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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Sketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sketch as a guide to producing graphs (with a graphing device) that display the major features of the curve. Use these graphs to estimate the maximum and minimum values.

f ( x ) = ( 2 x + 3 ) 2 ( x 2 ) x 3 ( x 5 ) 2

To determine

To Draw: The graph of the function f(x)=(2x+3)2(x2)5x3(x5)2 using asymptotes and intercepts and find the maximum and minimum values of the function using the graph.

Explanation

Given information:

The function is f(x)=(2x+3)2(x2)5x3(x5)2 (1)

Calculate the vertical axis:

Substitute x=0 in Equation (1)

f(x)=(0+3)2(02)5(0)(05)2=

Substitute x=5 in Equation (1)

f(x)=(5+3)2(52)5(5)3(55)=

Therefore, the function has vertical axis at x=0 and x=5 and the following conclusion arrived

  • limx0f(x)=
  • limx0+f(x)= and
  • limx5f(x)=

Calculate the horizontal axis:

Rewrite the Equation (1) as follows:

Dividing numerator and denominator of Equation (1) by x3

f(x)=(2x+3)2(x2)5x3(x5)2=x2(2+3x)2.x5(12x)5x5(15x)2=x7(2+3x)2(12x)5x5(15x)2=x2(2+3x)2(12x)5(15x)2 (2)

Substitute x=± in Equation (2)

f

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