Chapter 10

In Exercises, find the slope of the tangent line to the exponential function at the point \((0,1)\).

In Exercises, write the logarithmic equation as an exponential equation, or vice versa. $$ \ln 2=0.6931 \ldots $$

In Exercises, use the properties of exponents to simplify the expression. (a) \(\left(e^{3}\right)\left(e^{4}\right)\) (b) \(\left(e^{3}\right)^{4}\) (c) \(\left(e^{3}\right)^{-2}\) (d) \(e^{0}\)

In Exercises, evaluate each expression. (a) \(5\left(5^{3}\right)\) (b) \(27^{2 / 3}\) (c) \(64^{3 / 4}\) (d) \(81^{1 / 2}\) (e) \(25^{3 / 2}\) (f) \(32^{2 / 5}\)

In Exercises, find the slope of the tangent line to the exponential function at the point \((0,1)\).

In Exercises, write the logarithmic equation as an exponential equation, or vice versa. $$ \ln 9=2.1972 \ldots $$

In Exercises, use the properties of exponents to simplify the expression. (a) \(\left(\frac{1}{e}\right)^{-2}\) (b) \(\left(\frac{e^{5}}{e^{2}}\right)^{-1}\) (c) \(\frac{e^{5}}{e^{3}}\) (d) \(\frac{1}{e^{-3}}\)

In Exercises, evaluate each expression. (a) \(\left(\frac{1}{5}\right)^{3}\) (b) \(\left(\frac{1}{8}\right)^{1 / 3}\) (c) \(64^{2 / 3}\) (d) \(\left(\frac{5}{8}\right)^{2}\) (e) \(100^{3 / 2}\) (f) \(4^{5 / 2}\)

In Exercises, find the slope of the tangent line to the exponential function at the point \((0,1)\).

In Exercises, write the logarithmic equation as an exponential equation, or vice versa. $$ \ln 0.2=-1.6094 \ldots $$

In Exercises, use the properties of exponents to simplify the expression. (a) \(\left(5^{2}\right)\left(5^{3}\right)\) (b) \(\left(5^{2}\right)\left(5^{-3}\right)\) (c) \(\left(5^{2}\right)^{2}\) (d) \(5^{-3}\)

In Exercises, find the slope of the tangent line to the exponential function at the point \((0,1)\).

In Exercises, write the logarithmic equation as an exponential equation, or vice versa. $$ \ln 0.05=-2.9957 \ldots $$

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

In Exercises, find the derivative of the function. $$ y=e^{5 x} $$

In Exercises, write the logarithmic equation as an exponential equation, or vice versa. $$ e^{0}=1 $$

In Exercises, use the properties of exponents to simplify the expression. (a) \(\frac{5^{3}}{25^{2}}\) (b) \(\left(9^{2 / 3}\right)(3)\left(3^{2 / 3}\right)\) (c) \(\left[\left(25^{1 / 2}\right)\left(5^{2}\right)\right]^{1 / 3}\) (d) \(\left(8^{2}\right)\left(4^{3}\right)\)

In Exercises, find the derivative of the function. $$ f(x)=\ln 2 x $$

In Exercises, write the logarithmic equation as an exponential equation, or vice versa. $$ e^{2}=7.3891 \ldots $$

In Exercises, find the derivative of the function. $$ y=e^{1-x} $$

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

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

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, evaluate the function. If necessary, use a graphing utility, rounding your answers to three decimal places. \(g(x)=1.05^{x}\) (a) \(g(-2)\) (b) \(g(120)\) (c) \(g(12)\) (d) \(g(5.5)\)

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. $$ f(x)=\left(x^{2}+1\right) e^{4 x} $$

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

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, find the derivative of the function. $$ y=4 x^{3} e^{-x} $$

In Exercises, sketch the graph of the function. $$ f(x)=e^{2 x} $$

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?

In Exercises, find the derivative of the function. $$ f(x)=\frac{2}{\left(e^{x}+e^{-x}\right)^{3}} $$

In Exercises, sketch the graph of the function. $$ y=\ln (x-1) $$

In Exercises, sketch the graph of the function. $$ g(x)=e^{1-x} $$

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

In Exercises, sketch the graph of the function. $$ y=\ln |x| $$

In Exercises, sketch the graph of the function. $$ j(x)=e^{-x+2} $$

In Exercises, find the derivative of the function. $$ y=x e^{x}-4 e^{-x} $$

In Exercises, sketch the graph of the function. $$ y=\ln 2 x $$