## Calculus 8th Edition

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' $f'(100)= 10+0.1(100) = 10+10 = 20$