Title: Infinite Limits and Derivatives

Solve the following problems, providing detailed steps wherever required.

1. Calculate the limit analytically.

lim_h→0(√(5x + 5h) - √5x)/h, where x is constant.

2. Assume the function g satisfies the inequality 1 ≤ 5 g(x) sin²x+ 1 for x near 0. Use the Squeeze Theorem to find lim_x→0 g(x).

3. Evaluate the following limit or state that it does not exist.

lim_r→∞ 1/(COS r + 1)

4. Determine whether the following function is continuous at x = a using the continuity checklist to justify your answers.

g(x) = { X² - 16 if x = 4
           {   x - 4 if x = 4;

            {        9

5. Use the Intermediate Value Theorem to show that the equation x⁵ + 7x + 5 = 0 has a solution in the interval (-1, 0).

Find a solution to x⁵ + 7x + 5 = 0 in (-1, 0) using a root finder.

6. Find the derivative of the function f(s) = √s/4.

7. For the equation y = 1/2 x⁴ + x; a = 2:

a. Find an equation of the tangent line at x = a.

b. Use a graphing utility to graph the curve and the tangent line on the same set of axes.

8. Find f'(x), f"(x), and f⁽³⁾(x) for the following functions:

a. f(x) = 3x¹² + 4x³

b. f(x) = 1/8x⁴ - 3x² + 1

9. The position of a small rocket that is launched vertically upward is given by s(t) = -5t² + 40t + 100, for 0 ≤ t ≤ 10, where t is measured in seconds and s is measured in meters above the ground.

a. Find the rate of change in the position (instantaneous velocity) of the rocket, for 0 ≤ t ≤ 10.

b. At what time is the instantaneous velocity zero?

c. At what time does the instantaneous velocity have the greatest magnitude, for 0 ≤ t ≤ 10.

10. In 2008 the new social networking and microblogging service Twitter increases its number of unique visitors from 0.5 million to more than 4.5 million. A fit to the visitor data over several years using a quadratic polynomial gives V(t) = 0.0173t² + 0.1736t + 0.5, where V is measured in millions of visitors and t is measured in months, with t = 0 corresponding to January 1, 2008.

a. Compute V'(t). What units are associated with the derivative and what does it measure?

b. At what time during 2008 (on the interval [0, 12]) was the growth rate the greatest? What was the growth rate at that time?

c. At what time during 2008 was the growth rate the least? What was the growth rate at that time?