Download Notes - Ryan, Susan

3. Fundamental
Theorem of Calculus
Fundamental Theorem of
Calculus

We’ve learned two different branches of calculus so
far: differentiation and integration. Finding slopes
of tangent lines and finding areas under curves
seem unrelated, but in fact, they are very closely
related. It was Isaac Newton’s teacher at
Cambridge University, a man named Isaac Barrow
who discovered that these two processes are
actually inverse operations of each other in much
the same way division and multiplication are. It
was Newton and Leibniz who exploited this idea
and developed the calculus into its current form.
FTC Part 1

Definite integral from a to b can be found
from function’s antiderivative F as F(b) – F(a)

b
a

f ( x)dx  F (b)  F (a)
This means that the integral is the
antiderivative (integrals and derivatives are
inverses)
Example 1 - Evaluate
Example 2 – Evaluate and
confirm on calculator
Example 3 – Find the exact
value
1/ e
 10 x
1/2
4
2 1/2
 2(1  x )
5

x
Fundamental Theorem of
Calculus Part 2

The second part of the theorem deals with
integral equations of the form
x
F ( x)   f (t )dt
a
Example 4

The graph of f(t) is given. Let
x
F ( x)   f (t )dt
2
Example 5
x

F ( x)   (t 2  t  1)dt ,
If
find a simplified, expanded
version of F(x) by evaluating the definite
integral. Once you find F(x), find its
derivative F’(x). What do you notice?
3
FTC Part 2

x
d
f (t )dt  f ( x)

dx a
Example 6
Chain Rule anyone??

The FTC Part 2 in its most general form:
Example 7

Evaluate
Example 8

If
3x
F ( x) 

2  sin 2 x
ln tdt,
find F’(x)
Break!
The mean value theorem (for integrals)
(average value of a function)

If f is continuous on the closed interval [a,b],
then there exists a number x=c in the closed
interval [a,b] such that  f ( x)dx  f (c)  (b  a)
b
a

When f(c) is called the
average value , the
above equation can be
explicitly solved for f(c)
b
f (c ) 
 f ( x)dx
a
ba
What it means….
b
f (c ) 
 f ( x)dx
a
ba
This means that functions are different than a list of numbers.
If the average score on an exam is 84, then 84 lies somewhere
between the highest and lowest values, but no student had to
receive the score of 84. But a function will always take on its
mean value somewhere in the interval.
Example 9
In Kennesaw, the temperature (in oF) t hours
after 9 am was modeled by the function
t
T (t )  50  14sin
12 . Find the average temperature
during the 12 hour period from 9 am to 9 pm.
Show the integral setup then use your
calculator to solve.

Example 9

Find the value of c guaranteed by the MVT
for integrals for f ( x)  1  x 2 on [-1,2]. Interpret
the results graphically.
Example 10

The table below gives values for a continuous
function. Using a trapezoidal approximation
using 6 subintervals, estimate the average
value of f on [20,50].
Example 11

Find the number(s) b such that the average
value of f ( x)  2  6 x  3x on the interval [0,b] is
equal to 3.
2
Example 12 – Solve for the
indicated variable
Example 13

If f '( x)  2 x 2  2 and f(0)=5, find f(2) by a)
finding the particular solution to the
differential equation, then evaluating at x=2
and then by b) using a definite integral.


In the previous example, the first method
relied heavily upon our ability to find the
antiderivative of the integrand. This is not
always easy, possible, or prudent!
Being able to express a particular value of a
particular solution to a derivative as a definite
integral is of paramount importance,
especially when we don’t know how to find a
general antiderivative!
Important Fact

Where you are at any given time is a function
of 1) where you started and 2) where you’ve
gone from your starting point.

What you have at any given moment is a
function of 1) what you started with plus 2)
what you’ve accumulated since then.

When you accumulate at a variable rate, you
can use the definite integral to find your net
accumulation
Important fact mathematically
b
f (b)  f (a )   f '( x)dx
a
Example 14

f '( x)  4sin 2 (2 x)
If
and f(2)=-2, show the setup
to find f(3), f(5), f(-2) and an integral equation
for f(x)
Example 15

A man, at the start of Christmas break (t=0)
weighed 180 pounds. If the man gained
weight during the break at a rate modeled by
the function W (t )  10sin  8t  lbs per day, what was
 
the man’s weight at the end of the break, 14
days later?