32-FTC-last-problem

32-FTC-last-problem - Math 1a One Last Fundamental Theorem...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 1a One Last Fundamental Theorem Problem Fall, 2009 1 (a) Let f ( x ) = Z x t n dt for some fixed n > 0. Find f ( x ). (b) Let g ( x ) = Z x n t 1 /n dt for the same fixed n > 0. Find g ( x ). (c) Let F ( x ) = f ( x ) + g ( x ) = Z x t n dt + Z x n t 1 /n dt for this value of n > 0. Write down F ( x ). (d) What is F (0)? (e) Use your answers to parts (d) and (e) to find a simple expression for F ( x ). Explain this answer geometrically. For your convenience, we reprint here a few questions from Wednesdays worksheet. These questions (and answers) are from Maria Terrells GoodQuestions Project at the Cornell University Department of Mathematics: 2 True or False : If f is continuous on the interval [ a,b ], d dx Z b a f ( x ) dx = f ( x ). To the right is the graph of a function f . For the next two functions, we consider the function g ( x ) = Z x f ( t ) dt .- 1 1 2 ................................................................................................................................................................................................................... . . . . . . . . . . . . 1 2 3 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x y ............................................................................................................................................................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 For 0 < x < 2, g ( x ) is (a) increasing and concave up. (b) increasing and concave down. (c) decreasing and concave up. (d) decreasing and concave down. 4 For this g ( x ), find g (0), g (0), and g (2). 5 You are traveling with velocity v ( t ) that varies continuously over the interval [ a,b ] and your position at time t is given by s ( t ). Which of the following represent your average velocity for that time interval: (a) R b a v ( t ) dt ( b- a ) (b) s ( b )- s ( a ) b- a (c) v ( c ) for at least one c between a and b One Last Fundamental Theorem Problem Answers 1 (a) f ( x ) = x n (b) g ( x ) = ( x n ) ( 1 /n ) d dx ( x n ) = x nx n- 1 = nx n . (c) F ( x ) = ( n + 1) x n (d) F (0) = 0 (e) F ( x ) = x n +1 + C , and since F (0) = 0 we get F ( x ) = x n +1 . Heres a quick sketch of the areas described by the two functions f ( x ) and g ( x ): ...................................................................................................................................
View Full Document

This note was uploaded on 03/27/2012 for the course MATH 1a taught by Professor Benedictgross during the Fall '09 term at Harvard.

Page1 / 3

32-FTC-last-problem - Math 1a One Last Fundamental Theorem...

This preview shows document pages 1 - 3. Sign up to view the full document.

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