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Calculus I

- AntiDerivatives

AntiDerivatives

- Def Let f be a continuous function,
- we say that F is an antiderivative of f
- on some interval I, provided that Ff on I.
- Note that f has infinitely many antiderivatives.
- To express this we will say that given a

continuous - function f on I, we say that the most general

antiderivatives of - function f, is F c, where c is an arbitrary

constant and - F f.

AntiDerivative

- How do we do this? We read the
- derivative tables backwords ?!
- Example Given f(x) cos x, we can say
- the most general antiderivative of f is
- F(x) sin x C.

AntiDerivative

- Example Find the most general
- antiderivative of the following
- functions
- 1) f(x) 1
- 2) g(x) x
- 3) h(x) sec x tan x
- 4) k(x) ex

AntiDerivative

- Applications?
- Lets suppose that f(x) sec2 x x
- and suppose f(0) -3.

AntiDerivative

- Lets suppose that f(x) cos x 2ex 3x 5
- and suppose f(0) -3 and f(0) 2.

AntiDerivative

- More formally, the most general antiderivative
- of a continuous function f is defined to be the
- indefinite integral of f, and we introduce the
- notation
- Example

AntiDerivative

- Rules of Integration
- NOTE to find the indefinite integral is the
- same as to find the most general antiderivative
- of a function. Thus, we read the derivative
- Tables from right to left. In addition, we have
- Addition/Subtraction Law
- Power Law
- Constant Law

- Reference
- Stewart, Single Variable Calculus
- Early Transcendentals

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