MATH 1A

Midterm 1 | Fall '11 | Simic

1. (20 points) Let us write $\exp(x)$ for $e^x$. Define $f(x) = \exp(1 + \exp(2 + 3x))$ (a) Find the domain and range of $f$.

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(b) Show that $f$ is $1-1$.

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(c) Compute $f^{-1}$.

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2. (20 points) Sketch the graph of the following functions. Start with the graph of an "easy" function and apply the appropriate transformations.
(a) $f(x) = |x^2 + 2x|$

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(b) Sketch the graph of the following function. Start with the graph of an "easy" function and apply the appropriate transformations.
$g(x) = \frac{x}{x - 2}$

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3. (20 points) Compute the limit if it exists or explain why it doesn't exist.
(a) $\lim_{x \to \infty} \frac{1 + 2x + 3x^2 + 4x^3 + 5x^6}{x^2 + x^6}$

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(b) Compute the limit if it exists or explain why it doesn't exist.
$\lim_{x \to 0} \frac{e^{-1/x^2}}{1 + e^{-1/x^2}}$

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(c) Compute the limit if it exists or explain why it doesn't exist.
$\lim_{x \to 2} \frac{x^4 + 3x^3 - 10x^2}{x - 2}$

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(d) Compute the limit if it exists or explain why it doesn't exist. $[x]$ denotes the largest integer $\le x$.
$\lim_{x \to 1} x[x]$

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4. (20 points) (a) Using limit laws and theorems from the book, compute $\lim_{x \to 0}x^2 \cos \frac{1}{x}$

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(b) Using the precise ($\varepsilon - \delta$) definition of the limit, prove that your answer in (a) is correct. (Hint: cosine is bounded.)

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5. (20 points) Let $f(x) = \begin{cases} \arctan |x| & \text{if} \; x < 0, \\ \arcsin (x - 1) & \text{if} \; 0 \le x \lt 2, \\ \frac{\pi}{4} x & \text{if} \; x \ge 2 \end{cases}$ (a) Find all the points where $f$ is discontinuous, and explain why $f$ is continuous at all other points.

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(b) Approximately sketch the graph of $f$.

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