   Chapter 9.3, Problem 38E ### 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 Problems 35-38, (a) find the slope of the tangent to the graph of f ( x ) at any point, (b) find the slope of the tangent at the given point, (c) write the equation of the line tangent to the graph of f ( x ) at the given point, and (d) graph both f ( x ) and its tangent line (use a graphing utility). f ( x ) = 5 x 3 + 2 ; ( − 1 , − 3 )

(a)

To determine

To calculate: The slope of the tangent to the graph of the function f(x)=5x3+2

Explanation

Given Information:

The provided function is f(x)=5x3+2.

Formula used:

If a function f(x) is defined in the interval [x,x+h], then the slope of the tangent can be evaluated by the derivative as

f(x)=limh0f(x+h)f(x)h

Calculation:

Consider the function, f(x)=5x3+2

Substitute x=x+h in f(x) to get,

f(x+h)=5(x+h)3+2=5(x3+h3+3x2h+3xh2)+2=5x3

(b)

To determine

To calculate: The slope of the tangent to the graph of the function f(x) at (1,3) which is given as

f(x)=5x3+2

(c)

To determine

To calculate: The equation of the line which is tangent to the graph of f(x)=5x3+2 at point (1,3).

(d)

To determine

To graph: The function, f(x)=5x3+2 and the tangent line y=15x+12.

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