Math 3 Introduction to Calculus

Naomi Tanabe Department of Mathematics Dartmouth College

Lecture 23 - 2/26/2016 Fundamental theorem of calculus

Summary: We can approximate a function using tangent lines. This reveals a surprising relationship between Riemann sums and the derivative. This relationship is called the fundamental theorem of calculus (FTC).

Lecture 23 - 2/26/2016 Reminder - Riemann sums

How do we find the area A under a curve y = f(x)?

● We can approximate the area by dividing the interval into subintervals Example: of equal length. interval: [0,1], n=4

● We can then approximate the area using rectangles

● If we take smaller and smaller intervals and estimate approaches the real value A.

Lecture 23 - 2/26/2016 Definite integral

Def. integral of f(x) in the interval [a,b]:

● Recall: A sum of the form is called a Riemann sum and estimates the area.

● The definite integral calculates the net area, where areas under the x-axis are substracted.

Lecture 23 - 2/26/2016 Definite integral

Note: It is sufficient that the function f(x) is continuous in the given interval except for a finite number of jump discontinuities. Then the definite integral exists.

Lecture 23 - 2/26/2016 Integration and differentiation

Def. integral of f(x) in the interval [a,b]:

● How are integration and differentiation related?

● Let's approximate a function f(x) in a fixed interval with tangent lines.

● Let's take f(x) in the interval [0,4] in 4 steps we get:

● f(1) ≈ f(0) + f'(0) (point-slope form of tangent) f(2) ≈ f(1) + f'(1) ≈ f(0) + f'(0) + f'(1) f(3) ≈ f(2) + f'(2) ≈ f(0) + f'(0) + f'(1) + f'(2) f(4) ≈ f(3) + f'(3) ≈ f(0) + f'(0) + f'(1) + f'(2) + f'(3)

● Hence f(4) – f(0) ≈ f'(0) + f'(1) + f'(2) + f'(3).

● The term on the right hand side is a Riemann sum.

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Lecture 23 - 2/26/2016 Integration and differentiation

Def. integral of f(x) in the interval [a,b]:

Let's approximate a function f(x) in a fixed interval with tangent lines.

● Let's take f(x) in the interval [0,4]:

● Taking n subintervals we get: 4i 4

● The left hand side stays the same.

● The right hand side converges. Finally 4i 4 1 2 3 4

Lecture 23 - 2/26/2016 Fundamental theorem of calculus

This approach can be turned into a rigorous proof and generalized to show

Hence integration is equal to the infinitesimal version of approximating a function with tangent lines.

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Lecture 23 - 2/26/2016