Clicker Question 1
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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
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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
(9/10/10)
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Given a function f(x) on an interval [a, b]:
An antiderivative (or “indefinite integral”)
F(x) of f(x) is a function.
b
f ( x)dx (the “definite integral”) is a number

a
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(the adding-up of f 's value between a and b).
The FTC tells us how to get the first from the
second or the second from the first.
“Techniques of Integration”, or
“Techniques of Antidifferentiation”
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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”
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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” or “guess and check”
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Any ideas about x2(x3 + 4)5 dx ??
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How about x ex^2 dx ?
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Try ln(x) / x dx
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But we’ve been lucky! Try sin(x2) dx
The Substitution Technique
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It’s called a “technique”, not a “rule”, because
it may or may not work.
If there is a chunk, try calling the chunk u.
Compute du = (du/dx) dx
Replace all parts of the original expression
with things involving u (i.e., eliminate x).
If you were lucky/clever, the new expression
can be anti-differentiated easily.
Clicker Question 3
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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
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Read Section 5.5 of the text and go over
today’s class notes.
In Section 5.5, do Exercises 1-31 odd.