3.1.1 Rates of Change, Secants, and Tangents

Question 1

Rates of Change, Secants, and Tangent

Answer 1

• Approximate the instantaneous rate of change by finding the average rate of change on a small interval around the point in question.
• Represent the average rate of change graphically by a secant line. The average rate of change is equal to the slope of the secant line between the two points being considered.
• Represent the instantaneous rate of change graphically by a tangent line.
• To find the slope of a tangent line, take the limit of the function as the change in the independent variable approaches zero

Question 2

rate of change, secant, tangent

Answer 2

• Another way of studying Professor Burger’s bike ride is by graphing the position function. The result is a parabola.
• The line connecting the starting and ending points is called a secant line. Notice that the slope of the secant line is the same as the average rate of change of position over the entire trip.
• One way to turn the average rate of change into a better approximation of the instantaneous rate of change is to reduce the length of the interval.
• As you can see, this interval is so short that the secant line is almost tangent to the graph of the position function.
• Calculating the slope of this secant line produces an average rate of change of 32.73 mph.
• A smaller interval will produce an even better approximation.
• As the length of the interval becomes 0, the instantaneous rate becomes the limit of the change in the position function

Question 3

What is the average rate of change of the function y = 5x^ 2 + 1 between x = x1 and x = x2?

Answer 3

5 (x2 + x1 )

Question 4

The following graph describes the position of an object as a function of time. What can you say about the instantaneous velocity at the object at t = 3?

Answer 4

The instantaneous velocity of the object at t = 3 is zero.

Question 5

Mary is competing in the 100-meter dash.Suppose that her position is described by the position function p(t), where t is the time in seconds and p(t) is her position in meters. If she finishes the race (100 meters) in 15 seconds, and her instantaneous velocity at t=1.5 seconds is 6.2 meters/second, what is the slope of the line tangent to the graph of p(t) at t=1.5?

Answer 5

6.2

Question 6

Consider the curve f(x)=4x2, 0≤x≤3.What is the greatest possible slope of a secant line across an interval of width 0.1?

Answer 6

23.6

Question 7

Find the equation of the secant line to the curve f(x)=3x^2−2 on the interval [1.9,2.1].

Answer 7

y = 12x − 13.97

Question 8

Find the equation of the secant line to the curve f(x)=2x^2−3 on the interval [1,1.2].

Answer 8

y = 4.4x − 5.4

Question 9

What is the average rate of change of the function y = 7x^ 3 + 4 between x = x1 and x = x2?

Answer 9

7 (x2^2 + x2x1 + x1^2 )

Question 10

A biker rides along a horizontal straight line.Her location along the line is given by the function s(t)=1/10t^2, where t is measured in minutes and s is in miles. Estimate the instantaneous velocity at time t=2 minutes by computing the average rate of change from t=1.9 to t=2.0 minutes.

Answer 10

0.39 miles / minute

Question 11

An ant is crawling across a driveway. Its position is described by the position function p(t)=1/5t+2, where t is in seconds and p(t)is in feet. What is the ant’s instantaneous velocity at t=1?

Answer 11

1/5 feet/second

Question 12

Consider the curve f(x)=4x^2, 0≤x≤3.For which of the given intervals is the slope of the secant line equal to 16?

Answer 12

[1.9, 2.1]