   Chapter 11.3, Problem 72E Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

Solutions

Chapter
Section Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

In Exercises 71-76, find the equation of the line tangent to the graph of the given function at the point with the indicated x-coordinate. f ( x ) = ( x 0.5 + 1 ) ( x 2 + x ) ; x = 1

To determine

To calculate: The equation of the tangent to the graph of the function f(x)=(x0.5+1)(x2+x) at x=1.

Explanation

Given Information:

The function is f(x)=(x0.5+1)(x2+x) and x=1.

Formula used:

Product rule:

If f(x), g(x) and h(x) are differentiable functions, then

(fgh)'=f'gh+fg'h+fgh'

Power rule:

For a function f(x)=xn,

f'(x)=nxn1, where n is some constant.

Sum rule of derivative:

ddx[f(x)+g(x)]=ddx[f(x)]+ddx[g(x)]

Where, f(x) and g(x) are any two differentiable functions.

Slope of tangent of graph f(x) at point (a,b) is given by:

f'(a)

Equation of line is y=mx+b where m is the slope and b=y1mx1 when line passes through (x1,y1).

Calculation:

Consider the function, f(x)=(x0.5+1)(x2+x)

To find the slope of tangent of graph f(x)=(x0.5+1)(x2+x) determine the derivative of the function f(x).

Apply product rule to the function,

f'(x)=[ddx(x0.5+1)](x2+x)+(x0.5+1)[ddx(x2+x)]

Apply sum rule of derivative,

f'(x)=[ddx(x0.5)+ddx(1)](x2+x)+(x0.5+1)[ddx(x2)+ddx(x)]

Apply product rule and simplify further,

f'(x)=[0.5x0.51+0](x2+x)+(x0.5+1)[2x21+1x11]=(0

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