Question

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)). Consider the function f(x)=x^2+6 and find the following: (a) The average rate of change of f(x) from x=−1 to x=4. (b) The average rate of change of f(x) from x=a to x=b. (c) The average rate of change of f(x)f between the points (x,f(x)) and (x+h,f(x+h)). Assume h>0

Answer 1

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:

$\frac{f(b)-f(a)}{b-a}$

Part a)

$f(x) = x^{2}+6$

The average rate of change of f(x) from x = −1 to x = 4. Using these values in above formula, we get:

$\frac{f(4)-f(-1)}{4-(-1)}\\\\ =\frac{(4)^{2}+6-[(-1)^{2}+6]}{4+1}\\\\ =\frac{16+6-1-6}{5}\\\\ =\frac{15}{5}\\\\ =3$

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:

$\frac{f(b)-f(a)}{b-a}\\\\ = \frac{b^{2}+6-a^{2}-6}{b-a}\\\\ =\frac{b^{2}-a^{2}}{b-a}\\\\ =\frac{(b-a)(b+a)}{b-a}\\\\ =b+a$

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:

$\frac{f(x+h)-f(x)}{x+h-x}\\\\ =\frac{(x+h)^{2}+6-x^{2}-6}{h}\\\\ =\frac{(x+h)^{2}-x^{2}}{h}\\\\ =\frac{x^{2}+h^{2}+2xh-x^{2}}{h}\\\\ =\frac{h^{2}+2xh}{h}\\\\ =\frac{h(h+2x)}{h}\\\\ = h + 2x$

Thus, The average rate of change of f(x) between the points (x,f(x)) and (x+h,f(x+h)) is 2x + h.