Find the average rate of change of each function on the given interval. g(x) = 3x2 – 8x + 2; [4, 8]

Approximate to the nearest hundredth the relative or absolute extrema of each function. State the x-value(s) where they occur. $f(x)=-x^{5}+3 x^{2}+x-1$

a) A cup of coffee has temperature $95^{\circ} \mathrm{C}$ and takes 30 minutes to cool to $61^{\circ} \mathrm{C}$ in a room with temperature $20^{\circ} \mathrm{C}$. Show that the temperature of the coffee after minutes is $T(t)=20+75 e^{-k t}$ where $k \approx 0.02$.

Breathing is cyclic and a full respiratory cycle from the beginning of inhalation to the end of exhalation takes about 5 s. The maximum rate of air flow into the lungs is about 0.5 L/s. This explains, in part, why the function $f(t) = \frac{1}{2}\sin(2\pi t/5)$ has often been used to model the rate of air flow into the lungs. Use this model to find the volume of inhaled air in the lungs at time $t$.

Breathing is cyclic and a full respiratory cycle from the beginning of inhalation to the end of exhalation takes about 5 seconds. The function $f(t)=\frac{1}{2} \sin \left(\frac{2 \pi t}{5}\right)$ has often been used to model the rate of air flow into the lungs, in L/s. Use this model to compute the average volume of inhaled air in the lungs in one respiratory cycle.

Breathing is cyclic and a full respiratory cycle from the beginning of inhalation to the end of exhalation takes about 5s. The maximum rate of air flow into the lungs is about $0.5 \mathrm { L } / \mathrm { s }$. This explains, in part, why the function.

$$f ( t ) = \dfrac { 1 } { 2 } \sin ( 2 \pi t / 5 )$$

has often been used to model the rate of air flow into the lungs. Use this model to find the volume of inhaled air in the lungs at time $t$.

For a person at rest, the velocity v (in liters per second) of airflow during a respiratory cycle (the time from the beginning of one breath to the beginning of the next) is given by v = 0.85 sin πt / 3 where t is the time (in seconds). Find the time for one full respiratory cycle.

Breathing is cyclic and a full respiratory cycle from the beginning of inhalation to the end of exhalation takes about 5 seconds. The function $f(t)=\frac{1}{2} \sin \left(\frac{2 \pi t}{5}\right)$ has often been used to model the rate of air flow into the lungs, in $\mathrm{L} / \mathrm{s}$. Use this model to compute the average volume of inhaled air in the lungs in one respiratory cycle.

A cup of coffee has temperature 95 °C and takes 30 minutes to cool to 61 °C in a room with temperature 20 °C. Use Newton’s Law of Cooling to show that the temperature of the coffee after t minutes is T(t) = 20 + 75e^-kt where k ≈ 0.02.

What is the average temperature of the coffee during the first half hour?

Explain the relationship between the slope and the derivative of $f(x)$ at $x=a$ .

The temperature (in ${ }^{\circ} \mathrm{F}$ ) in a certain city $t$ hours after 9 A.M. was approximated by the function

$$T(t)=50+14 \sin \frac{\pi t}{12}$$

Find the average temperature during the period from 9 A.M. to 9 P.M.

Find c in the given interval such that $f_{\text {avg }}=f(c)$.

$$f(x)=2 x e^{-x^{2}}, \quad[0,2]$$

Find the average value of the function on the given interval. f(t)=e^sint*cost, [0, pi/2]