Chapter 5:

Applications of the Derivative

Chapter 4:

Derivatives

Chapter 5:

Applications

Objectives:

To be able to use the derivative to analyze function

Draw the graph of the function based on the analysis

Apply the principles learned to problem situations

Example 1: A stone is thrown upward from the ground with

an initial velocity of 32 ft per sec; find

a. the average velocity in the interval 3/4 < t < 5/4

b. the instantaneous velocity at t = 3/4 and t = 5/4

c. how high will the stone rise?

d. how many seconds will it take the stone to reach the ground?

Example 2: The positions of 2 particles A and B are given

by the equations:

SA = 3t3 – 12t2 + 18t+ 5 and SB = -t3 + 9t2 – 12t

When do they have the same velocity?

Example 3: A point along a horizontal coordinate in such a

way that, s(t) = t3 - 3t2- 24t- 6. When is the

point slowing down?

Example 4: An object thrown directly upward at a

height, s(t) = -16t2 + 48t +256 after t seconds.

a. What is its initial velocity?

b. When did it reach the maximum height?

c. What is its maximum height?

d. When did it hit the ground?

e. At what speed did it hit the ground?

Example 5: Analyze and sketch the motion of

s = t3 + 3t2 - 9t+4

Example 6: The equation of motion of a particle is

s(t)= 2t3 - 7t2 + 4t + 1. Make an analysis of

motion.

Example 7: Analyze and sketch the motion of

s = t / (9+t2)

Example 8: A stone is dropped from a building 256 ft high.

a. Write the equation of motion of the stone.

b. Find the instantaneous velocity of the stone at

1 sec and 2 sec.

c. Find how long it takes the stone to reach the ground.

Example 9: A ball is thrown vertically upward from the

top of a building 35 ft high beside which is a covered walk with a vertical clearance of 7 ft. Its

equation of motion is given by s = -16t2 + 48ta. How high from the top of the building will

the ball go?

b. When will it land on the roof of the covered walk?

Example 10: A rocket is fired vertically upward from the

ground with an initial velocity of 560 fps.

a. Write the equation of motion of the rocket.

b. Estimate how high the rocket will go and how long it takes the rocket to reach its highest point.

c. Find the instantaneous velocity of the rocket at 10 sec. and 25 sec.

d. Find the speed of the rocket at 10 sec and 25 sec.

e. Find the speed of the rocket when it reaches the ground.

Example 11: Suppose a sprinter running in a 100-meter

race is s meters from the finish line t seconds after the start of the race where:

Find the sprinters' speed

a. At the start of the race.

b. When the sprinter crosses the finish line.

Example 12: A billiard ball is hit and travels a line. If an s centimeter is the distance of the ball from its

initial position at t seconds, then s = 100t2 + 100t. If the ball hit a cushion that is 39 cm from its initial position, at what velocity does it hit the cushion?