Question

given f(x)=5x2+4x−4, find the average rate of change of f(x) from x=−1 to x=t. Give your answer in terms of t. Simplify your answer.

Answer 1

The average rate of change of f(x) from x = -1 to x = t is given by (5t^2 + 4t + 1) / (t + 1).

Step-by-step explanation:

The average rate of change of a function, in this case f(x), is the difference in the function's values at two points divided by the difference in their corresponding x-values. To find the average rate of change of f(x) from x = -1 to x = t, we need to substitute the given values into the function and simplify the expression.

f(x)= 5x^2 + 4x - 4

Let's first find f(-1) and f(t):

f(-1) = 5(-1)^2 + 4(-1) - 4 = 5 - 4 - 4 = -3

f(t) = 5t^2 + 4t - 4

The average rate of change of f(x) from x = -1 to x = t is given by:

(f(t) - f(-1)) / (t - (-1)) = (5t^2 + 4t - 4 - (-3)) / (t + 1) = (5t^2 + 4t + 1) / (t + 1)

Answer 2

The average rate of change of
`f(x) = 5x^2 + 4x - 4` from x = -1 to x = t is
`(5t^2 + 4t + 3)/(t + 1)`

Step-by-step explanation:

The average rate of change of a function from x=a to x=b is given by the formula:

Average rate of change = (f(b) - f(a))/(b - a)

In this case, the function is
`f(x) = 5x^2 + 4x - 4`, and we need to find the average rate of change from x = -1 to x = t.

Substituting the values into the formula, we get:

Average rate of change = (f(t) - f(-1))/(t - (-1))

Expanding the brackets and simplifying the expression:

Average rate of change =
`(5t^2 + 4t - 4 - 5(-1)^2 - 4(-1) + 4)/(t + 1)`

Combining like terms:

Average rate of change =
`(5t^2 + 4t + 4 - 5 + 4)/(t + 1)`

Simplifying further:

Average rate of change =
`(5t^2 + 4t + 3)/(t + 1)`