Flashcards in 3.1.1 Rates of Change, Secants, and Tangents Deck (12):
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?
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?
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?