MATH 21A

Final | Fall '13

1. (15 points) Calculate the following limits. Then include the precise definition that is satisfied for the limit to be true.
a. $\lim_{x \to 2} (5x^2 - 3x)$

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b. $\lim_{x \to \infty} \frac{1}{x}$

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c. $\lim_{x \to 3^-} \frac{x}{x-3}$

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2. (15 points) Consider the function $f(x) = \frac{x^2 + 2x - 2}{x - 1}$.
a. Perform polynomial long division on $f(x)$ to rewrite $f(x)$ into a simpler form.

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b. Does this function have any horizontal asymptotes? If yes, what are they? In any case, for there to an asymptote what limit is satisfied?

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c. Does this function have any vertical asymptotes? If yes, what are they? In any case, for there to an asymptote what limit is satisfied?

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d. Does this function have any slant asymptotes? If yes, what are they? In any case, for there to an asymptote what limit is satisfied?

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3. (14 points)
a. What is the definition of the derivative of $f(x)$ at a point $x = x_0$?

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b. Use the definition in part a to calculate the derivative of $f(x) = \sqrt{x}$ at $x_0 = 9$ using algebra and limit laws.

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c. Find the tangent line to $f(x)$ at $x = 9$. Use this to estimate $\sqrt{9.12}$.

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d. Graph $f(x)$ and the tangent line calculated in part (c) on the same set of axes.

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4. (12 points) When a person stands on a scale, their weight pushes down against a spring. The weight of a person, $W$, in pounds, is given by how far the scale displaces downward, $x$, in milimeters, by the function $W(x) = 20x^2$. Suppose your weight is exactly $180$ pounds. What are the exact bounds on $x$ that give a value of $W(x)$ between $179$ and $181$ pounds?

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Find the Linearization of $W(x)$ around the value $x = 3$.

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Find the differential of $W(x)$ for general $x$.

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If the absolute error on the weight is $1$ pound when $x = 3$, what does the differential approximate the absolute error on the displacement to be?

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5. (14 points) Calculate the following derivatives. There is no need to simplify long polynomials, but clearly state the rule you are using.
a. $\frac{d}{dx} \left[ (3 - x^2)(x^3 - x + 1) \right]$

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b. $\frac{d}{dx} \left[ \frac{3x^2 - 4}{2x^2 - 8} \right]$

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c. Calculate $\frac{d}{dx}[\tan(x)]$. Don't state a memorized answer. Give the simplest form as a final solution.

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d. $\frac{d}{dx} \left[ e^{(-x^2)} \right]$

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6. (16 points) The population of fish in the Lake Spafford and Putah creek is a function of time $P(t)$, where $P$ is the number of fish and $t$ is in years.
a. What is an appropriate range of populations? What does the derivative $\frac{dP}{dt}$ represent?

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b. The function $P(t)$ satisfies the following differential equation $\frac{dP}{dt} = P(500 - P)$ which means that $\frac{dP}{dt}$ depends on the current population of fish. Use implicit differentiation to calculate the second derivative of $P$ with respect to $t$.

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c. Calculate $\frac{dP}{dt}$ and $\frac{d^2P}{dt^2}$ when the population is $400$ fish. What do these quantities mean for the fish population? Sketch the approximate local graph of $P(t)$ on axes labeled $P$ and $t$, given that the population is approximately $400$ fish at time $t = 10$.

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7. (20 points) Let $f(x) = xe^{(-x^2)}$.
a. What are the critical points of $f(x)$?

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b. On what intervals is $f(x)$ increasing? Where is it decreasing? Classify the critical points of $f(x)$.

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c. Justify and use L'Hospital's Rule to calculate the limit of $f(x)$ as $x$ goes to $\infty$.

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d. Where is $f(x)$ concave up? Where is it concave down? Where are the points of inflection of $f(x)$?

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e. Graph $f(X)$. Label critical points, points of inflection, and concavity.

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8. (14 points)
a. What is the definition for a function $f(x)$ to be continuous at $x_0$, an interior point of its domain?

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b. Show that $g(x) = x^3 + x + 1$ is continuous at any real value $x = x_0$.

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c. Because $g(x)$ is continuous on the interval $[-1, 1]$, what values must $g(x)$ take on that interval?

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d. What does the Mean Value Theorem tell you about $g(x)$ on $[-1, 1]$?

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