MATH 1501

Midterm 1 | Summer '15 | Barone

Find the domain of the function. Express your answer in interval notaion. (10 pts.)

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Find the value of a that makes the function continuous at $x=2$. (20 pts.)

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Find the derivative $f'(x)$. $f(x) = \frac{2x - 1}{x^2 + 1}$

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A particles position in $cm$ after $t$ seconds is given by the function

Find the velocity of the particle at time $t=0$.

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Find the limits if they exist. Otherwise, write $+ \infty \; \text{DNE}, -\infty \; \text{DNE}$, or just $\text{DNE}$.
(a) $\lim_{x \to \infty} \frac{3x^2 - 3x + 3}{2x^3 + 10x}$

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(b) $\lim_{x \to 5^{+}} \frac{4x - 5}{30 - 6x}$

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(c) $\lim_{x \to -1} \frac{2x \cos(\pi x)}{x^2 + 1}$

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(d) $\lim_{x \to 3} \frac{2x - 6}{9 - x^2}$

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6. Use the figure below to answer the questions. The figure depicts the graph of the function y=f(x). (4 pts. each)

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(d) Give an estimate for $f'(4)$ and sketch the tangent line at $x = $ in the figure above.

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(e) For what values of $x$ is $f(x)$ continuous? Give your answer in interval notation.

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