121616949-math.162

# 121616949-math.162 - the beginning of the interval namely...

This preview shows page 1. Sign up to view the full content.

148 Chapter 7 Integration Proof sketch for Theorem 7.4 . We want to compute G ( x ), so we start with the definition of the derivative in terms of a limit: G ( x ) = lim Δ x 0 G ( x + Δ x ) - G ( x ) Δ x = lim Δ x 0 1 Δ x ˆ Z x x a f ( t ) dt - Z x a f ( t ) dt ! = lim Δ x 0 1 Δ x ˆ Z x a f ( t ) dt + Z x x x f ( t ) dt - Z x a f ( t ) dt ! = lim Δ x 0 1 Δ x Z x x x f ( t ) dt. Now we need to know something about Z x x x f ( t ) dt when Δ x is small; in fact, it is very close to Δ xf ( x ), but we will not prove this. Once again, it is easy to believe this is true by thinking of our two applications: The integral Z x x x f ( t ) dt can be interpreted as the distance traveled by an object over a very short interval of time. Over a sufficiently short period of time, the speed of the object will not change very much, so the distance traveled will be approximately the length of time multiplied by the speed at
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: the beginning of the interval, namely, Δ xf ( x ). Alternately, the integral may be interpreted as the area under the curve between x and x + Δ x . When Δ x is very small, this will be very close to the area of the rectangle with base Δ x and height f ( x ); again this is Δ xf ( x ). If we accept this, we may proceed: lim Δ x → 1 Δ x Z x +Δ x x f ( t ) dt = lim Δ x → Δ xf ( x ) Δ x = f ( x ) , which is what we wanted to show. It is still true that we are depending on an interpretation of the integral to justify the argument, but we have isolated this part of the argument into two facts that are not too...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern