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

Question

Demetrius Kaufman · Accepted Answer

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 1b−a∫abg(x)dx.We want to find the average slope of the function f(x)=1−x2 on x∈[−1,0].Its slope at any given point is ddx(1−x2). Thus, the average slope fromx=-1 to x=0 is given by10−(−1)∫−10ddx(1−x2)dxBy the Fundamental Theorem of Calculus, we know that integration and differentiation are inverse operations. Thus, we have the above expression equal to=[1−x2]−10=1

umgebautv6v2 · Accepted Answer

Note that the &quot;interval [0, -1] should mean the set of all x with 0≤x 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:f(0)−f(−1)0−(−1)=1−01=1If the intended interval is [0,1], then the average slope isf(1)−f(0)1−0=0−11=−1

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

Answered question

Answer & Explanation

New Questions in Calculus 1