MATH 2B Lecture Notes - Lecture 1: Antiderivative, Mean Value TheoremPremium
Course CodeMATH 2B
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MATH 2B - Lecture 1 - Anti-derivatives
Anti-differentiation is exactly what it sounds like: the opposite of differentiation. That is,
given a function f, can we nd a function F whose derivative is f.
Denition. An anti-derivative of a function f is a function F such that F’(x) = f(x) for all x.
1. is an anti-derivative of (x) sin xF = (x)os xf =c
2. is an anti-derivative of (x)x)F= ( 2+ 3 3/2 (x) 3xf = √x2+ 3
3. is an anti-derivative of (x)F=x−3
cos x(x)f=cos x
We can easily check the veracity of these statements by differentiating F(x). But what if
you are asked to nd an anti-derivative, not just check something you’ve been given? In
general this is a hard problem.
● Differentiation is often described as easy in the sense that nice functions can
usually be differentiated via familiar rules
● Anti-differentiation is hard: very few functions have anti-derivatives that can
easily be computed
: It is easy to find the derivative of , but can you find an(x)f=e−x2
anti-derivative of f(x)?
The only method that really exists for explicitly computing anti-derivatives is guess and
differentiate! Every famous rule that you’ll study in Integration (substitution, integration
by parts, etc.) is merely the result of guessing a general anti-derivative and checking
that your guess is correct
Anti-derivatives of Common Functions
● Guessing is, of course, easier if you have familiarity with differentiation. With
each of the following functions f(x) you should be able to guess the chosen
anti-derivative F(x) just from what you know about derivatives
1. has an anti-derivative of (x)f= 3 + x
2(x)x ln|x|f= 3 + 2
2. has an anti-derivative (x)sec x ef = 3 2− 2 3x(x)tan x ef = 3 − 3
3. Find an anti-derivative of (x)os xg =c+x2
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