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?

Log in or sign up to see discussion or post a question.

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

Log in or sign up to see discussion or post a question.

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

Log in or sign up to see discussion or post a question.

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) $f(x) = \frac{\ln(x + 4)}{(x^2 + 1)(x + 3)}$

Log in or sign up to see discussion or post a question.

b) (3 points) Hint:use logarithmic differentiation


Log in or sign up to see discussion or post a question.

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.

Log in or sign up to see discussion or post a question.

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

Log in or sign up to see discussion or post a question.

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

Log in or sign up to see discussion or post a question.

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.

Log in or sign up to see discussion or post a question.

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.

Log in or sign up to see discussion or post a question.

c) (2 points) Let $f(x) = x^3 -2$ and let $g(x)$ be the inverse of $f(x)$. Compute $g'(6)$.

Log in or sign up to see discussion or post a question.