Antidifferentiation TS: Making decisions after reflection and review.

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Presentation transcript:

Antidifferentiation TS: Making decisions after reflection and review

Objectives  To define antidifferentiation.  To investigate antiderivatives, indefinite integrals, and all of their parts.  To use basic integration rules to find antiderivatives.

CALCULUS

$200 Its derivative is What is ?

$400 Its derivative is What is ?

$600 Its derivative is What is ?

$800 Its derivative is What is ?

$1000 Its derivative is What is ?

Antidifferentiation  Up to this point in calculus, you have been concerned primarily with this problem: given a function, find its derivative  Many important applications of calculus involve the inverse problem: given the derivative of a function, find the function

Antidifferentiation  This operation of determining the original function from its derivative is the inverse operation of differentiation and is called antidifferentiation.  Antidifferentiation is a process or operation that reverses differentiation.

Antidifferentiation What is the antiderivative of ? Notice: G is an antiderivative of F.

Antiderivatives & Indefinite Integrals Differential (variable of integration) Integral sign Integrand Antiderivative  The antidifferentiation process is also called integration. Indefinite Integral The derivative of F is f.

Antiderivatives & Indefinite Integrals Q: What function has the derivative ? A: ln x This absolute value prevents you from having to find the natural log of a negative number. The Power Rule for Integration

Antiderivatives & Indefinite Integrals What if we were to shift the graph up 1 unit?

Antiderivatives & Indefinite Integrals Do the slopes change?

Antiderivatives & Indefinite Integrals The slopes stay the same.

Antiderivatives & Indefinite Integrals If a function has an antiderivative, then it has an infinite number of antiderivates.

Antiderivatives & Indefinite Integrals To capture the fact that there are infinitely many antiderivatives we add a constant. Constant of Integration

Basic Integration Rules

Constant Rule for Integration

Basic Integration Rules The integral of a function times a constant is equal to the constant times the integral of the function.

Basic Integration Rules Q: How do you know if you have found the correct antiderivative? A: Take the derivative of your answer to check.

Basic Integration Rules Constant Multiple Rule for Integration Sum & Difference Rules for Integration

Basic Integration Rules

You Try these three.

Conclusion  Antidifferentiation is a process or operation that reverses differentiation.  The antidifferentiation process is also called integration.  Similar to differentiation, integration has a variety of rules that we must remember, recall, and be able to use.