Suppose a stone thrown vertically upward from the edge of a

Question

Solve the following problems, providing detailed steps wherever required.

1. Suppose a stone thrown vertically upward from the edge of a cliff on mars (where the acceleration due to gravity is only about 12 ft/s²) with an initial velocity of 64 ft/s from a height of 192 ft above the ground. The height s of the stone above the ground after t seconds is given by s = -6t²+ 64t + 192.

a. Determine the velocity v of the stone after t seconds.

b. When does the stone reach its highest point?

c. What is the height of the stone at the highest point?

d. When does the stone strike the ground?

e. With what velocity does the stone strike the ground?

2. Use the chain rule to evaluate dy/dx.

y = (5x²+ 11x)²⁰

3. Use the chain rule to evaluate dy/dx.

y = sin√x

4. Use the chain rule to evaluate dy/dx.

y = ((x + 2) (3x³ + 3x))⁴

5. Find d²y/dx² for the following function.

y = xcosx²

6. Find d²y/dx² for the following function.

y = √(3x³ + 4x + 1)

7. Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur.

8. For the given function f(x) = 1/8 x³ - ½x; [-1, 3]:

a. Find the critical points of the function on the domain or on the given interval.

b. Use a graphing utility to determine whether each critical point corresponds to a local maximum, or neither.

9. For the given function f(x) = sinx cosx; [0, 2π]:

a. Find the critical points of the following functions on the domain or on the given interval.

b. Use a graphing utility to determine whether each critical point corresponds to a local maximum, or neither.

10. For the given function f (x) = x√(2 - x²); [-√2, √2]

a. Find the critical points of f on the given interval.

b. Determine the absolute extreme values off on the given interval (if they exist).

c. Use the graphing utility to confirm your conclusions.