Clicker Question 1

What is the derivative of f(x) = 7x4 + ex sin(x)?– A. 28x3 + ex cos(x)– B. 28x3 – ex cos(x)– C. 28x3 + ex (cos(x) + sin(x))– D. 28x3 + ex (cos(x) – sin(x))– E. (7/5)x5 – ex cos(x)

Clicker Question 2

What is the derivative of g(x) = tan(x2 + 5) ?– A. tan(2x)– B. sec2(2x)– C. sec2(x2 + 5)– D. 2x tan(x2 + 5)– E. 2x sec2(x2 + 5)

The Fundamental Theorem of Calculus (1/24/14)

Given a function f(x) on an interval [a,b], the Fundamental Theorem of Calculus tells us how the definite integral (a number) and the antiderivative (a function) are related.

Part 1 says you can get an antiderivative of f by turning the definite integral into a function of the right-hand endpoint. (This part is less used – more theoretical.)

Part 2 says that we can compute the definite integral provided we can find an antiderivative. Then we just evaluate that at the endpoints and subtract. (Used!)

FTC - A Quick Outline

Given a function f (x ) on [a, b ]:

F (x ) an antiderivative of f

Part 1: (by freeing up right endpoint) Part 2: (by evaluating F (b) – F (a ) )

b

adxxf )(

Examples

Example of Part 1:Q: What is an antiderivative of cos(x 2)?

A: Example of Part 2 (you’ve seen lots!):

Q: What is ?

A: arctan(6) – arctan(2)

x

adtt )cos( 2

dxx

6

2 21

1

Clicker Question 3

According to the Fundamental Theorem of Calculus, what is the integral (i.e., the sum of the values) of the function f(x) = 1/x2 on the interval from 1 to 4?– A. 1/4– B. 3/4– C. -3/4– D. 15/16– E.

“Techniques of Integration”, or“Techniques of Antidifferentiation”

Finding derivatives involves facts and rules; it is a completely mechanical process.

Finding antiderivatives is not completely mechanical. It involves some facts, a couple of rules, and then various techniques which may or may not work out.

There are many functions (e.g., f(x) = ex^2) which have no known antiderivative formula.

There are a Couple of “Rules”

Sum and Difference Rule: Antiderivatives can be found working term by term (just like derivatives).

Constant Multiplier Rule: Constant multipliers just get carried along as you get antiderivatives (just like derivatives).

HOWEVER, there is no Product Rule, Quotient Rule, or Chain Rule for Antiderivatives!

Reversing the Chain Rule:“Substitution”

Any ideas about x2(x3 + 4)5 dx ??

How about x ex^2 dx ?

Try ln(x) / x dx

But we’ve been lucky! Try sin(x2) dx

The Substitution Technique

It’s called a “technique”, not a “rule”, because it may or may not work.

If there is an (inner) “chunk”, call the chunk u and compute its derivative du/dx. Does this derivative appear as a multiplier in the original (up to a constant coefficient)? If so,

Replace all parts of the original expression with things involving u (i.e., eliminate x entirely). Now consider this new function of u.

Clicker Question 4

What is an antiderivative of f(x) = x2 cos(x3) ?– A. (1/3) sin(x3)– B. (1/3)x3 sin(x3)– C. sin(x3)– D. (-1/3) sin(x3)– E. (1/3)x3 sin((1/4)x4)

Assignment for Monday

Read Section 5.5 of the text and go over today’s class notes.

In Section 5.5, do Exercises 7, 9, 11 13, 15, 17, 21, 23, 25, 29, 31, 33.