2.1.1 Finding Rate of Change over an Interval

Question 1

Finding Rate of Change over an Interval

Answer 1

• The position function relates time and location. Setting the position function equal to a distance indicates the time an object is at that location.
• When solving a difficult problem, it is a good idea to approximate the answer first. Approximating can give insight into how to work the problem.

Question 2

position function

Answer 2

• In order to analyze the rate of change of an object, it is necessary to know certain things about the object’s location at different times.
• The position function is a mathematical relation that connects location to time. By setting time equal to a specific value, the position function will tell you where the object was at that time.
• The position function can also be used to determine when an object is at a specific location. By setting the function equal to that distance you can solve the resulting equation for a value of time.
• Notice that the situation a problem models can affect which answers are plausible. For example, in most problems time cannot be negative.

Question 3

rate of change

Answer 3

• We have already determined that it is not possible to find an instantaneous rate of changestrictly through algebra because of the 0/0 issue.
• However, approximating the answer often leads to an insight. Consider the example shown here. Instead of finding the instantaneous rate of change, try finding the average rate of changeover a small part of the bike ride.
• Recall that Professor Burger’s average rate of change over the entire bike ride was 20 miles per hour. However, his average rate of change over this smaller interval is almost 30 miles per hour. Shrinking the interval makes it possible to refine the approximation even further.
• If the function for Professor Burger’s position is not known, but a table of values is, his rate of change can still be approximated. The best approximation of his instantaneous rate of change is the slope of the secant joining the points to the left and to the right of 1.2 hours.
• Based on the table, Professor Burger’s approximate rate of change when he passed the speed limit sign was 32 miles per hour. He was definitely breaking the law!

Question 4

Let f (x) = x^2 − 3.
What is the change in f
as x changes from 1 to 1.1?

Answer 4

Δf = .21

Question 5

Gary drives his car from San Francisco to Los Angeles, a total distance of 400 miles, leaving at 9:30 am and arriving at 2:15 pm the same day. What is his average speed, in miles / hour?

Answer 5

84

Question 6

If f (x) = 3x + 2, g (x) = x  62 − 2, and h (x) = f (g (x)), what is the average rate of change in h along the x-interval (1, 1.2)?

Answer 6

6.6

Question 7

The position of a rocket in miles is given by the position function p (t) = 2t ^2 + 5t − 2, where t is the time in seconds. What is the rocket’s average speed over the time interval [3, 6]?

Answer 7

23.0 miles / second

Question 8

The velocity of the cyclist in feet per second as a function of time is given in the table below.
t 0 1 2 3 4
f(t) 5 10 12 11 9
The approximate acceleration (rate of change of the velocity with respect to time) of the cyclist at time t = 2 seconds is which of the following?

Answer 8

None of the above

Question 9

A cougar is running in a straight line at a constant speed of 35 miles / hour. Which of the following position functions might describe its motion?

Answer 9

p (t) = 35t + 2, where t is the time in hours and p (t) gives the position in miles.

Question 10

You can approximate the instantaneous rate of change by:

Answer 10

Examining the average rates of change over smaller and smaller intervals.

Question 11

The position (in feet) of an object moving in a straight line is given by the position function p(t)=t^2−1/2t+7, where t is the time measured in seconds. What is the object’s average speed over the time interval [2,4]?

Answer 11

5.5 feet / second

Question 12

Mickie is blowing up a spherical balloon. What is the average rate of change of the volume of the balloon (with respect to the radius of the balloon) as the radius increases from 4 inches to 7 inches?

Answer 12

389.6 in^3 / in.

Question 13

The velocity of the cyclist in feet per second as a function of time is given in the table below.
t 0 1 2 3 4
f(t)10 20 24 22 18
The approximate acceleration (rate of change of the velocity with respect to time) of the cyclist at time t = 2 seconds is which of the following?

Answer 13

1 ft / s^2

Question 14

After jumping out of a plane at t=0, a skydiver’s altitude in the air in meters is given by the position function p(t)=−50t^2+200t+2350, where t is the time in seconds. At what time will the parachuter be 100 meters from the ground?

Answer 14

t = 9

Question 15

A dragster’s position down the track in meters is given by the function p(t)=3^t+8t^2+5t−1, where t is the time in seconds. How far down the track is the dragster at t=2.4 seconds

Answer 15

71 meters

Question 16

Let f (x) = −2x^ 2 + 4. What is the average rate of change of f over the interval (1, 1.1)?

Answer 16

−4.2

Question 17

Let f (x) = x^ 2 + 1. What is the average rate of change of f over the interval (2, 2.1)?

Answer 17

4.1

Question 18

While driving home from work on the freeway, Sandra calculates that her average speed from 5:20 pm to 5:30 pm is 57 miles / hour, while her average speed from 5:22 pm to 5:26 pm is 62 miles / hour. What is her instantaneous speed at 5:25 pm?

Answer 18

There is not enough information