Consider the function \(\displaystyle{f{{\left({x}\right)}}}=\sqrt{{{1}-{x}^{{{2}}}}}\) on the interval



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Consider the function f(x)=1x2 on the interval [0, -1], how do you find the average or mean slope of the function on this interval?

Answer & Explanation

Demetrius Kaufman

Demetrius Kaufman

Beginner2022-03-26Added 10 answers

The slope of a function f(x) at any point is equal to ddx(f(x)).
The average value of a function g(x) from x[a,b] is given by 1baabg(x)dx.
We want to find the average slope of the function f(x)=1x2 on x[1,0].
Its slope at any given point is ddx(1x2). Thus, the average slope from
x=-1 to x=0 is given by
By the Fundamental Theorem of Calculus, we know that integration and differentiation are inverse operations. Thus, we have the above expression equal to


Beginner2022-03-27Added 10 answers

Note that the "interval [0, -1] should mean the set of all x with 0x and i will assume that you mean [-1,0].
The slope is the same as the rate of change.
So the average slope on [-1, 0] is equal to the average rate of change on the interval:
If the intended interval is [0,1], then the average slope is

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