Tangent Lines and Rates of Change - College of Arts and Sciences
Tangent Lines and Rates of Change
Math 121 - Week 3
Motivation: Two Central Questions
How do I find the slope of a line tangent to a curve at a given point? (Calc. 1)
How do I find the exact area of the region bounded between a given curve and the x-axis? (Calc. 2)
Start with average rate of change
Consider the following table of distances and times for a short road trip.
t (in hrs.) 0 D (in mi.) 0
12345 30 80 100 160 200
What was the driver's average speed over the course of the trip?
What was the average speed from t=1 to t=3?
How about t=3 to t=4? How do we make these
calculations?
Average rate of change formula
Based on our work from the previous slide, we can say that the average rate of change for a function f(x) over the interval [a,b] is given by...
This should be very reminiscent of a formula you have lots of experience with.
The slope of the line that passes through points (x1,y1) and (x2,y2) is given by:
Secant line
A secant line is a line that passes through two given points of the graph of a function. How is this related to the average rate of change of a function?
The average rate of change of f(x) over [a,b] is also the slope of the secant line through (a,f(a)) and (b,f(b)).
Ex: Compute the average rate of change for the function f(x) = 1/x over the interval [1,3].
Secant line for 1/x
We found that the average rate of change for f(x) = 1/x on the interval [1,3] is -.
Now find an equation for the secant line that corresponds to this average rate of change.
Tangent line
What happens as the distance between the two points connected by the secant line gets smaller?
Instantaneous rate of change
Given a function f, the instantaneous rate of change for f at x=a is given by...
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