Q. 91 - Calculus

Question

To save up for a car, you take a job working 10 hours a week at the school library. For the first six weeks, the library pays you $\backslash (8.00$ an hour. After that, you earn $\backslash )11.50$ an hour. You put all of the money you earn each week into a savings account. On the day you start to work your savings. account already holds $\$200.00$. Let S(t) be the function that describes the amount in your savings account t weeks after your library job begins.

(a) Find the values of $S\left(3\right), S\left(6\right), S\left(8\right), S\text{'}\left(3\right), S\text{'}\left(6\right) and S\text{'}\left(8\right)$,  if possible, and describe their meanings in practical terms. If it is not possible to find one or more of these values, explain why.

(b) Write an equation for the function $S\left(t\right)$. (Hint: $S\left(t\right)$ will be a piecewise-defined function.) Be sure that your equation correctly produces the values you calculated in part (a).

(c) Sketch a labeled graph of $S\left(t\right)$. By looking at this graph, determine whether $S\left(t\right)$ is continuous and whether $S\left(t\right)$ is differentiable. Explain the practical significance of your answers.

(d) Show algebraically that $S\left(t\right)$ is a continuous function, but not a differentiable function.

Step-by-Step Solution

Answer

Ans: Part (a).

$$\begin{gathered} S\left(3\right)=\$440 \\ S\left(6\right)=\$680 \\ S\left(8\right)=\$910 \\ S\text{'}\left(3\right)=\$80 \\ S\text{'}\left(8\right)=\$115 \\ S\text{'}\left(6\right) =undefined \end{gathered}$$

Part (b). The equation for the function $S\left(t\right)$ is $S\left(t\right)=\left\{\begin{array}{ll}200+8\left(10\right)t& t\le 6\\ 680+11.5\left(10\right)(t-6)& t>6\end{array}\right.$

Part (c). The graph is clear that it is continuous, but not a differentiable function.

Part (d). It is clear that the function is a continuous function, but not a differentiable function.

Step 1. Given information:

It is given that the job is for 10 hours a week at the school library. For the first six weeks the library pays$\$8$ an hour and after that $\$11.50$ an hour. You put all of the money you earn each week into a savings account. On the day you start work your savings account already holds $\$ 200$. Let $S\left(t\right)$ be the function that describes the amount in your saving account $t$ weeks after your library job begins.

Step 2. Solving part (a):

The functions and values are

$$\begin{gathered} S\left(t\right) =200+8\left(10\right)t \\ S\left(3\right) =200+8\left(10\right)3 \\ =\$440 \\ S\left(6\right) =200+8\left(10\right)6 \\ =\$680 \\ S\left(8\right) =200+8\left(10\right)6+11.50\left(10\right)t \\ =200+8\left(10\right)6+11.50\left(10\right)2 \\ =\$910 \\ S\text{'}\left(t\right) =8\left(10\right) \\ =80 \\ S\text{'}\left(3\right) =\$80 \\ S\text{'}\left(8\right) =11.50\left(10\right) \\ =\$115 \\ S\text{'}\left(6\right) =undefined \end{gathered}$$

Step 3. Solving part (b):

$$\begin{gathered} From the given information, the equation of the function is \\ S\left(t\right)=\left\{\begin{array}{ll}200+8\left(10\right)t& t\le 6\\ 680+11.5\left(10\right)(t-6)& t>6\end{array}\right. \end{gathered}$$

Step 4. Solving part (c):

The graph is clear that it is continuous, but not a differentiable function

Step 5. Solving part (d):

$$\begin{gathered} Here \\ \underset{x\to 6}{\lim }S\left(t\right)=680 \\ But, \\ \underset{x\to 0^{+}}{\lim }\frac{S(6+h)-S\left(6\right)}{h}=115 \\ \underset{x\to 0^{-}}{\lim }\frac{S(6+h)-S\left(6\right)}{h}=80 \end{gathered}$$

Thus it is clear that the function is a continuous function, but not a differentiable function