Verified Answer :
Answer:
The average rate of change for f(x) from x=−1 to x = 4 is, 1
Step-by-step explanation:
Average rate A(x) of change for a function f(x) over [a, b] is given by:
\(A(x) = \frac{f(b)-f(a)}{b-a}\)
As per the statement:
\(f(x) = x^2-2x-4\)
we have to find the average rate of change from x = -1 to x = 4
At x = -1
\(f(-1) = (-1)^2-2(-1)-4 = 1+2-4 = -1\)
and
at x = 4
\(f(4) = (4)^2-2(4)-4 = 16-8-4 = 4\)
Substitute these in [1] we have;
\(A(x) = \frac{f(4)-f(-1)}{4-(-1)}\)
⇒\(A(x) = \frac{4-(-1)}{4+1}\)
⇒\(A(x) = \frac{5}{5}\)
Simplify:
A(x) = 1
Therefore, the average rate of change for f(x) from x=−1 to x = 4 is, 1
You can use the definitions of average rate of change to find the intended value.
The average rate of change for the quadratic function from x = -1 to x = 4
is given by
\(\text{Average rate of } f(x)|_{x=-1 \: to \: x = 4}= \dfrac{f(4) - f(-1)}{4 - (-1)} = \dfrac{4-(-1)}{4-(-1)} = 1\)
What is average rate of change of a function over a given interval?
Average rate of change is ratio of change in output to the change in input for the given function.
Using above definition to find the intended rate
\(f(4) = 4^2 - 2 \times 4 - 4 = 4\)
\(f(-1) = (-1)^2 -2 \times -1 - 4 = -1\)
\(\text{Average rate of } f(x)|_{x=-1 \: to \: x = 4}= \dfrac{f(4) - f(-1)}{4 - (-1)} = \dfrac{4-(-1)}{4-(-1)} = 1\)
Thus, the average rate needed is 1 (this denotes that input and output increased equally averagely in given interval [-1.4] on input axis)
Learn more about rate of change of functions here:
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