Sections 5.5 & 5.FT
The Fundamental Theorems of Calculus
Applied Calculus, 3/E by Deborah Hughes-Hallet
Copyright 2006 by John Wiley & Sons. All rights reserved.
Review Example
The following is a graph of
the derivative of a function f
defined on the closed
interval [0, 4].
What are the critical numbers (x values)?
Where are all the local maximums and minimums?
Where are all the global maximums and minimums?
Where are the inflection points (x values)?
0.6, 3.3
Max: 0.6, Min: 0, 4
Max: 0.6, Min: 4
1.7, 3.3
Analyzing the graph of the derivative
Analyzing the
Graph of the Derivative
flattening
increasing
decreasing
still decreasing
flattening
maximum rate of decrease
still decreasing
The Fundamental Theorem of Calculus
•
The Fundamental Theorem of Calculus provides the bridge
between two seemingly unrelated branches of calculus:
–
•
–
Differential calculus, which arose from the tangent
problem, and
Integral calculus, which arose from the area problem.
The Fundamental Theorem gives the precise relationship
between these.
The Fundamental Theorem of Calculus
•
•
Given the rate of change, the definite integral
gives total change in a quantity.
Suppose F(t) is the quantity. Then F'(t) is the
rate of change of that quantity.
• So the total change in F(t) from t = a to
t = b, that is F(b) – F(a), is the definite
integral of F′(t) from t=a to t=b.
The Fundamental Theorem of Calculus:
Derivation
• Suppose a quantity F(t) is given. Then F′(t) is the rate of change of that
quantity with respect to t.
• Compute the total change from t=a to t=b.
• Change in F = Rate × Time.
• Compute the total change by computing a Riemann Sum.
• Break up interval into segments of length ∆ t = (b–a)/n.
• Consider the first subinterval. Approximate the rate of change in that
•
•
•
•
•
•
subinterval by F ′(t1). Then the change in that subinterval is approximately F
′t1) × ∆ t.
Continue - adding up all the changes in each subinterval.
Total Change in F between a and b is
F ′(t1) × ∆ t + F ′(t2) × ∆ t + F ′(t3) × ∆ t + ... + F ′(tn) × ∆ t.
As n →∞, this sum becomes a definite integral.
But the total change in F between a and b is also F(b) - F(a).
So ...
b

a
F  t dt  F b   F a 
The First Fundamental Theorem of Calculus
also called
The Evaluation Theorem
Graphically, the signed area between the graph of 𝐹′(𝑡) and the horizontal axis
from 𝑎 and 𝑏 is equal to the change in 𝐹 between 𝑎 and 𝑏.
Example
Suppose the following is
a graph of the derivative
of a function f.
If 𝑓 0 = 3, what is 𝑓(1)? 𝑓(2)? 𝑓(3)? 𝑓(𝑥)?
𝑓 1 =3+1=4
𝑓 𝑥 = 3 + 𝑥, if 0 ≤ 𝑥 ≤ 1
𝑓 2 = 4 + 0.5 = 4.5
𝑓 𝑥 = 4 + 0.5(2 − 𝑥)2 , if 1 ≤ 𝑥 ≤ 2
𝑓 3 = 4.5 + 0.5 = 5
𝑓 𝑥 = 4.5 + 0.5(𝑥 − 2)2 , if 2 ≤ 𝑥 ≤ 3
Example
1.5 0.5 -0.5 -1 -0.5 0.5
Area Function
x
g x    f t dt
a
Example
𝑥
Suppose g x = 0 𝑓 𝑡 𝑑𝑡 where
𝑓(𝑡) is given by the graph.
Find 𝑔(0), 𝑔(1), 𝑔(2), 𝑔(3),
𝑔(4), and 𝑔(5). Graph 𝑔.
g 0 =0
g 1 =1
g 2 =1+2=3
g 3 = 3 + 1.3 = 4.3
g 4 = 4.3 − 1.3 = 3
g 5 = 3 − 1.3 = 1.7
Second Fundamental Theorem of Calculus
So …
In other words …
Examples
f x  
G y  
y

3x
0

x

y
0
1
f  x  
t 2 dt
t
2
t 1
2
1  t dt
3
dt
G  y  
dy

dx
Complete Fundamental Theorem of Calculus
1.
Suppose f is a continuous function on an interval [a,b]. Then
If g x  
2.
x
 f t dt then g x   f x  for a  x  b.
a
Suppose f  is a continuous function on an interval [a,b].
b
Then  f  t dt  f b   f a .
a