91Ó°ÊÓ

In Exercises, use the properties of exponents to simplify the expression. (a) \(\left(4^{3}\right)\left(4^{2}\right)\) (b) \(\left(\frac{1}{4}\right)^{2}\left(4^{2}\right)\) (c) \(\left(4^{6}\right)^{1 / 2}\) (d) \(\left[\left(8^{-1}\right)\left(8^{2 / 3}\right)\right]^{3}\)

In Exercises, write the logarithmic equation as an exponential equation, or vice versa. $$ e^{-3}=0.0498 \ldots $$

In Exercises, evaluate the function. If necessary, use a graphing utility, rounding your answers to three decimal places. \(f(x)=2^{x-1}\) (a) \(f(3)\) (b) \(f\left(\frac{1}{2}\right)\) (c) \(f(-2)\) (d) \(f\left(-\frac{3}{2}\right)\)

In Exercises, find the derivative of the function. $$ y=e^{-x^{2}} $$

In Exercises, use the given information to write an equation for \(y\). Confirm your result analytically by showing that the function satisfies the equation \(d y / d t=C y .\) Does the function represent exponential growth or exponential decay? $$ \frac{d y}{d t}=2 y, \quad y=10 \text { when } t=0 $$

In Exercises, write the logarithmic equation as an exponential equation, or vice versa. $$ e^{0.25}=1.2840 . $$

In Exercises, evaluate the function. If necessary, use a graphing utility, rounding your answers to three decimal places. \(f(x)=3^{x+2}\) (a) \(f(-4)\) (b) \(f\left(-\frac{1}{2}\right)\) (c) \(f(2)\) (d) \(f\left(-\frac{5}{2}\right)\)

In Exercises, use the given information to write an equation for \(y\). Confirm your result analytically by showing that the function satisfies the equation \(d y / d t=C y .\) Does the function represent exponential growth or exponential decay? $$ \frac{d y}{d t}=-\frac{2}{3} y, \quad y=20 \text { when } t=0 $$

In Exercises, find the derivative of the function. $$ f(x)=e^{1 / x} $$

In Exercises, find the derivative of the function. $$ f(x)=e^{-1 / x^{2}} $$