The Fundamental Theorems of Calculus Lesson 5 4

The Fundamental Theorems of Calculus Lesson 5. 4

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Don't let this happen to you! 4

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

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

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)? 0

The Area Function f(x) A(x+h) a x x+h 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 Divide both sides by h • Now divide by h … then take the limit 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 Since this is a constant … • Then • What about ?

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