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Calculus I

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indefinite integral of f, and we introduce the. notation ... NOTE to find the indefinite integral is the. same as to find the most general antiderivative ... – PowerPoint PPT presentation

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Title: Calculus I


1
Calculus I
  • AntiDerivatives

2
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.

3
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.

4
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

5
AntiDerivative
  • Applications?
  • Lets suppose that f(x) sec2 x x
  • and suppose f(0) -3.

6
AntiDerivative
  • Lets suppose that f(x) cos x 2ex 3x 5
  • and suppose f(0) -3 and f(0) 2.

7
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

8
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

9
  • Reference
  • Stewart, Single Variable Calculus
  • Early Transcendentals
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