Suppose f(x) = 2x^{3}. Write an expression in terms of x and h that represents the average rate of change of f over any interval of length h. [That is, over any interval (x, x + h)] Simplify your answer as much as possible.

Jaya Legge

Jaya Legge

Answered question

2021-02-01

Suppose f(x)=2x3. Write an expression in terms of x and h that represents the average rate of change of f over any interval of length h. [That is, over any interval (x, x + h)] Simplify your answer as much as possible.

Answer & Explanation

Obiajulu

Obiajulu

Skilled2021-02-02Added 98 answers

Step 1
Consider the function:
f(x)=2x3
The average rate of change of function over any intervals of length “h” is given by the formula,
Average rate =f(x + h)  f(x)h
Step 2
The average rate of changes of the given function is,
Average rate =2(x + h)3  2x3h
=2(x3 + h3 + 3x2h + 3xh2)  2x3h
=2x3 + 2h3 + 6x2h + 6xh2  2x3h
=h(2h2 + 6x2 + 6xh)h
=2h2 + 6x2 + 6xh
=6x2 + 6xh + 2h2
Hence the average rate of changes of the given function is 6x2 + 6xh + 2h2
Andre BalkonE

Andre BalkonE

Skilled2023-05-26Added 110 answers

Given that f(x)=2x3, we can find the function values at the endpoints of the interval (x,x+h) by substituting x and x+h into the function. Let's calculate those values:
f(x)=2x3
f(x+h)=2(x+h)3
The difference in function values is:
f(x+h)f(x)=2(x+h)32x3
The length of the interval is h, so the average rate of change is:
f(x+h)f(x)h=2(x+h)32x3h
We can simplify this expression further. Expanding (x+h)3 using the binomial formula, we get:
(x+h)3=x3+3x2h+3xh2+h3
Substituting this into our expression, we have:
2(x+h)32x3h=2(x3+3x2h+3xh2+h3)2x3h
Simplifying the numerator, we get:
2x3+6x2h+6xh2+2h32x3h
Cancelling out the 2x3 terms, we have:
6x2h+6xh2+2h3h
Factoring out h from each term in the numerator, we get:
h(6x2+6xh+2h2)h
Finally, we can cancel out the h terms:
6x2+6xh+2h2
Hence, the expression in terms of x and h that represents the average rate of change of f over any interval of length h is 6x2+6xh+2h2.
Jazz Frenia

Jazz Frenia

Skilled2023-05-26Added 106 answers

Answer:
6x2+6xh+2h2
Explanation:
Let's denote the average rate of change as AVG(f,h). The function values at the endpoints of the interval are f(x) and f(x+h), respectively. Therefore, the average rate of change can be expressed as:
AVG(f,h)=f(x+h)f(x)(x+h)x
Substituting the given function f(x)=2x3 into the expression, we have:
AVG(f,h)=2(x+h)32x3h
Expanding (x+h)3 using the binomial formula, we get:
AVG(f,h)=2(x3+3x2h+3xh2+h3)2x3h
Simplifying the expression further:
AVG(f,h)=2x3+6x2h+6xh2+2h32x3h
Canceling out the 2x3 terms:
AVG(f,h)=6x2h+6xh2+2h3h
Finally, we can simplify the expression by factoring out an h from each term:
AVG(f,h)=h(6x2+6xh+2h2)h
Simplifying further:
AVG(f,h)=6x2+6xh+2h2
Thus, the expression representing the average rate of change of f over any interval of length h is 6x2+6xh+2h2.

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