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

Learn to evaluate the integral with functions as bounds
Learn to evaluate the integral with functions as bounds

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warmup 1) 2)

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5.4: Fundamental Theorem of Calculus

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

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First Fundamental Theorem: 1. Derivative of an integral.

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

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

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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:

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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:

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

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

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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.

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Group Problem:

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The graph above is g(t)

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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.

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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)

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is a general antiderivative so…

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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

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Group Work

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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.

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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

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Group Problem

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

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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

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Ex. Given

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1) 2)

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

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the end

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