Question

For the graph of f shown next at the left, list the following quantities in order from least to greatest:
(a) the average rate of change of f on [−1, 1]
(b) the instantaneous rate of change of f at x = 1
(c) f ′(−1)
(d) f(2)− f(−1)
2
15. For the graph of f shown next at the left, list the following quantities in order from least to greatest: (a) the average rate of change of f on [-1,1] (b) the instantaneous rate of change of f at x1 f(2) f(-1)
−(−1)

Answer 1

The average rate of change of f(x) over the interval [a,b] is

Part A

Therefore average rate of change of f(x) over the interval [-1,1] is

$AvgRateOfChange=\frac{f(1)-f(-1)}{1-(-1)}=\frac{1}{2}[f(1)-f(-1)]$

From the graph,

$AvgRateOfChange=\frac{1}{2}[1-(-1)]=1$

Part B

Instanteneous rate of change at x=1 is the slope of the tangent line at x=1 which is

$f'(1)=\frac{f(2)-f(1)}{2-1}=\frac{2-1}{1}=1$

Part C

Instanteneous rate of change at x=-1 is the slope of the tangent line at x=-1 which is

$f'(1)=\frac{f(-0.6)-f(-1)}{-0.6-(-1)}=\frac{-2-(-1)}{0.4}=\frac{-1}{0.4}=\frac{-5}{2}$

Part D

$\frac{f(2)-f(-1)}{2-(-1)}=\frac{2-(-1)}{3}=1$