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 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. 1 2 3 4 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. 1 2 3 4 Lecture 23 - 2/26/2016.