   Chapter 3.9, Problem 44E

Chapter
Section
Textbook Problem

# Find a function f such that f ′ ( x ) = x 3 and the line x   +   y   = 0 is tangent to the graph of f.

To determine

To find:

The function f such that f'x=x3 and the line x+y=0 is tangent to the graph of f

Explanation

From the given information f'x=x3 first find fx  by using antiderivative theorem. From given tangent line we find point of contact. And from this point we find the constant C in f(x)

1) Concept:

The slope of function fx is m=f'x

2) Given:

f'x=x3, x+y=0 is tangent line to the function.

3) Calculation:

f'x=x3

The general antiderivative of f'x using rules of antiderivative is,

fx=14x4+C Here C is the arbitrary constant.

Given x+y=0 is tangent line to the function .

Solving for y,

y=-x

Slope of this tangent line is -1

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