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)

This means that the integral is the antiderivative (integrals and derivatives are inverses)

( ) ( ) ( )b

af x dx F b F a

Example 1 - Evaluate

Example 2 – Evaluate and confirm on calculator

Example 3 – Find the exact value

1/4 2 1/2

1/2

510 2(1 )e

x xx

Example 4

Be careful when asked to find area! Find the area under the parabola y = x2-1

bounded by the x-axis from x=0 to x=3. Verify on the calculator.

Example 5

Find the area bounded by the curves y=0, y=2/x, x=1 and x=3. Verify on the calculator.

Example 6

Find the area bounded by x-axis and the function y=20/(x2+1) on the interval [-1,1]. Explore the function’s symmetry. Verify on the calculator.

Fundamental Theorem of Calculus Part 2

The second part of the theorem deals with integral equations of the form

( ) ( )x

a

F x f t dt

Example 7

The graph of f(t) is given. Let 2

( ) ( )x

F x f t dt

Example 8

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?

2

3

( ) ( 1) ,x

F x t t dt

FTC Part 2

( ) ( )x

a

df t dt f x

dx

Example 9

Chain Rule anyone??

The FTC Part 2 in its most general form:

Example 10

Evaluate using FTC Part 2, then verify by doing the Looooooooooooooong way.

Example 11

If , find F’(x) 3

2 sin 2

( ) ln

x

x

F x tdt

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 ( ) ( ) ( )

b

a

f x dx f c b a

When f(c) is called the average value , the above equation can be explicitly solved for f(c)

( )

( )

b

a

f x dx

f cb a

What it means….

This means that functions are different than a list of numbers.If the average score on an exam is 84, then 84 lies somewherebetween the highest and lowest values, but no student had to receive the score of 84. But a function will always take on itsmean value somewhere in the interval.

( )

( )

b

a

f x dx

f cb a

Example 12

In Kennesaw, the temperature (in oF) t hours after 9 am was modeled by the function

. 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.

( ) 50 14sin12

tT t

Example 13

Find the value of c guaranteed by the MVT for integrals for on [-1,2]. Interpret the results graphically.

2( ) 1f x x

Example 14

Show that the average rate of change in a car’s position over a time interval [t1, t2] is the same as the average value of its velocity function over the same interval.

Example 15

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 16

Find the number(s) b such that the average value of on the interval [0,b] is equal to 3.

2( ) 2 6 3f x x x

Example 17 – Solve for the indicated variable

Example 18

If 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.

2'( ) 2 2f x x

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

a

f b f a f x dx

Example 19

If and f(2)=-2, show the setup to find f(3), f(5), f(-2) and an integral equation for f(x)

2'( ) 4sin (2 )f x x

Example 20

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 lbs per day, what was the man’s weight at the end of the break, 14 days later?

( ) 10sin8

tW t