Answer:
b=4
Step-by-step explanation:
So, we have the function [tex]f(x)=1/x[/tex]. We need to find b such that the average rate of change or the slope is -1/8 between the intervel [2, b]. First, let's find f(2).
f(2) = 1/(2) = 1/2
So, we have the point (2, 1/2)
At point b, f(b) = 1/b.
Let's plug this into the slope formula:
[tex]\frac{y_2-y_1}{x_2-x_1}=\frac{.5-\frac{1}{b} }{2-b} =-1/8[/tex]
Now, we just need to solve for b. First, let's multiply both the numerator and denominator by b (to get rid of the annoying fraction in the numerator).
[tex]\frac{.5b-1}{2b-b^2} =\frac{-1}{8}[/tex]
Now, cross multiply.
[tex]4b-8=b^2-2b[/tex]
[tex]b^2-6b+8=0[/tex]
Solve for b. Factor using the numbers -4 and -2.
[tex]=(b-4)(b-2)=0[/tex]
Thus, b=4 or b=2.
However, b=2 is not a possible solution since the interval [2,2] means nothing. Thus, b=4.
We want to find an interval such that the given equation, f(x) = 1/x, has an average rate of change of -1/8 in that interval.
We will see that the interval is [2, 4]
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For a function f(x), the average rate of change in the interval [a, b] is given by:
[tex]r = \frac{f(b) - f(a)}{b - a}[/tex]
Here we have:
[tex]f(x) = 1/x[/tex]
And the interval is [2, b] such that r in that interval is -1/8, so we need to solve:
[tex]r = -1/8 = \frac{f(b) - f(2)}{b - 2} = \frac{1/b - 1/2}{b - 2}[/tex]
We can rewrite it to:
[tex]-1/8 *(b - 2)= 1/b - 1/2\\\\-1/8 *(b - 2)= 2/2b - b/2b = (2 - b)/2b = -(b - 2)/2b[/tex]
Now we can remove the term (b - 2) because it appears on both sides, so we get:
[tex]-1/8 = -1/2b\\1/8 = 1/2b\\2/8 = 1/b\\1/4 = 1/b\\b = 4[/tex]
Then we found that b must be equal to 4, so the interval is [2, 4]
If you want to learn more, you can read:
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