The Fundamental Theorems of Calculus Lesson 5.4

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Average Value Theorem Consider function f(x) on an interval Area = a b Consider function f(x) on an interval Area = Consider the existence of a value for x = c such that a < x < b and c

Average Value of Function a b Also The average value is c

Area Under a Curve f(x) a b We know the area under the curve on the interval [a, b] is given by the formula above Consider the existence of an Area Function

Area Under a Curve f(x) A(x) a x b The area function is A(x) … the area under the curve on the interval [a, x] a ≤ x ≤ b What is A(a)? What is A(b)?

The Area Function If the area is increased by h units f(x) A(x+h) x x+h a b h If the area is increased by h units New area is A(x + h) Area of the new slice is A(x + h) – A(x) It's height is some average value ŷ It's width is h

The Area Function The area of the slice is Now divide by h … then take the limit Divide both sides by h Why?

The Area Function The left side of our equation is the derivative of A(x) !! The derivative of the Area function, A' is the same as the original function, f

Fundamental Theorem of Calculus We know the antiderivative of f(x) is F(x) + C Thus A(x) = F(x) + C Since A(a) = 0 Then for x = a, 0 = F(a) + C or C = -F(a) And A(x) = F(x) – F(a) Now if x = b A(b) = F(b) – F(a)

Fundamental Theorem of Calculus We have said that Thus we conclude The area under the curve is equal to the difference of the two antiderivatives Often written

First Fundamental Theorem of Calculus Given f is continuous on interval [a, b] F is any function that satisfies F’(x) = f(x) Then

First Fundamental Theorem of Calculus The definite integral can be computed by finding an antiderivative F on interval [a,b] evaluating at limits a and b and subtracting Try

Area Under a Curve Consider Area =

Area Under a Curve Find the area under the following function on the interval [1, 4]

Second Fundamental Theorem of Calculus Often useful to think of the following form We can consider this to be a function in terms of x View Movie on next slide

Second Fundamental Theorem of Calculus This is a function View Demo

Second Fundamental Theorem of Calculus Suppose we are given G(x) What is G’(x)?

Second Fundamental Theorem of Calculus Note that Then What about ? Since this is a constant …

Second Fundamental Theorem of Calculus Try this

Second Fundamental Theorem of Calculus An application from Differential Equations Consider Now integrate both sides And

Assignment Lesson 5.4 Page 327 Exercises 1 – 49 odd