MATH 21A

Final | Fall '15

1. (15 points) Compute the following limits:
a) $\lim_{x \to 2} \ln(x^2 + 2x)$

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b) $\lim_{x \to 1} \frac{\sin(x) - \sin(1)}{x^2 - 1}$

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c) $\lim_{x \to \infty} \frac{7x - 8x^2 + 5x^3}{2x^3 - x^2 + 6x - 1}$

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2. (15 points) Consider the function $f(x) = e^{x - x^2}$.
a) Find the domain and the equations of vertical and horizontal asymptotes.

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b) Find the first derivative and determine the intervals where the function is increasing/decreasing.

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c) Find the second derivative and determine the intervals where the function is concave up/down.

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d) Sketch the graph of this function.

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3. (15 points) Compute the derivatives of the following functions:
a) $f(x) = \sin(x^3 - x)$

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b) $f(x) = x \ln (x - 1)$

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c) $f(x) = \frac{\sin x}{x + 1}$

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4. (15 points) Find the equation of the tangent line to the graph of the function $f(x) = \ln(2x + 3)$ at $x = -1$.

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5. (15 points) Find $y'$ given the equation $x^3 + y^3 + x + y = 1$.

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6. (15 points) Find the minimal and maximal values of the function $f(x) = x^4 + 3x^2 + 5$ on the interval $[-2, 2]$.

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7. (10 points) Find the maximal area of a rectangle that fits between the graph of the function $f(x) = \frac{1}{x^2 + 1}$ and the $x$-axis. Assume that one side of the rectangle belongs to the $x$-axis.

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8. (10 points) (Bonus Problem) Find the number of solutions to the equation $x^3 + 2x^2 - x - 1 = 0$ Justify your answer.

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