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AP Calculus AB: 3.1.1 Rates of Change, Secants, and Tangents

Mathematics12 CardsCreated 3 months ago

This flashcard set covers the relationship between average and instantaneous rates of change, using secant and tangent lines respectively. It explains how limits are used to find the slope of the tangent line and provides examples of calculating slopes of secant lines and interpreting instantaneous velocity from graphs and functions.

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Rates of Change, Secants, and Tangent

  • 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

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Key Terms

Term
Definition

Rates of Change, Secants, and Tangent

  • Approximate the instantaneous rate of change by finding the average rate of change on a small interval around the point in question.

rate of change, secant, tangent

  • Another way of studying Professor Burger’s bike ride is by graphing the position function. The result is a parabola.

  • The lin...

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

5 (x2 + x1 )

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?

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

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?

6.2

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?

23.6

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AP Calculus AB: 3.1.1 Rates of Change, Secants, and Tangents
TermDefinition

Rates of Change, Secants, and Tangent

  • 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

rate of change, secant, tangent

  • 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

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

5 (x2 + x1 )

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?

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

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?

6.2

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?

23.6

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

y = 12x − 13.97

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

y = 4.4x − 5.4

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

7 (x2^2 + x2x1 + x1^2 )

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.

0.39 miles / minute

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?

1/5 feet/second

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?

[1.9, 2.1]