4-4: Fun Theorems, 2
Objectives:
Assignment:
1. To find the average value • P. 291-292: 47-50, 54-60,
of a function
63
2. To understand and use
the Second Fundamental • P. 293: 75-91 odd
Theorem of Calculus
Warm Up
Find the value of 𝑐
guaranteed by the
MVT for integrals for
𝑓(𝑥) = 1 + 𝑥 2 on the
interval −1,2 .
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.
Objective 1
You will be able to find
the average value of a
function
Average Value
If 𝑓 is integrable
on 𝑎, 𝑏 , then the
average value of
𝑓 on 𝑎, 𝑏 is
1
𝑏−𝑎
𝑏
𝑓 𝑥 𝑑𝑥
𝑎
Average Value
If 𝑓 is integrable
on 𝑎, 𝑏 , then the
average value of
𝑓 on 𝑎, 𝑏 is
1
𝑏−𝑎
𝑏
𝑓 𝑥 𝑑𝑥
𝑎
Average Value
Partition 𝑎, 𝑏 such that ∆𝑥 =
value in the 𝑖th subinterval.
𝑏−𝑎
,
𝑛
where 𝑥𝑖∗ is any
𝑓 𝑥1∗ + 𝑓 𝑥2∗ + 𝑓 𝑥3∗ + ⋯ + 𝑓 𝑥𝑛∗
Average =
𝑛
Almost a
Riemann sum
1
=
𝑛
𝑛
𝑓
𝑏−𝑎
𝑏−𝑎
𝑥𝑖∗
𝑖=1
1
=
𝑏−𝑎
𝑛
𝑓
𝑖=1
𝑥𝑖∗
𝑏−𝑎
𝑛
Take the limit as 𝑛 → ∞
=
1
𝑏−𝑎
1
=
𝑏−𝑎
𝑛
𝑓 𝑥𝑖∗ ∆𝑥
𝑖=1
𝑏
𝑓 𝑥 𝑑𝑥
𝑎
Exercise 1
Find the average value
of 𝑓 𝑥 = 3𝑥 2 − 2𝑥 on
1,4 .
Exercise 2
If the function 𝑓 𝑥 = 𝑥 3 has an average
value of 9 on the closed interval 0, 𝑘 , then
what is the value of 𝑘?
Exercise 3
Show that the average velocity of a car over
a time interval 𝑡1 , 𝑡2 is equal to the average
of value of its velocity function.
The average
value of a function
𝑓 is equivalent to
the slope of the
secant line of the
antiderivative of 𝑓.
Objective 2
You will be able to understand
and use the Second Fundamental
Theorem of Calculus
Definite Versus Indefinite
Remember that a definite integral yields a
number; however…
Definite Integral
Indefinite Integral
Number
Family of functions
Net area above/below 𝑓
Antiderivative of 𝑓′
Definite Versus Indefinite
Remember that a definite integral yields a
number; however…
Definite Integral
Number
Net area above/below 𝑓
There is a way
to define a
function based
on a definite
integral
Exercise 4
Evaluate 𝐹 𝑥 =
your answer.
𝑥
0
𝑡 + 1 𝑑𝑡 and interpret
𝑓 𝑡 =𝑡+1
𝐹(𝑥) is an accumulation function:
𝐹 2 =4
Area over 0,2 = 4
𝐹 4 = 12
Area over 0,4 = 12
𝐹(𝑥)
Number Versus Function
Remember that a definite integral yields a
number; however…
Definite Integral as a Number:
𝑏
𝑓 𝑥 𝑑𝑥
Variable
Definite Integral as a Function:
𝑥
𝐹 𝑥 =
𝑓 𝑡 𝑑𝑡
𝑎
𝑎
Constant
Constants
Function of 𝑥
Function of 𝑡
Exercise 5
Evaluate 𝐹 𝑥 =
and 𝜋/2.
𝑥
cos 𝑡 𝑑𝑡
0
at 𝑥 = 0, 𝜋/6, 𝜋/4, 𝜋/3,
Now, what do
you get if you
take the
derivative of
𝐹(𝑥)?
2nd Fundamental Theorem of Calculus
If 𝑓 is continuous
on and open
interval containing
𝑎, then for every 𝑥
in the interval
Insert Proof Here
𝑑
𝑑𝑥
𝑥
𝑓 𝑡 𝑑𝑡 = 𝑓 𝑥
𝑎
Exercise 6
Evaluate
𝑑
𝑑𝑥
𝑥
0
𝑡 2 + 1 𝑑𝑡
Exercise 7
Find the derivative of 𝐹 𝑥 =
𝑥3
cos
𝑡
𝑑𝑡.
𝜋/2
4-4: Fun Theorems, 2
Objectives:
Assignment:
1. To find the average value • P. 291-292: 47-50, 54-60,
of a function
63
2. To understand and use
the Second Fundamental • P. 293: 75-91 odd
Theorem of Calculus