P32 - inite integral using FTC and properties of the denite...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Project 32 MAC 2311 YOU MUST SHOW YOUR WORK TO RECEIVE FULL CREDIT!! 1. Let’s show that the Fundamental Theorem of Calculus (FTC) does what it claims with an example. Examine the function f ( t ) = t 2 . Fix an unspecified number x 1 (as shown on the axes below), and then draw and shade the region represented by the definite integral R x 1 t 2 d t . 0 1 x Show that the integral R x 1 t 2 d t is represented by the limit lim n →∞ n X i =1 h 1 + ( x - 1 n ) i i 2 ( x - 1 n ) . Evaluate the limit above WITHOUT using FTC ( x is just some unspecified number—treat it like a constant in the sum). Is your answer an antiderivative for f ( x ) as FTC states it should be (you might need to expand and combine like terms to see it)? So what is d dx Z x 1 t 2 d t ?
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2. Sketch the following piecewise-defined function, and evaluate the given def-
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: inite integral using FTC and properties of the denite integral. Shade the regions whose areas contribute to this denite integral. f ( x ) = 2 sin( x ) x < x x Z 1-1 f ( x ) d x = Is your answer positive or negative? What has more area: one hump of the function sin( x ), or the area under the square root function from 0 to 1 ? How much bigger is it? Just looking at your picture and thinking in terms of areas, is the function R x-1 f ( t ) d t increasing or decreasing at x =-1 2 ? at x = 0? at x = 1 2 ? Is it changing more rapidly at x = 1 2 or at x = 1?...
View Full Document

This note was uploaded on 01/13/2010 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

Page1 / 2

P32 - inite integral using FTC and properties of the denite...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online