# Warmup 1) 2). 5.4: Fundamental Theorem of Calculus.

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warmup 1) 2)

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.

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:

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

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.

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. It does not matter what constant we use! (Limits are reversed.) (Chain rule is used.) We split the integral into two parts.

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

a)Find g(-5) b)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.

a)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

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