   Chapter 4.R, Problem 52E

Chapter
Section
Textbook Problem

# The Fresnel function S ( x ) = ∫ 0 x sin ( 1 2 π t 2 )   d t was introduced in Section 4.3. Fresnel also used the function C ( x ) = ∫ 0 x cos ( 1 2 π t 2 )   d t in his theory of the diffraction of light waves.(a) On what intervals is C is increasing?(b) On what intervals is C concave upward?(c) Use a graph to solve the following equation correct to two decimal places: ∫ 0 x cos ( 1 2 π t 2 )   d t = 0.7 (d) Plot the graphs of C and S on the same screen. How are these graphs related?

To determine

(a)

To find:

The interval in which C is increasing

Explanation

Concept:

To find the interval in which a function is increasing, recall that the function is increasing if the first derivative C'x>0

Given:

Cx=0xcos12πt2dt

Calculation:

The given integral is

Cx=0xcos12πt2dt

Applying fundamental theorem of calculus

ddxCx=ddx0xcos12πt2dt

C'x =cos12πx2 The function C is increasing when C' is positive

To determine

b)

To find:

The intervals in which C is concave upward of function

To determine

c)

0xcos12πt2dt=0.7 correct up to two decimal places by using the graph.

To determine

d)

To plot the graph of C and S to compare

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