MATH 2A

Final - Practice 1 | Fall '16 | Campbell

1. For what value of $a$ is the function $f(x) = \begin{cases} x^2 & x < 3 \\ 2ax & x \ge 3 \end{cases}$ continuous at every $x$?

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2. The theory of relativity predicts that an object whose mass is $m_0$ when it is at rest will appear heavier when it is moving at speeds near the speed of light. When the object is moving at speed $v$, its mass $m$ is given by $m = \frac{m_0}{\sqrt{1 - (v^2/c^2)}}$ where $c$ is the speed of light. Find $\frac{dm}{dv}$ and explain in terms of physics what this quantity tells you.

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3. Show that the equation $3x + 2 \cos x + 5 = 0$ has exactly one real root.

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4. A hyperbola is given by th equation $x^2 + 2xy - y^2 + x = 2$. Use implicit differentiation to find an equation of the tangent line to this curve at the point $(1, 2)$.

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5. Little Susie is enjoying a nice spherical lollipop. She sucks the lollipop in such a way that the circumference decreases by $1$ centimeter per minute. How fast is the volume of her lollipop changing when the lollipop has a radius of $5$ centimeters?

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6. Find the linear approximation of the function $f(x) = x^{3/4}$ at the point $a = 16$.

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7. If $f(3) = 4, g(3) = 2, f'(3) = -5, g'(3) = 6$, find the following values
a) $(f + g)'(3)

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b) $(fg)'(3)$

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c) $\left( \frac{f}{g} \right)'(3)$

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8. Find the absolute maximum and minimum values of the function $f(x) = 3x^4 - 4x^3$ on the interval $[-1, 2]$.

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9. A balloon ascending at a rate of $12$ ft/s is at a height of $80$ ft above the ground when a package is dropped. How long does it take the package to reach the ground? (Hint: the acceleration due to gravity is $32 \; ft/s^2$ downward. Use antiderivatives.) You may leave your answer in radical form.

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10. The graph of $f(x)$ is below. Sketch graphs for $f'(x)$ and $f''(x)$.


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11. Complete each of the following definitions and statements.
a) A function $f$ is continuous at a number $a$ if ________.

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b) The derivative of a function $f$ at a number $a$ is $f'(a) =$ ________ if this limit exists.

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c) The Intermediate Value Theorem says ________

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d) ________ says "If $f$ is continuous on a closed interval $[a, b]$ , then $f$ attains an absolute maximum value $f(c)$ and an absolute minimum value $f(d)$ at some numbers $c$ and $d$ in $[a, b]$."

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e) A function $F$ is called an antiderivative of $f$ on an interval $I$ if ________ for all $x$ in $I$.

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12. Find the dimensions of the rectangle with largest area that can be inscribed in a semicircle of radius $2$ inches.

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13. For the following problems, find the limit if it exists or explain why the limit does not exist.
a) $\lim_{x \to 0} \frac{\sqrt{x^2 + 9} - 3}{x^2}$

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b) $\lim_{x \to \infty} \frac{5x + 2}{7x^2 - 4x + 8}$

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c) $\lim_{x \to 3^-} \frac{x}{x - 3}$

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d) $\lim_{x \to 1} \frac{x - 1}{x^4 - 1}$

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e) $\lim_{x \to 1} \frac{x^2 - 1}{|x - 1|}$

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14. Compute each of the following.
a) $\frac{dy}{dt}$ for $y = \frac{1}{\sqrt{t}} + 5t + 3e^t$

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b) $f'(4)$ for $f(x) = \sqrt{9 + 4x}$

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c) $f'(x)$ for $f(x) = \sin(x \tan^{-1} x)$

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d) $h'(r)$ for $h(r) = r \ln 3r$

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e) $\frac{dy}{dx}$ for $y = x^{\tan x}$.

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15. Consider the function $f(x) = \frac{(x + 1)^2}{1 + x^2}$ a) Find the domain of $f(x)$.

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b) Find the $x$ and $y$ intercepts.

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c) Determine if $f(x)$ is even, odd, or periodic.

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d) Find any vertical, horizontal or slant asymptotes of $f(x)$.

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e) Find intervals on which $f(x)$ is increasing and on which it is decreasing.

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f) Find any local maximum and minimum values.

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g) Find intervals on which $f(x)$ is concave up and on which it is concave down.

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h) Find any points of inflection.

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i) Sketch a graph of $f(x)$.

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