91Ó°ÊÓ

In Exercises, find the derivative of the function. $$ y=\ln \sqrt{x-4} $$

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

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? $$ \frac{d y}{d t}=-4 y, \quad y=30 \text { when } t=0 $$

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? $$ \frac{d y}{d t}=5.2 y, \quad y=18 \text { when } t=0 $$

In Exercises, find the derivative of the function. $$ g(x)=e^{\sqrt{x}} $$

In Exercises, evaluate the function. If necessary, use a graphing utility, rounding your answers to three decimal places. \(g(x)=1.075^{x}\) (a) \(g(1.2)\) (b) \(g(180)\) (c) \(g(60)\) (d) \(g(12.5)\)

In Exercises, find the derivative of the function. $$ f(x)=\left(x^{2}+1\right) e^{4 x} $$

After \(t\) years, the remaining mass \(y\) (in grams) of 16 grams of a radioactive element whose half-life is 30 years is given by \(y=16\left(\frac{1}{2}\right)^{n / 30}, \quad t \geq 0\)

In Exercises, sketch the graph of the function. $$ h(x)=e^{x-3} $$

After \(t\) years, the remaining mass \(y\) (in grams) of 23 grams of a radioactive element whose halflife is 45 years is given by \(y=23\left(\frac{1}{2}\right)^{1 / 45}, \quad t \geq 0\) How much of the initial mass remains after 150 years?