BIOMEDICAL: Blood Flow It follows from Poiseuille's Law that blood flowing through certain arteries will encounter a resistance of $R(x)=0.25(1+x)^4$, where x is the distance (in meters) from the heart. Find the instantaneous rate of change of the resistance at:

a. 0 meters.

b. 1 meter.

The derivative $f^{\prime}(x)$ of a function is in prendles per blarg. What are the units of x ? What are the units of f ? [Hint: One is prendles and the other is blargs.]

BUSINESS: Marginal Average Cost A company can produce LCD digital alarm clocks at a cost of $6 each while fixed costs are$45. Therefore, the company's cost function is C(x) = 6x + 45.

a. Find the average cost function $A C(x)=\frac{C(x)}{x}$.

b. Find the marginal average cost function $M A C(x)$.

c. Evaluate $M A C(x)$ at x=30 and interpret your answer.

BIOMEDICAL: Drug Sensitivity (63 continued) For the reaction function given in Exercise 63, find the dose at which the sensitivity is 25 . [Hint: On a graphing calculator, enter the reaction function $y_1=4 x \sqrt{11+0.5 x}$, and use NDERIV to define $y_2$ to be the derivative of $y_1$. Then graph $y_2$ on the window $[0,140]$ by $[0,50]$ and TRACE along $y_2$ to find the x-coordinate (rounded to the nearest whole number) at which the y-coordinate is 25 . You may have to ZOOM IN to find the correct x-value.]

For the piecewise linear function:

$$f(x)= \begin{cases}5-x & \text { if } x \leq 3 \\ x-1 & \text { if } x>3\end{cases}$$

a. Draw its graph (by hand or using a graphing calculator).

b. Find the limits as x approaches 3 from the left and from the right.

c. Is it continuous at x=3 ? If not, indicate the first of the three conditions in the definition of continuity that is violated.

GENERAL: Population The population of a city x years from now is predicted to be $P(x)=\sqrt[4]{x^2+1}$ million people for $1 \leq x \leq 5$. Find when the population will be growing at the rate of a quarter of a million people per year. [Hint: On a graphing calculator, enter the given population function in $y_1$, use NDERIV to define $y_2$ to be the derivative of $y_1$, and graph both on the window [1,5] by [0,3]. Then TRACE along $y_2$ to find the x-coordinate (rounded to the nearest tenth of a unit) at which the y-coordinate is 0.25. You may have to ZOOM IN to find the correct x-value.]

BUSINESS: Marginal Average Cost A company can produce computer flash memory devices at a cost of $6 each, while fixed costs are$50 per day. Therefore, the company's cost function is C(x) = 6x + 50. Find the average cost function

$$A C(x)=\frac{C(x)}{x} .$$

a. Graph the average cost function $A C(x)$ that you have found on the window [0,50] by [0,50]. TRACE along the average cost curve to see how the average cost falls as the number of devices increases. Note that although average cost falls, it does so more slowly as the number of units increases.

b. Evaluate MAC(x) at x = 25 and interpret your answer. To check your answer, use the " $d y / d x^n$ or SLOPE feature of your calculator to find the slope of the average cost curve at x=25. This slope gives the rate of change of the cost. Find the slope (rate of change) for other x-values to see that the rate of change of average cost tends toward zero (the law of diminishing returns).

Velocity A rocket rises $s(t)=8 t^{5 / 2}$ feet in $t$ seconds. Find its velocity and acceleration after 25 seconds.

For a function $f(x)$, if $f$ is in widgets and $x$ is in blivets, what are the units of the derivative $f^{\prime}(x)$, widgets per blivet or blivets per widget?

The tuition for a public college is approximated by the function $f(x)=400 x^2+500 x+2700$, where $x$ is the number of five-year intervals since the academic year 1995-96 .

a. Use this function to predict tuition in the academic year 2020-21.

b. Find the derivative of this function for the x-value that you used in part (a) and interpret it as a rate of change in the proper units.

c. From your answer to part (b), estimate how rapidly tuition will be increasing per year in 2020-21.

Population The population of a city t years from now is predicted to be $P(t)=0.25 t^3-3 t^2+5 t+200$ thousand people. Find $P(10), P^{\prime}(10)$, and $P^{\prime \prime}(10)$ and interpret your answers.

When we calculate the derivative using the formula $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$, we eventually evaluate the limit by direct substitution of h=0. Why don't we just substitute h=0 into the formula to begin with?

The tuition for a private college is approximated by the function $f(x)=650 x^2+3000 x+12,000$, where x is the number of five-year intervals since the academic year 1995-96 .

a. Use this function to predict tuition in the academic year 2020-21.

b. Find the derivative of this function for the $x$-value that you used in part (a) and interpret it as a rate of change in the proper units.

c. From your answer to part (b), estimate how rapidly tuition will be increasing per year in 2020-21.

ENVIRONMENTAL SCIENCE: Water Purification If the cost of purifying a gallon of water to a purity of x percent is

$$C(x)=\frac{100}{100-x} \text { cents for } 50 \leq x<100$$

a. Use a graphing calculator to graph the cost function C(x) on the window [50,100] by [0,20]. TRACE along the curve to see how rapidly costs increase for purity ( x-coordinate) increasing from 50 to near 100.

b. Evaluate this rate of change for a purity of 95% & 98% and interpret your answer. To check your answers, use the " $d y / d x^{\prime \prime}$ or SLOPE feature of your calculator to find the slope of the cost curve at $x=95$ and at $x=98$. Note that further purification becomes increasingly expensive at higher purity levels.