Chapter 4.1, Problem LC

(a)

To determine

To calculate: The values of $f\left(a\right)$, $A\left[a,b\right]$, $\frac{A\left[a,b\right]}{f\left(a\right)}$ in the table given below.

(b)

To determine

The unit of the quantity $A\left[a,b\right]$ if the function $f\left(t\right)=300e^{\left(0.5t\right)}$ gives the number of bacteria at time $t$ hours. The bacteria are growing at a rate of 50% per hour. $A\left[a,b\right]$ represents the average rate of change in the number of bacteria on the time interval $\left[a,b\right]$.

(c)

To determine

The conclusion that can be drawn from the values of the ratio $\frac{A\left[a,b\right]}{f\left(a\right)}$ if the function $f\left(t\right)=300e^{\left(0.5t\right)}$ gives the number of bacteria at time $t$ hours. The bacteria are growing at a rate of 50% per hour. $A\left[a,b\right]$ represents the average rate of change in the number of bacteria on the time interval $\left[a,b\right]$.

(d)

To determine

The effect of the length of the interval on the ratio $\frac{A\left[a,b\right]}{f\left(a\right)}$. Test for some more arbitrary intervals by using the information is given below:

The function $f\left(t\right)=300e^{\left(0.5t\right)}$ gives the number of bacteria at time $t$ hours. The bacteria are growing at a rate of 50% per hour. $A\left[a,b\right]$ represents the average rate of change in the number of bacteria on the time interval $\left[a,b\right]$.

(e)

To determine

The conjecture in terms of variation for the ratio $\frac{A\left[a,b\right]}{f\left(a\right)}$ and the constant of proportionality if the function, $f\left(t\right)=300e^{\left(0.5t\right)}$ gives the number of bacteria at time $t$ hours. The bacteria are growing at a rate of 50% per hour. $A\left[a,b\right]$ represents the average rate of change in the number of bacteria on the time interval $\left[a,b\right]$.

(f)

To determine

To calculate: The values of $f\left(a\right)$, $A\left[a,b\right]$, $\frac{A\left[a,b\right]}{f\left(a\right)}$ in the table given below and by using the information given below.

The function $f\left(t\right)=800t+300$ gives the number of bacteria at time $t$ hours. The bacteria are growing at a rate of 50% per hour. $A\left[a,b\right]$ represents the average rate of change in the number of bacteria on the time interval $\left[a,b\right]$.

(g)

To determine

Justify the exponential model with function, $f\left(t\right)=300e^{\left(0.5t\right)}$ is better than the linear model with the function, $f\left(t\right)=800t+300$.