Suppose that P dollars is invested at an annual interest rate of

$$r \times 100 \%$$

. If the accumulated interest is credited to the account at the end of the year, then the interest is said to be compounded annually; if it is credited at the end of each 6-month period, then it is said to be compounded semiannually; and if it is credited at the end of each 3-month period, then it is said to be compounded quarterly. The more frequently the interest is compounded, the better it is for the investor since more of the interest is itself earning interest. One can imagine interest to be compounded each day, each hour, each minute, and so forth. Carried to the each hour, each minute, and so forth. Carried to the instant of time; this is called continuous compounding. Thus the value A of P dollars after t years when invested at an annual rate of

$$r \times 100 \%$$

, compounded continuously, is

$$A = \lim _ { n \rightarrow + \infty } P \left( 1 + \frac { r } { n } \right) ^ { n t }$$

. Use the fact that

$$\lim _ { x \rightarrow 0 } ( 1 + x ) ^ { 1 / x } = e$$

to prove that

$$A = P e ^ { r t }$$

.

Suppose that a quantity y has an exponential growth model

$$y = y _ { 0 } e ^ { k t }$$

or an exponential decay model

$$y = y _ { 0 } e ^ { - k t }$$

, and it is known that y=

$$y_1$$

if t=

$$t_1$$

. In each case find a formula for k in terms of

$$y_0$$

,

$$y_1$$

, and

$$t_1$$

, assuming that

$$t _ { 1 } \neq 0$$

.

Review the subsection of the previous section entitled Error Propagation, and then estimate the percentage error that results in the computed age of an artifact from an r%r error in the half-life of carbon-14.

It is currently accepted that the half-life of carbon-14 might vary \pm 40 years from its nominal value of 5730 years. Does this variation make it possible that the Shroud of Turin dates to the time of Jesus of Nazareth?

Use the graph to estimate the percentage of carbon-14 that would have to have been present in the 1988 test of the Shroud of Turin for it to have been the burial shroud of Jesus of Nazareth.

In 1950, a research team digging near Folsom, New Mexico, found charred bison bones along with some leaf-shaped projectile points (called the “Folsom points”) that had been made by a Paleo-Indian hunting culture. It was clear from the evidence that the bison had been cooked and eaten by the makers of the points, so that carbon-14 dating of the bones made it possible for the researchers to determine when the hunters roamed North America. Tests showed that the bones contained between 27% and 30% of their original carbon-14. Use this information to show that the hunters lived roughly between 9000 b.c. and 8000 b.c.

In 1950, a research team digging near Folsom, New Mexico, found charred bison bones along with some leaf-shaped projectile points (called the “Folsom points”) that had been made by a Paleo-Indian hunting culture. It was clear from the evidence that the bison had been cooked and eaten by the makers of the points, so that carbon-14 dating of the bones made it possible for the researchers to determine when the hunters roamed NorthAmerica. Tests showed that the bones contained between 27% and 30% of their original carbon-14. Use this information to show that the hunters lived roughly between 9000 B.C. and 8000 B.C.

Find a formula for the tripling time of an exponential growth model.

The initial-value problem

$$\frac { d y } { d x } = - \frac { x } { y } , \quad y ( 0 ) = 1$$

has solution y(x)=___.

The initial-value problem

$$\frac{d y}{d x}=-\frac{x}{y}, \quad y(0)=1$$

has solution $y(x)=$ _______.

Solve the differential equation by the method of integrating factors.

$$2 \frac { d y } { d x } + 4 y = 1$$

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of x.

$$\left(1+x^4\right) \frac{d y}{d x}=\frac{x^3}{y}$$

The initial-value problem

$$\frac { d y } { d x } = - \frac { x } { y } , \quad y ( 0 ) = 1$$

has solution y(x)=_.

Solve the initial-value problems in the previous exercise, and use a graphing utility to confirm that the integral curves for these solutions are consistent with the sketches you obtained from the slope field.