Problem Name

Problem Description

Notes

derivative_51

When Sam throws his beach ball straight up in the air, its height at time is given by .
(a) Find the velocity of the ball at time .
(b) Find the velocity of the ball when .
(c) Find out how long the ball takes to reach its highest point. (i.e., at what value of does ?).

derivative_52

When Sam drops his ball off the Mingus Mountain Viewpoint, its height above the ground at time is given by .
(a) What is the velocity at time ? (Is the velocity positive or negative?)
(b) How long will the ball take to hit the ground (i.e., at what value of does )?
(c) How fast is the ball going the instant before it hits the ground?
(d) The quantity is known as the acceleration of gravity and is measured in meters per second per second, or meters per second squared. Find a numerical value for if the ball takes 3.61 seconds to fall to the ground.

derivative_53

Find the values of where the slope of the curve is equal to 3.

derivative_54

For what values of is the line tangent to the curve ?

derivative_55

The mean value theorem states that, if a function has a derivative defined everywhere between and , then there is some value of (call it ) such that and ) equals the slope of the secant line between the points and .
Consider the function , and two points on the graph of that function: (2, 1) and (6, 9). Find the value of that is predicted by the mean value theorem (i.e., find such that for and ).

derivative_56

Newton's method provides an iterative method for estimating the x-intercept of complicated functions. The goal of the method is to find such that . First, make a guess () that is reasonably close to the true value of . Then calculate a better guess according to the formula . The method can be repeated to yield a still better guess, . Keep going until you are satisfied that the result is close enough to the true answer. Now, use Newton's method to estimate the cube root of 7. Start with , and find the x-intercept of the function . Perform a total of 3 iterations and compare with a calculator value.

derivative_57

Find formulas for the derivative and the second derivative, and determine the value of corresponding to maximum or minimum points and say which it is.

derivative_58

Find formulas for the derivative and the second derivative, and determine the value of corresponding to maximum or minimum points and say which it is.

derivative_59

Find formulas for the derivative and the second derivative, and determine the value of corresponding to maximum or minimum points and say which it is.

derivative_60

Find formulas for the derivative and the second derivative, and determine the value of corresponding to maximum or minimum points and say which it is.