Answer
a)i)20.25
a)ii)20.05
b)20
Work Step by Step
a)i) First calculate the y value for the function at x=100 and x=105:
$f(x)= 5000+10x+0.05x^{2}$
$f(100)= 5000+10(100)+0.05(100)^{2}=6500$
$f(105)= 5000+10(105)+0.05(105)^{2}=6601.25$
Next, the average rate of change is the same as the slope. Slope is calculated by $\frac{rise}{run}$ in other words, $\frac{change in y}{change in x}$ so:
change in y: $6601.25-6500=101.25$
change in x:$105-100=5$
Therefore average rate of change: $\frac{101.25}{5}=20.25$
ii) First calculate the y value for the function at x=100 and x=101:
$f(x)= 5000+10x+0.05x^{2}$
$f(100)= 5000+10(100)+0.05(100)^{2}=6500$
$f(101)= 5000+10(101)+0.05(101)^{2}=6520.05$
Next, the average rate of change is the same as the slope. Slope is calculated by $\frac{rise}{run}$ in other words, $\frac{change in y}{change in x}$ so:
change in y: $6520.05-6500=20.05$
change in x:$101-100=1$
Therefore average rate of change: $\frac{20.05}{1}=20.05$
b)First, find the derivative of the overall function:
$f(x)= 5000+10x+0.05x^{2}$
Apply chain rule to find derrivative:
$f'(x)= 10+0.1x$
Plug in 100 for 'x'
$f'(100)= 10+0.1(100) = 10+10 = 20$
