#### MATH 21A

##### Midterm 2 | Fall '16| Fuchs
1) (10 points) A 50 foot long ladder is leaning on a flat wall which is perpendicular to the floor, and sliding down. Let $\theta$ be the angle that the ladder makes with the floor. This angle $\theta$ is changing a constant rate of -0.2 radians per second. How fast is the ladder slipping down the wall when it is touching the wall 25 feet above the ground?

2) a) (5points)Let $\sin(x + y) = x^2$. Find $\frac{dy}{dx}$.

b) (5 points) Again, letting $\sin(x + y) = x^2$, find $\frac{d^2y}{dx^2}$.

3) Use any differentiation rules (sum/product/quotient/chain/etc) or logarithmic differentiation to find derivatives of the following functions. No need to simplify answers.
a) (4 points) $f(x) = \frac{\ln(x + 4)}{(x^2 + 1)(x + 3)}$

b) (3 points) Hint:use logarithmic differentiation

c) (3 points) Use your answer in part (b) to estimate 4.01$^{2.01}$. No need to simplify. If you didn't do part (b) explain how part (b) can be used to do this.

4) Compare the following derivatives $f'(x)$ using the LIMIT DEFINITION. Be sure to show all your work.
a) (5 points) $f(x) = x^3 - x$.

b) (5 points) $f(x) = 9 - \sqrt{2x + 3}$

5) a) (4 points) Compute the derivative $f(x) = x^2 \sin^{-1}x$ , At each step in your solution, clearly explain what differentiation rules you're using.
b) (4 points) Compute the derivative of $f(x) = x(\tan^{-1}x)^2$. At each step in your solution, clearly explain what differentiation rules you're using.
c) (2 points) Let $f(x) = x^3 -2$ and let $g(x)$ be the inverse of $f(x)$. Compute $g'(6)$.