   Chapter 9.3, Problem 8E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# In Example 6 in this section, we were given f ( x ) =   3 x 2 + 2 x +   11  and found  f ' ( x ) =   6 x   +   2 . Find(a) the instantaneous rate of change of f(x) at x = 6.(b) the slope of the tangent to the graph of y = f(x) at x = 6.(c) the point on the graph of y = f(x) at x = 6.EXAMPLE 6 Tangent LineGiven y =   f ( x ) = 3 x 2 + 2 x   + 11 , find(a) the derivative of f(x) at any point (x, f(x)).(b) the slope of the tangent to the curve at (1, 16).(c) the equation of the line tangent to y =   3 x 2 +   2 x +   11 at (1.16).Solution(a) The derivative of f(x) at any value x is denoted by f'(x) and is y ' = f ' ( x ) = lim h → 0 f ( x + h ) − f ( x ) h = lim h → 0 [ 3 ( x + h ) 2 + 2 ( x + h ) + 11 ] − ( 3 x 2 + 2 x + 11 ) h = lim h → 0 3 ( x 2 + 2 x h + h 2 ) 2 x + 2 h + 11 − 3 x 2 − 2 x − 11 h = lim h → 0 6 x h + 3 h 2 + 2 h h = lim h → 0 ( 6 x + 3 h + 2 ) = 6 x + 2 (b) The derivative is f ' ( x ) =   6 x   +   2 , so the slope of the tangent to the curve at (1,16) is f ' ( 1 )  =  6 ( 1 )   +   2   =   8 .(c) The equation of the tangent line uses the given point (1, 16) and the slope m = 8. Using y −   y 1 = m ( x   − x 1 ) gives y −   16   =   8 ( x − 1 ) ,  or  y   =   8 x   +   8 .The discussion in this section indicates that the derivative of a function has several interpretations.

(a)

To determine

To calculate: The instantaneous rate of change of function f(x) at x=6 if f(x)=3x2+2x+11 and f(x)=6x+2.

Explanation

Given Information:

The function f(x) is f(x)=3x2+2x+11 and f(x)=6x+2.

Formula used:

If a function f(x), the derivative of the function f(x) is the instantaneous rate of change of function

(b)

To determine

To calculate: The slope to the tangent of the graph y=f(x) at x=6 if f(x)=3x2+2x+11 and f(x)=6x+2.

(c)

To determine

The point on the graph of y=f(x) at x=6 if f(x)=3x2+2x+11 and f(x)=6x+2.

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