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See attached file for full problem description.

1. Use the definition of the derivative to find f(x) given f(x) = square root (x).

2. A ball thrown vertically upward at time t = 0 (s) with initial velocity 80 ft/s and with initial height 96 ft has height function:

y(t) = -16t^2 + 80t + 96.

a) What is the maximum height attained by the ball?

b) When and with what impact speed does the ball hit the ground?

3. Write an equation of the line tangent to the curve y = (5/x^2) - (2/x^3) at the point P(-1, 7). Express the answer in the form ax + by = c.

4. Given f(x) = ((x^2 + 3x + 1)^5) / ((x + 3)^5), identify a function u of x and an integer n =/= 1 such that f(x) = u^n. Then compute f'(x).

5. Differentiate the function f(x) = (1 + x)^(3/2)(2 + x)^(2/3).

6. Write the equation of the line tangent to the curve y = 3 square root (x) at the point P where x = 4. Write the equation in the form ax + by = c.

7. Find the maximum and minimum values attained by the function h(x) = (x - 1) / (x + 1) on the interval [0, 2].

8. A mass of clay of volume 432 in.^3 is formed into two cubes. What is the minimum possible total surface area of the two cubes? What is the maximum?

9. Find dx/dt given x = square root (1 + cot 3t).

10. The equation f(x) = x^3 - 3x + 1 has three distinct real roots. Approximate their locations by evaluating f at -2, -1, 0, 1, and 2. Then use Newton's method to approximate each of the three roots to four-place accuracy.

11. Sand falling from a hopper at 10 pi ft^3/s forms a conical sand pile whose radius is always equal to its height. How fast is the radius increasing when the radius is 5ft?

12. Find the open intervals on the x-axis on which the function f(x) = x^2 / x-1 is increasing and those on which it is decreasing.

13. What is the maximum possible volume of a right circular cylinder with a total surface area 600 pi in.^2 (including the top and the bottom)?

14. Find the interval on which the function f(x) = (x - 2)^2(x + 3)^2 is increasing and decreasing. Sketch the graph of y = f(x), and identify any local maxima and minima. Any global extrema should also be identified.

15. Find the exact coordinates of the inflection points and critical points of the function f(x) = 2x^3 + 3x^2 - 180x + 150 on the interval (-10, 10).

16. Find the first three derivatives of the function f(x) = 2cos x sin 2x.

17. Sketch, by hand, the graph of f(x). Identify all extrema, inflection points, intercepts, and asymptotes. Show the concave structure clearly and note any discontinuities.

f(x) = x^2 / x - 1

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Solution Summary

This is a series of calculus problems that involves derivatives, velocity and height, tangent lines, maximum and minimum values, and Newton's method.

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