Question

Use the following graph of the function f(x) = 3x4 − x3 + 3x2 + x − 3 to answer this question: graph of 3 x to the fourth, minus x cubed, plus 3 x squared, plus x minus 3. What is the average rate of change from x = 0 to x = 1?

Answer 1

Average rat of change of function is 6 from x = 0 to x = 1.

Explanation:

We are given a function:

`f(x) = 3x^4 - x^3 + 3x^2 + x - 3`

The attached image shows the graph for the given function.

Average rate of change of function =

`\text{Average rate of change} = \displaystyle(\delta y)/(\delta x) = (f(b)-f(a))/(b-a)`

Now we will evaluate f(1) and f(0).

`f(x) = 3x^4 - x^3 + 3x^2 + x - 3\\f(1) = 3(1)^4 - (1)^3 + 3(1)^2 + (1) - 3 = 3\\f(0) = 3(0)^4 - (0)^3 + 3(0)^2 + (0) - 3 = -3`

Putting the values, we get,

`\text{Average rate of change} = \displaystyle(f(1)-f(0))/(1-0)\\\\= (3-(-3))/(1-0) = (6)/(1) = 6`

Thus, the average rat of change of function is 6 from x = 0 to x = 1.

Answer 2

For this case we have the following function:

`f (x) = 3x ^ 4 - x ^ 3 + 3x ^ 2 + x - 3`
By definition we have that the average rate of change is given by:

`AVR = (f(x2)-f(x1))/(x2-x1)`
We evaluate the function for the given interval:
For x = 0:

`f (0) = 3 (0) ^ 4 - (0) ^ 3 + 3 (0) ^ 2 + (0) - 3 f (0) = - 3`
For x = 1:

`f (1) = 3 (1) ^ 4 - (1) ^ 3 + 3 (1) ^ 2 + (1) - 3 f (1) = 3`
Substituting values we have:

`AVR = (3-(-3))/(1-0)`
Rewriting we have:

`AVR = (3+3)/(1-0)`

`AVR = (6)/(1)`

`AVR =6`
Answer:
the average rate of change from x = 0 to x = 1 is:

`AVR =6`