CP Calculus
Section 0.4
Part I:
1. Example 1: Christine is driving down to the shore on the Parkway. At 9:48 she passes through
the Raritan toll, which is at mile marker 126. At 10:23, she passes through the Toms River toll,
which is at mile marker 85. If the speed limit is 65 mph, was she speeding between the two toll
barriers? How do you know?
2. Definition: the Average Rate of Change for a function f(x) from x=a to x=b is
𝑓(𝑏) − 𝑓(𝑎)
𝑏−𝑎
y=f(x)
10
f(b)8
6
f(b)-f(a)
4
2
f(a)
b-a
0.5
a
1
b2
1.5
2.5
3. Example 2: Find the average rate of change on the function 𝑓(𝑥) = 𝑥 2 − 4𝑥 + 1 on the interval
[2,5].
4. Example 3: A toy rocket is fired into the air. Its height can be modeled with the equation below:
ℎ(𝑡) = −16𝑡 2 + 300𝑡
What is the average speed of the rocket over the first 2 seconds? Between 1 and 3 seconds?
5. Example 4: Find the average rate of change for each of the following functions over the given
interval.
a.
b.
CP Calculus
Section 0.4
Part II: Domain
There are three main function types that have a restricted domain. For an expression A that contains a
variable:
1
𝐴
⇒𝐴≠0
√𝐴 ⇒ 𝐴 ≥ 0
log(𝐴) ⟹ 𝐴 > 0
6. Example 5: Find the domain of the following functions.
x2  x  2
x2  x  6
a.
f  x 
c.
f ( x)  2 x  3  2
e.
f  x 
3
x2
b. f  x  
x 2  x  12
x2  9
d. f ( x)  2 x 2  x  2  2
f. f  x  
x2  4
x3