Answer:
a) 3
b) b + a
c) 2x + h
Step-by-step explanation:
Since, The average rate of change of a function f(x)f from x=a to x=b is the slope of the line which passes through (a,f(a) and (b,f(b)), we can write the formula for Average Rate of Change as:
![\frac{f(b)-f(a)}{b-a}](https://tex.z-dn.net/?f=%5Cfrac%7Bf%28b%29-f%28a%29%7D%7Bb-a%7D)
Part a)
![f(x) = x^{2}+6](https://tex.z-dn.net/?f=f%28x%29%20%3D%20x%5E%7B2%7D%2B6)
The average rate of change of f(x) from x = −1 to x = 4. Using these values in above formula, we get:
![\frac{f(4)-f(-1)}{4-(-1)}\\\\ =\frac{(4)^{2}+6-[(-1)^{2}+6]}{4+1}\\\\ =\frac{16+6-1-6}{5}\\\\ =\frac{15}{5}\\\\ =3](https://tex.z-dn.net/?f=%5Cfrac%7Bf%284%29-f%28-1%29%7D%7B4-%28-1%29%7D%5C%5C%5C%5C%20%3D%5Cfrac%7B%284%29%5E%7B2%7D%2B6-%5B%28-1%29%5E%7B2%7D%2B6%5D%7D%7B4%2B1%7D%5C%5C%5C%5C%20%3D%5Cfrac%7B16%2B6-1-6%7D%7B5%7D%5C%5C%5C%5C%20%3D%5Cfrac%7B15%7D%7B5%7D%5C%5C%5C%5C%20%3D3)
Thus, the average rate of change of f(x) from x = −1 to x = 4 is 3
Part b)
The average rate of change of f(x) from x=a to x=b. Using the values in the above formula, we get:
![\frac{f(b)-f(a)}{b-a}\\\\ = \frac{b^{2}+6-a^{2}-6}{b-a}\\\\ =\frac{b^{2}-a^{2}}{b-a}\\\\ =\frac{(b-a)(b+a)}{b-a}\\\\ =b+a](https://tex.z-dn.net/?f=%5Cfrac%7Bf%28b%29-f%28a%29%7D%7Bb-a%7D%5C%5C%5C%5C%20%3D%20%5Cfrac%7Bb%5E%7B2%7D%2B6-a%5E%7B2%7D-6%7D%7Bb-a%7D%5C%5C%5C%5C%20%3D%5Cfrac%7Bb%5E%7B2%7D-a%5E%7B2%7D%7D%7Bb-a%7D%5C%5C%5C%5C%20%3D%5Cfrac%7B%28b-a%29%28b%2Ba%29%7D%7Bb-a%7D%5C%5C%5C%5C%20%3Db%2Ba)
Thus, the average rate of change of f(x) from x = a to x = b is b + a
Part c)
The average rate of change of f(x) between the points (x,f(x)) and (x+h,f(x+h)). Using the values in above formula, we get:
![\frac{f(x+h)-f(x)}{x+h-x}\\\\ =\frac{(x+h)^{2}+6-x^{2}-6}{h}\\\\ =\frac{(x+h)^{2}-x^{2}}{h}\\\\ =\frac{x^{2}+h^{2}+2xh-x^{2}}{h}\\\\ =\frac{h^{2}+2xh}{h}\\\\ =\frac{h(h+2x)}{h}\\\\ = h + 2x](https://tex.z-dn.net/?f=%5Cfrac%7Bf%28x%2Bh%29-f%28x%29%7D%7Bx%2Bh-x%7D%5C%5C%5C%5C%20%3D%5Cfrac%7B%28x%2Bh%29%5E%7B2%7D%2B6-x%5E%7B2%7D-6%7D%7Bh%7D%5C%5C%5C%5C%20%3D%5Cfrac%7B%28x%2Bh%29%5E%7B2%7D-x%5E%7B2%7D%7D%7Bh%7D%5C%5C%5C%5C%20%3D%5Cfrac%7Bx%5E%7B2%7D%2Bh%5E%7B2%7D%2B2xh-x%5E%7B2%7D%7D%7Bh%7D%5C%5C%5C%5C%20%3D%5Cfrac%7Bh%5E%7B2%7D%2B2xh%7D%7Bh%7D%5C%5C%5C%5C%20%3D%5Cfrac%7Bh%28h%2B2x%29%7D%7Bh%7D%5C%5C%5C%5C%20%3D%20h%20%2B%202x)
Thus, The average rate of change of f(x) between the points (x,f(x)) and (x+h,f(x+h)) is 2x + h.