201 Sample C (Sp26)

(a)

$\displaystyle y = (x + \frac{1}{x})^9$

(b)

$\displaystyle y = \frac{x^3 - 5x^4 + x^8}{e^{3x}}$

(c)

$\displaystyle y = 3^x \cos(2x)$

(d)

$\displaystyle x^2 + 6xy - 2y^2 = 3$ (Use implicit differentiation.)

(a) $\displaystyle \int x\left( \sqrt{x} + \frac{\sec^2 (x)}{x} + x^{-3} \right)\, dx$

(b) $\displaystyle \int \sin^3(x)\cos(x)\, dx$

(c) $\displaystyle \int \frac{\sin(\ln x)}{x}\, dx$

(d) $\displaystyle \int_0^1 \frac{3}{(2x+1)^3}\, dx$

(a) $\displaystyle \lim_{x\to\infty} \left( \sqrt{x^2 + 3x} - \sqrt{x^2 - 2x} \right)$

(b) $\displaystyle \lim_{x\to\infty} (1 + 2x)^{1/(2\ln x)}$

(c) $\displaystyle \lim_{x\to 0} \frac{\sin x}{\tan(4x)}$

(a) (6 pts)

Let

$$f(x) = \begin{cases} x^2, & x < 2 \\ 8 - 2x, & x \ge 2 \end{cases}$$

1. Use the definition to show $f(x)$ is continuous at $x = 2$.
2. Is $f(x)$ differentiable at $x = 2$? Explain.

(b) (4 pts)

Using the Fundamental Theorem of Calculus, Part I, find

$$F'(x) \quad \text{if} \quad F(x) = \int_{\arctan x}^{0} \frac{1}{t^3 + 1}\, dt.$$

(a) (5 pts)

Find $\frac{dx}{dt}$ at $x = 1$ if

$$y = x^3 + x^2 - 1, \qquad \frac{dy}{dt} = -1.$$

(b) (3 pts)

State the Intermediate Value Theorem.

(a) (3 pts)

State the Mean Value Theorem.

(b) (5 pts)

Suppose $f(x)$ is differentiable, $f(1)=2$, and $f'(x)\le 5$.
What is the largest possible value of $f(4)$?

(a) (3 pts)

Find the linear approximation $L(x)$ at $x = e^2$.

(b) (2 pts)

Find the differential $df$ at $x = e^2$.

(c) (3 pts)

Use either of the previous answers to estimate the change in $f$ when $x$ increases to $e^2 + 0.2$.

(a) (4 pts)

Approximate

$$\int_1^3 \ln x\, dx$$

using a Riemann sum with $n=6$ equal length subintervals and left endpoints as the sample points.
(You may leave the answer as an unsimplified sum.)

(b) (5 pts)

Find all points on $y = x^{-1}$ where the tangent line is parallel to $y = -4x + 7$.

(a)

Find equations of any vertical asymptotes of the graph of $f$.

(b)

Does the graph of $f$ have horizontal asymptotes? If so, in which direction(s)?

(a)

Find the coordinates of all intercepts of its graph and the limits of $f(x)$ as $x\to\pm\infty$.
(There are exactly two x‑intercepts and they are integers that lie in $[-2,2]$.)

(b)

Find the intervals of increase and the intervals of decrease for $f(x)$ and the coordinates of any local maxima and local minima.

(c)

Find the intervals of concavity and the coordinates of any inflection points.

(d)

Sketch the graph of $f(x)$ including all features from (a)–(c).