When interest is compounded continuously, the amount of money increases at a rate proportional to the amount S present at time t, that is, d S / d t=r S, where r is the annual rate of interest.

Find the amount of money accrued at the end of 5 years when $5000 is deposited in a savings account drawing$5 $\frac{3}{4}$ %$ annual interest compounded continuously.

A 30-volt electromotive force is applied to an LR-series circuit in which the inductance is 0.1 henry and the resistance is 50 ohms. Find the current i(t) if i(0) = 0. Determine the current as $t \rightarrow \infty$.

a. Use a computer or calculator to plot the graph of the function $f$.

b. Plot some level curves of $f$ and compare them with the graph obtained in part (a).

$$f(x,y)=ye^{1-x^2-y^2}$$

(a) Use the first derivative and a rough sketch of f to determine the location and the values of the local maxima and minima of $f(x)=-x^{3}+3 x-2.$ Use the second derivative test as necessary to justify your answer. (b) Find the coordinates of any points of inflection.

A thermometer reading $70^{\circ} \mathrm{F}$ is placed in an oven preheated to a constant temperature. Through a glass window in the oven door, an observer records that the thermometer reads $110^{\circ} \mathrm{F}$ after $\frac{1}{2}$ minute and $145^{\circ} \mathrm{F}$ after 1 minute . How hot is the oven?

An amount of invested money is said to draw interest compounded continuously if the amount of money increases at a rate proportional to the amount present. Suppose $1000 is invested and draws interest compounded continuously, where the annual interest rate is 6%. (a) How much money will be present 10 years after the original amount was invested? (b) How long will it take the original amount of money to double?

When a vertical beam of light passes through a transparent medium, the rate at which its intensity $I$ decreases is proportional to $l(t)$. where $t$ represents the thickness of the medium (in feet). In clear seawater, the intensity 3 feet below the surface is 25% of the initiali ntensity $I_0$ of the incident beam. What is the intensity of the beam 15 feet below the surface?

When a vertical beam of light passes through a transparent medium, the rate at which its intensity I decreases is proportional to I(t), where t represents the thickness of the medium (in feet). In clear seawater, the intensity 3 feet below the surface is 25% of the initial intensity $I_{0}$ of the incident beam. What is the intensity of the beam 15 feet below the surface?

The population of bacteria in a culture grows at a rate proportional to the number of bacteria present at time t. After 3 hours it is observed that there are 400 bacteria present. After 10 hours there are 2000 bacteria present. What is the initial number of bacteria?

Find the equation of the line tangent to the graph of $x^{2}-4 y^{2}=0$ at the point (4, 2)

Initially, 100 milligrams of a radioactive substance was present. After 6 hours the mass had decreased by 3%. If the rate of decay is proportional to the amount of the substance present at time $t$, find the half-life of the radioactive substance.

The radioactive isotope of lead, Pb-209, decays at a rate proportional to the amount present at time t and has a half-life of 3.3 hours. If 1 gram of lead is present initially, how long will it take for 90% of the lead to decay?

Initially there were 100 milligrams of a radioactive substance present. After 6 hours the mass had decreased by 3%. If the rate of decay is proportional to the amount of the substance present at time t, find the amount remaining after 24 hours.

Initially, 100 milligrams of a radioactive substance was present. After 6 hours the mass had decreased by 3%. If the rate of decay is proportional to the amount of the substance present at time $t$, find the amount remaining after 24 hours.