MATH 2A

Final - Practice 2 | General

1. (10 points) Determine the value of each of the following limits.
a. $\lim_{x \to 1} \frac{2 - x}{(x - 1)^1}$

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b. $\lim_{x \to 3} f(x)$ where $f(x) = \begin{cases} x^2 + 5 & \text{if} \; x \ne 3 \ 7 & \text{if} \; x = 3 \end{cases}$

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c. $\lim_{x \to 0} \frac{\sqrt{x + 1} - 1}{x}$

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d. $\lim_{r \to \infty} \frac{\ln \sqrt{r}}{r^2}$

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e. $\lim_{x \to -\infty} \frac{\sqrt{7x^2 + 3x}}{3x - 5}$

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2. (10 points) Compute the indicated derivative of each of the following functions. (You do not need to simplify the result algebraically.)
a. Find $\frac{dy}{dx}$, for $y = 2x + 6 - 4x^2 + \frac{5}{x^2} + \ln x$

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b. Find $f'(4)$, for $f(\theta) = 2\sqrt{\theta} + \frac{2}{\sqrt{\theta}}$

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c. Find $y'$, for $y = \frac{3t - 1}{t^2 + t - 2}$

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d. Find $r'(t)$, for $r(t) = 5^t \sin t$.

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e. Find $y''$, for $y = \sec(x) = 3\cos(x)$

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3. (10 points) Compute the indicated derivative of each of the following functions.
a. For $y = (5x^2 - 2x)^{\frac{3}{4}}$, find $\frac{dy}{dx}|_{x=2}$

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b. For $f(x) = (3x)^{\tan^{-1}(x)}$, find $f'(x)$.

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4. (10 points)
a. Given that the tangent line to $y = f(x)$ at $(4, 3)$ passes through the point $(0, 2)$, find $f(4)$ and $f'(4)$.

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b. Sketch a graph of a continuous, differentiable function $g(x)$ which satisfies
  • $g'(x) > 0$ for $x < -2$ and $x > 3$
  • $g'(x) < 0$ for $-2 < x < 3$
  • $g'(x) = 0$ for $x = -2$ and $x = 3$

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5. (10 points) Use implicit differentiation to find the equation of the tangent line to the curve $e^y \sin(x) + x - xy = \pi$ at the point $(\pi, 0)$.

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6. (10 points) For the function $f(x) = \frac{2x^2 - 1}{x^2 - 4x - 21}$, answer each of the following.
a. Find all of the points of discontinuity of $f(x)$.

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b. Use the Intermediate Value Theorem to verify that $f(x)$ has a zero on the interval $(0, 1)$.

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c. Find the equations for any horizontal and vertical asymptotes of this function.

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7. (10 points) Answer True or False to each of the following and explain your answer. (Unjustified answers will not receive credit.)
a. True/False: Let $f(x) = \tan(x)$. By the Mean Value Theorem, there exists some $c$ in the interval $(0, \pi)$ such that $f'(c) = 0$.

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b. True/False: Assume the half-life of a certain radioactive substance is $200$ years. If we begin with $n$ grams of the substance, then after $600$ years we will have $\frac{1}{8}n$ grams.

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c. True/False: $F(x) = \frac{\sin^3(x)}{3} + 10$ is an antiderivative of $\sin^2(x)$.

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8. (10 points) A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius $r$ of the outer ripple is increasing at a constant rate of $2$ ft/sec. When the total area of the disturbed water is $16\pi$ square feet, at what rate is the total area of the disturbed water changing? Remember to include units.

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9. (10 points) According to postal regulations, a carton is classified as "oversized" if the sum of its height and girth (the perimeter of its base) exceesd $108$ inches. Find the dimensions of a carton with a square base that is not oversized and has maximum volume.

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10. (10 points) For the function $f(x) = 3x^4 - 8x^3 + 6x^2 + 1$, answer each of the following.
a. Find all intervals on which $f$ is increasing and all itnervals on which $f$ is decreasing.

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b. Find any local maximum and minimum values.

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c. Find all intervals on which $f$ is concave up and all intervals on which $f$ is concave down.

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d. Find any points of inflection.

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e. Graph the function $f(x)$.

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