Calc Homework , math homework help

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Mth 252 Lab: The Fundamental Theorem of Calculus The purpose of this lab is to deepen your understanding of antiderivatives presented in integral form as area functions. This will require that you recall how to glean information about a function based on the graph of its derivative, i.e., you will need to recall some basic Mth 251 concepts. The graph of f illustrated below is made up of line segments and two semicircles (of radius 1). x Let A(x) = ∫ x f (t)dt and F(x) = −4 ∫ f (t)dt be two area functions for f. (Before you start, −1 you need to discuss the relationship between the function f and the functions A and F.) Do all work on another sheet of paper. 1. Determine the intervals over which A and F are increasing and those over which A and F are decreasing. Discuss how you are using the graph of f to determine this, referring to concepts from Mth 251. 2. Determine the intervals over which the graphs of A and F are concave up and those over which the graphs of A and F are concave down. Discuss how you are using the graph of f to determine this, referring to concepts from Mth 251. 3. State the x coordinates of any local maxima and local minima for A and F. Discuss how you are using the graph of f to determine this, referring to concepts from Mth 251. 4. State the x coordinates of any inflection points for A and F. Discuss how you are using the graph of f to determine this, referring to concepts from Mth 251. 5. Use the formula given for A and the graph of f to complete the table below. As you find each value, write down the appropriate integral expression for A and then show/explain how you are finding each value. Use geometric formulas to evaluate any definite integrals. First, find the exact value and then round to one decimal place. x -4 -3 -2 -1 0 A(x)   Page  90   1 2 3 4 6. Use the formula given for F and the graph of f to complete the table below. As you find each value, write down the appropriate integral expression for F and then show/explain how you are finding each value. Use geometric formulas to evaluate any definite integrals. First, find the exact value and then round to one decimal place. x -4 -3 -2 -1 0 1 2 3 4 F(x) 7. Plot all the points found in #5. Sketch the graph of A carefully noting the increasing/decreasing behavior and the concavity. Label any local maxima as M, any local minima as m, and any inflection points as ip. 8. On the same axes, plot all the points found in #6. Sketch the graph of F carefully noting the increasing/decreasing behavior and the concavity. Label any local maxima as M, any local minima as m, and any inflection points as ip. 9. Describe what you notice about the graphs of A and F. Does this support what you think you should happen with regard to the graphs of the functions A and F? Why or why not? x 10. Lastly, symbolically, find A′(x) = d ∫ f (t)dt = __________ and dx −4 F ′(x) = d ∫ f (t)dt = __________. dx −1 x Be careful that your answer is written with the appropriate variable. 11. Extra thought. We see that A(−4) = 0 . Can you think of a way to easily write another antiderivative of f where A(−4) = 5 ? This should require no work – only some thought.   Page  91   ...
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Tutor Answer

pallveechem123
School: UC Berkeley

please see and let me know if any correction required... i tried to finish as soon as possible..

Given that

x

A( x) 



x

f (t )dt and F ( x) 

4

 f (t )dt

1

f (t )   (1  (t  3) 2

- 4  t  2

t2

- 2  t  -1

1

-1  t  0

 1 t

0t 1

 1-t

1t 2

 -1  1 - (3 - t) 2

2t4

As f (t ) is continuous over [4,4] , then

A( x)  f ( x) for  4  x  4
As f (t ) is continuous over [1,4] , then

F ( x)  f ( x) for  1  x  4

1. Determine the intervals over which A and F are increasing and those over which A and F are
decreasing.
x

A( x) 

 f (t )dt

is decreasing over the following intervals

4

(4,2) and (1,4)
x

A( x) 

 f (t )dt

is increasing over the following inter...

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