MATH 1A

Midterm 2 | Fall '11 | Simic

1. (20 points) Let f : R → R be a differentiable function and define

for

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2. (20 points) $A$ curve $C$ is defined by the equation

Find the equation of the tangent line to $C$ at the point of intersection of $C$ with the positive x-axis.

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3. (20 points) (a) Show that the equation has a unique root and that it lies in the interval (−1, 0).
(b) Find the absolute extrema of the function

on the interval [−1, 2].

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4. (20 points) (a) If f : $R$ → $R$ is a differentiable function andwhere $c$ is a constant, what can be said about $f$?
(b) Assume compute $f$.

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5. (20 points) Let

(a) Find the intervals of monotonicity and extrema of $f$.
(b) Find the intervals of concavity and inflection points of $f$.
(c) Find the horizontal asymptotes of $f$.
(d) Sketch the graph of $f$.

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