Antidifferentiation TS Making decisions after reflection and review

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

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

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

Basic Integration Rules

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.