4-4: Fun Theorems, 1
Objectives:
Assignment:
1. To evaluate definite
integrals with
• P. 291-294: 5-37 odd, 97,
Fundamental Theorem of
101-103
Calculus
2. To understand and use
the Mean Value Theorem • P. 291: 43, 45
for Integrals
Warm Up
4
,
𝑥
Given 𝑓(𝑥) = 5 − find all values of 𝑐 in the
open interval (1, 4) such that
′
𝑓 𝑐 =
𝑓 4 −𝑓(1)
.
4−1
Mean Value Theorem (Derivatives)
If 𝑓 is continuous on the closed interval 𝑎, 𝑏
and differentiable on the open interval 𝑎, 𝑏 ,
then there exists a number 𝑐 in 𝑎, 𝑏 such
′
that 𝑓 𝑐 =
𝑓 𝑏 −𝑓 𝑎
𝑏−𝑎
.
At some point, the
instantaneous rate of
change is equal to the
average rate of change.
Objective 1
You will be able to evaluate
definite integrals with
Fundamental Theorem of Calculus
Exercise 1
Given 𝑓 𝑥 =
1 2
𝑥
2
+ 2𝑥, find 𝑓′(𝑥).
Exercise 2
Given 𝑓 ′ 𝑥 = 𝑥 + 2, find 𝑓(𝑥) if 𝑓 0 = 0.
Exercise 3
Evaluate
4
𝑥
2
+ 2 𝑑𝑥.
Exercise 4
Relate the value of
𝑓 𝑥 =
1 2
𝑥
2
4
𝑥
2
+ 2 𝑑𝑥 to the graph of
+ 2𝑥.
Or at least a
part of it
This is the
Fundamental
Theorem of
Calculus
Calculus: Two Kinds
Tangent Line Problem
Area Problem
5
4
3
2
1
f (x )
2
A
–1.00
x
2
1
4
B
5.00
2
3
Differential Calculus
Integral Calculus
At first thought to be unrelated
6
Calculus: Two Kinds
Tangent Line Problem
Area Problem
Inverse Operations
Fundamental Theorem of Calculus
If 𝑓 is a continuous function on 𝑎, 𝑏 and 𝐹 is
an antiderivative of 𝑓, then
𝑏
𝑓(𝑥) 𝑑𝑥 = 𝐹 𝑏 − 𝐹 𝑎
𝑎
= 𝐹(𝑥)
One
Find Antiderivative
𝑏
𝑎
If 𝑓 is nonnegative, then the area
under 𝑓 is equal to the difference
of the 𝑦-values in the
antiderivative of 𝑓.
Plug limits into
Two
Antiderivative
Subtract:Three
Upper - Lower
You don’t have to
worry about the
constant of
integration.
Insert Proof Here
Exercise 5
Evaluate each definite integral.
1.
2 2
𝑥
1
− 3 𝑑𝑥
2.
4
3
1
𝑥 𝑑𝑥
3.
𝜋/4
2
sec
𝑥 𝑑𝑥
0
Exercise 6
Evaluate
2
0
2𝑥 − 1 𝑑𝑥.
Exercise 7
Find the area bounded by the graph of
𝑦 = 2𝑥 2 − 3𝑥 + 2, the 𝑥-axis, 𝑥 = 0, and 𝑥 = 2.
Objective 2
You will be able to understand
and use the Mean Value
Theorem for Integrals
Goldilocks
Recall that when considering the area under a
curve, the inscribed rectangle was too small, while
the circumscribed rectangle was too big.
The Mean
Value
Rectangle,
however, is just
right.
Mean Value Theorem (Integrals)
If 𝑓 is a continuous
function on 𝑎, 𝑏 , then
there exists a number
𝑐 in 𝑎, 𝑏 such that
𝑏
𝑓(𝑥) 𝑑𝑥 = 𝑓 𝑐 𝑏 − 𝑎
𝑎
Mean Value Rectangle
There exists a
rectangle whose
base is the width
of the interval,
whose height is a
function value and
whose area is
equal to the area
under the curve.
Exercise 8
Find the value of 𝑐 guaranteed by the MVT
3 9
for integrals for 1 3 𝑑𝑥.
𝑥
4-4: Fun Theorems, 1
Objectives:
Assignment:
1. To evaluate definite
integrals with
• P. 291-294: 5-37 odd, 97,
Fundamental Theorem of
101-103
Calculus
2. To understand and use
the Mean Value Theorem • P. 291: 43, 45
for Integrals