\tan x , \quad (\pi/3, \sqrt{3})$$

   3. $$y=x \sqrt{1+x^{2}} , \quad (1, \sqrt{2})$$

4. Find $y^{\prime}$ if $x \tan y = y-1$ .
5. Find $y^{\prime\prime}$ if $x^{6}+y^{6}=1$ .
6. Find the equation of the tangent line for the equation $y^3-xy=-6$ at $(7,2)$ .
7. Find the local linearization of $f(x)=\ln(1+x)$ near $x=0$ .
8. Find the local linearization of $f(x)=\sqrt[3]{1+3x}$ near $x=0$ .
9. Exercises 2, 4 and 6 on page 158.
10. Find the limit.

    1. $$\lim_{x \rightarrow 0} \frac{\sec x}{1 - \sin x}$$

    2. $$\lim_{t \rightarrow 0} \frac{t^{3}}{\tan^{3} 2 t}$$

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