Math 142 Lecture Notes for Section 3.2
Section 3.2 -
1
Rates of Change
Example 3.2.1:
Suppose that you are on a road trip to Las Vegas. The total distance is 1,325 miles and it takes you
exactly 25 hours to get there. Answer the following questions:
(a) What was your average velocity on the trip?
(b) After 4 and a half hours of driving, you pass a cop with a radar gun. Assuming the speed limit is
55 mph, were you speeding?
(c) Suppose that the following table list the positions of your car for the first five hours,
0
0
1
65
2
135
3
200
4
255
5
325
Based on the table, what was the average velocity of the 5th hour of your trip?
(d) Use the table below to estimate your instantaneous velocity when you passed the cop. The table
below represents the car’s position near the 4 hour mark.
3.9
249
3.99
254.4
3.999
254.94
4
255
4.001
254.93
4.01
254.3
4.1
248
Math 142 Lecture Notes for Section 3.2
2
Definition 3.2.2:
The average velocity of an object on the interval time t = a to t = a + h is given by
vavg =
f (a + h) − f (a)
.
h
Definition 3.2.3:
The instantaneous velocity of an object at time t = a is given by
vi = lim
h−→0
f (a + h) − f (a)
.
h
Example 3.2.4:
If a rock is thrown up in the air on the moon, with an initial velcoity of 11 m/s, its height in meters t
seconds later is given by h(t) = 11t − 3.29t2 .
(a) Find the average velocity over the given intervals:
(i) [1, 1.1]
(ii) [1, 1.01]
(iii) [1, 1.001]
(b) Use the previous information to estimate the instantaneous velocity when time t = 1.
(c) Find the exact value of the instantaneous velocity when time t = 1.
Math 142 Lecture Notes for Section 3.2
3
Definition 3.2.5:
For y = f (x), the average rate of change from x = a to x = a + h is
f (a + h) − f (a)
∆y
=
.
∆x
h
Definition 3.2.6:
For y = f (x), the instantaneous rate of change at x = a is
dy
f (a + h) − f (a)
= lim
.
dx h−→0
h
Example 3.2.7:
First estimate and then find exactly the instantaneous rate of change of f (x) = x2 − 4 at 2.
Math 142 Lecture Notes for Section 3.2
4
Definition 3.2.8:
For a function y = f (x), The line tangent to f (x) at a is the line with slope equal to the instantaneous
rate of change of f (x) at a that intersects the point (a, f (a)).
Example 3.2.9:
√
Find the equation of the line tangent to f (x) = x + 6 at a = 10.