MATH 16A

Final | Fall '15 | Kouba

1. (6 pts each) Evaluate the following limits.
a) $\lim_{x \to 2} \frac{\frac{1}{x} - \frac{1}{2}}{x - 2}$

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

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c) $\lim_{x \to 2^-} \frac{x^3 + 1}{x^2 - 4}$

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2. (5 pts each) Differentiate. Do not simplify answers.
a) $y = \frac{\sqrt{5 + 2x}}{\cos(3x + 1)}$

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b) $y = \tan^3 (\sin(5x))$

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3. A baseball is hit straight upward. Its height above the ground after $t$ seconds is $h(t) = 128t - 16t^2$ feet.
a) (4 pts) How long is the ball in the air?

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b) (4 pts) What is the ball's maximum height?

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c) (4 pts) What is the ball's velocity as it strikes the ground?

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d) (4 pts) What is the ball's acceleration as it strikes the ground

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4. (12 pts) Function $g(x) = 3 + \frac{x}{2 + x}$. Find another function $f(x)$ so that $g(f(x)) = \sin x$. This problem is not difficult. Just do it.

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5. (20 pts) For the following function $f$ state its domain and determine all absolute and relative maximum and minimum values, inflection points, and intercepts. State clearly the $x$-values for which $f$ is increasing, decreasing, concave up, and concave down. Neatly sketch the graph of $f$. $f(x) = x(x - 6)^2$

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6. (10 pts) Find two numbers $x$ and $y$ so that $x + y = 10$ and the sum $S = x^2 + 4y^2$ is a minimum. Your answer should include $x, y$, and the minimum $S$.

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7. (10 pts) Find the point $(x, y)$ on the graph of $y = 1 + \sqrt{x}$ which is nearest the point $(9/2, 1)$. Your answer should include $x, y$, and the minimum distance.

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8. (10 pts) There are $30$ pear trees in an orchard. Each tree produces $500$ pounds of pears. For each additional three (3) trees which are planted in the orchard, the output per tree drops by 25 pounds. How many pear trees should be added to the existing orchard in order to have the largest total output (pounds) of pears? Your ansewr should use calculus and include the number of trees, the output per tree, and the maximum total output.

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9. (10 pts) A closed rectangular box with a square base is to be constructed from two different materials. Material for the top and bottom costs $\text{\$}3/ft^2$ and material for the sides costs $\text{\$}2/ft^2$. Find the dimensions of the box of largest volume which can be constructed for exactly $\text{\$}72$. Your answer should include the length, width, height, and volume of the box.

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10. (12 pts) The circumference of a circle is increasing at the rate of $2$ ft/sec. At what rate is the area of the circle changing when the radius of the circle is $6$ ft.?

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11. (12 pts) Use limits to determine all horizontal asymptotes for $y = \frac{x}{\sqrt{x^2 + 25}}$

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12. (12 pts) A tank contains $100$ gallons of solution in which $4$ lbs. of salt is dissolved. A solution containing $3/4$ lbs. of salt per gallon beings flowing into the tank at the rate of $8$ gal/min. After how many minutes $t$ will the concentration of salt in the tank be $2/3$ lbs/gal?

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13. (12 pts) Assume that $x \sec y = y^2 + xy$. Find $y' = dy/dx$.

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14. (12 pts) Use $\lim_{x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$ to differentiate the function $f(x) = 3x - x^2$.

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15. (12 pts) Use differentials to estimate the value of $\sqrt{1.2} + (1.2)^{1/3}$.

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16. (12 pts) Assume that the maximum absolute percentage error in measuring the volume of a sphere is $27%$. Use differentials to determine the resulting maximum absolute percentage error in computing the radius of the sphere? (The volume of a sphere is $V = (4/3)\pi r^3$)

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