Calculus I

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Calculus I

• AntiDerivatives

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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.

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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.

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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

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AntiDerivative

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

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AntiDerivative

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

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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

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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

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• Reference
• Stewart, Single Variable Calculus
• Early Transcendentals