An expression for the given graph.

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 1.1, Problem 52E
To determine

To find: An expression for the given graph.

Expert Solution

The expression for the function of the given graph is f(x)={32x3if 4x24x2if 2<x<232x3if 2x4_ .

Explanation of Solution

Formula used:

Point slope form, yy1=m(xx1) .

The equation of the circle with center (0,0) and radius r is, x2+y2=r2 .

Calculation:

The graph has an upper half of a circle and two lines.

Obtain the expression for an upper half of a circle with center (0,0) and radius 2 on the domain 2<x<2 .

The equation of the circle with center (0,0) and radius 2 is x2+y2=22 .

Find the value of y to obtain the equation of the upper half of the circle.

x2+y2=22y2=22x2

y=4x2 (1)

Obtain the expression for a line with slope 32 passing through (2,0) on the domain 4x2 .

Use the point slope form and obtain the equation of the line with slope 32 and the point (2,0) .

y0=32(x(2))

y=32(x+2)

y=32x3 (2)

Obtain the expression for the other line with slope 32 and passing through (2,0) on the domain 2x4 .

Use point slope form and obtain the equation of the line with slope 32 and the point (2,0) .

y0=32(x2)y=32x322

y=32x3  (3)

Combine the equations (1), (2), and (3) and the domains to obtain the required function.

The piecewise function of the given graph is, f(x)={32x3 if 4x24x2if 2<x<232x3if 2x4 .

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