MATH 1A

Midterm 1 | Fall '09 | Christ

(1a) 7 points. Use limit rules to evaluate $\lim_{x \to 9} \frac{\sqrt{x} - 3}{x - 9}$. Show your steps.

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(1b) 3 points. Let $f(x) = \sqrt{x}$, with its natural domain. Does $f'(9)$ exist? Justify very briefly.

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(2a) 6 points. Let $f(x) = \frac{(x - 1)(x - 4)(x - 6)}{5(x - 2)(x - 3)(x - 4)}$. With its natural domain. Find all asymptotes of the graph $f$. (you need not sketch the graph, just indicate the types and locations of the asymptotes.)

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(2b) 6 points. Find $\lim_{x \to 0}x \cos(e^{2/x})$. Justify your answer briefly, using any limit rules or theorems from this course.

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(3) 6 points. Show that there is at least one real number $x$ which satisfies $x^6 = 1 + \sin(x)$.

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(4a.) Let $t > 0$. How is $\log_t(3)$ defined?

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(4b.) Let $f(x) = \tan(x)$ with domain $( \frac{\pi}{2}, \pi)$. Does $f$ has an inverse? If so, what are the domain and range of the inverse function?

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(4c.) If some vertical line intersects a graph at a more than one point, what does it say about the graph?

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(4d.) Simplify: $\ln(5e\sqrt{x})$, assuming that $x > 0$.

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(4e.) If the domain of $f$ contains $(-1,1)$, and if $f$ is continuous at $0$, must $f’(0)$ exist?
Either explain in words why it must exist, or give an example of a function for which it does not exist.

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(5a) 5 points. Let $f(x) = x^2$. Find $\delta > 0$ such that $|f(x) - 36| < \frac{1}{1000}$ whenever $|x - 6| < \delta$, and show your reasoning in full detail. You need not simplify any numbers which arise, and there is no penalty if your $\delta$ is smaller than necessary.

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(5b) 7 points. Show, using the precise definition of limit, that $\lim_{x \to \frac{1}{2}} (x - \frac{1}{4x}) = 0$ (Here "show" means prove".)

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