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Presentation transcript:

warmup 1)

5.4: Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus, Part 1 If f is continuous on , then the function has a derivative at every point in , and

First Fundamental Theorem: 1. Derivative of an integral.

First Fundamental Theorem: 1. Derivative of an integral. 2. Derivative matches upper limit of integration.

First Fundamental Theorem: 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.

First Fundamental Theorem: New variable. 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.

The long way: First Fundamental Theorem: 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.

1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.

The upper limit of integration does not match the derivative, but we could use the chain rule.

The lower limit of integration is not a constant, but the upper limit is. We can change the sign of the integral and reverse the limits.

Group Problem:

The graph above is g(t)

Neither limit of integration is a constant. We split the integral into two parts. It does not matter what constant we use! (Limits are reversed.) (Chain rule is used.)

The Fundamental Theorem of Calculus, Part 2 If f is continuous at every point of , and if F is any antiderivative of f on , then (Also called the Integral Evaluation Theorem)

is a general antiderivative so…

Remember, the definite integral gives us the net area Net area counts area below the x-axis as negative The net area, or if this were a definite integral, would =5-3+4=6 The area, or “total area”, or area to the x-axis, would be 5+3+4=12

Group Work

Find g(-5) Find all values of x on the open interval (-5,4) where g is decreasing. Justify your answer. c) Write an equation for the line tangent to the graph of g at x = -1 d) Find the minimum value of g on the closed interval [-5,4]. Justify your answer.

Solution Find g(-5) Find all values of x on the open interval (-5,4) where g is decreasing. Justify your answer. c) Write an equation for the line tangent to the graph of g at x = -1 d) Find the minimum value of g on the closed interval [-5,4]. Justify your answer.

Group Problem

Group Problem

Using FTC with an initial condition: IF the initial condition is given, it accumulates normally and then adds the initial condition.

Ex. If oil fills a tank at a rate modeled by and the tanker has 2,500 gallons to start. How much oil is in the tank after 50 minutes pass? f(a) a is the lower limit

Ex. Given

1) 2)

2) Where does is the particle at t=5 ?

the end