Miss Battaglia AB/BC Calculus.  Connects differentiation and integration.  Integration & differentiation are inverse operations. If a function is continuous.

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Presentation transcript:

Miss Battaglia AB/BC Calculus

 Connects differentiation and integration.  Integration & differentiation are inverse operations. If a function is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then

1. Provided you can find an antiderivative of f, you now have a way to evaluate a definite integral without having to use the limit of a sum. 2. When applying the FTC, the following notation is convenient For instance, to evaluateyou can write 3. It is not necessary to include a constant of integration C in the antiderivative because

a. b. c.

Evaluate

Find the area of the region bounded by the graph of y=x 3 +x, bounded by x=2 and y=0

If f is continuous on the closed interval [a,b], then there exists a number c in the closed interval [a,b] such that

If f is integrable on the closed interval [a,b], then the average value of f on the interval is

Find the average value of f(x)=9-x 2 the interval [-3,3]

Evaluate the function at x=0, π/6, π/4, π3, and π/2

If f is continuous on an open interval I containing a, then, for every x in the interval,

Evaluate

Find the derivative of

The definite integral of the rate of change of a quantity of F’(x) gives the total change, or net change in that quantity on the interval [a,b].

A chemical flows into a storage tank at a rate of 180+3t liters per min, where 0 < t < 60. Find the amount of the chemical that flows into the tank during the first 20 min.

Displacement on [a,b] Total distance traveled on [a,b] =

A particle is moving along a line so that its velocity is v(t)=t 3 -10t 2 +29t-20 ft/sec at time t. a. What is the displacement of the particle on the time interval 1 < t < 5? b. What is the total distance traveled by the particle on the time interval 1 < t < 5?

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