Chapter 5

For each probability density function, over the given interval, find \(\mathrm{E}(x), \mathrm{E}\left(x^{2}\right),\) the mean, the variance, and the standard deviation. $$ f(x)=\frac{1}{5}, \quad[3,8] $$

Find the general solution and three particular solutions. \(y^{\prime}=10 x^{2}\)

Find the volume generated by rotating the area bounded by the graphs of each set of equations around the \(x\) -axis. $$ y=x, x=0, x=1 $$

Find the future value \(P\) of each amount \(P_{0}\) invested for time period t at interest rate \(k\), compounded continuously. $$ P_{0}=\$ 100,000, \quad t=6 \mathrm{yr}, \quad k=3 \% $$

Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. $$ \int_{5}^{\infty} \frac{d x}{x^{2}} $$

\(D(x)\) is the price, in dollars per unit, that consumers will pay for \(x\) units of an item, and \(S(x)\) is the price, in dollars per unit, that producers will accept for \(x\) units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. \(D(x)=-3 x+7, \quad S(x)=2 x+2\)

For each probability density function, over the given interval, find \(\mathrm{E}(x), \mathrm{E}\left(x^{2}\right),\) the mean, the variance, and the standard deviation. $$ f(x)=\frac{1}{4}, \quad[3,7] $$

Find the general solution and three particular solutions. \(y^{\prime}=5 x^{6}\)

Find the volume generated by rotating the area bounded by the graphs of each set of equations around the \(x\) -axis. $$ y=x, x=0, x=2 $$

Verify Property 2 of the definition of a probability density function over the given interval. $$ f(x)=\frac{1}{4} x, \quad[1,3] $$

Find the future value \(P\) of each amount \(P_{0}\) invested for time period t at interest rate \(k\), compounded continuously. $$ P_{0}=\$ 55,000, \quad t=8 \mathrm{yr}, \quad k=4 \% $$

Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. $$ \int_{3}^{\infty} \frac{d x}{x^{2}} $$

\(D(x)\) is the price, in dollars per unit, that consumers will pay for \(x\) units of an item, and \(S(x)\) is the price, in dollars per unit, that producers will accept for \(x\) units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. \(D(x)=-\frac{5}{6} x+9, \quad S(x)=\frac{1}{2} x+1\)

For each probability density function, over the given interval, find \(\mathrm{E}(x), \mathrm{E}\left(x^{2}\right),\) the mean, the variance, and the standard deviation. $$ f(x)=\frac{1}{8} x, \quad[0,4] $$

Find the general solution and three particular solutions. \(y^{\prime}=2 e^{-x}+x\)

Find the volume generated by rotating the area bounded by the graphs of each set of equations around the \(x\) -axis. $$ y=2 x, x=1, x=3 $$

Find the future value \(P\) of each amount \(P_{0}\) invested for time period t at interest rate \(k\), compounded continuously. $$ P_{0}=\$ 140,000, \quad t=9 \mathrm{yr}, \quad k=5.8 \% $$

Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. $$ \int_{3}^{\infty} \frac{d x}{x} $$

\(D(x)\) is the price, in dollars per unit, that consumers will pay for \(x\) units of an item, and \(S(x)\) is the price, in dollars per unit, that producers will accept for \(x\) units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. \(D(x)=(x-3)^{2}, \quad S(x)=x^{2}+2 x+1\)

For each probability density function, over the given interval, find \(\mathrm{E}(x), \mathrm{E}\left(x^{2}\right),\) the mean, the variance, and the standard deviation. $$ f(x)=\frac{2}{9} x, \quad[0,3] $$

Find the general solution and three particular solutions. \(y^{\prime}=e^{4 x}-x+2\)

Find the volume generated by rotating the area bounded by the graphs of each set of equations around the \(x\) -axis. $$ y=\sqrt{x}, x=1, x=4 $$

Verify Property 2 of the definition of a probability density function over the given interval. $$ f(x)=\frac{1}{5}, \quad[3,8] $$

Find the future value \(P\) of each amount \(P_{0}\) invested for time period t at interest rate \(k\), compounded continuously. $$ P_{0}=\$ 88,000, \quad t=13 \mathrm{yr}, \quad k=4.7 \% $$

Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. $$ \int_{4}^{\infty} \frac{d x}{x} $$

\(D(x)\) is the price, in dollars per unit, that consumers will pay for \(x\) units of an item, and \(S(x)\) is the price, in dollars per unit, that producers will accept for \(x\) units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. \(D(x)=(x-4)^{2}, \quad S(x)=x^{2}+2 x+6\)

For each probability density function, over the given interval, find \(\mathrm{E}(x), \mathrm{E}\left(x^{2}\right),\) the mean, the variance, and the standard deviation. $$ f(x)=\frac{2}{3} x, \quad[1,2] $$

Find the general solution and three particular solutions. \(y^{\prime}=\frac{4}{x}-\frac{1}{x^{2}}\)

Find the volume generated by rotating the area bounded by the graphs of each set of equations around the \(x\) -axis. $$ y=e^{x}, x=-2, x=5 $$

Verify Property 2 of the definition of a probability density function over the given interval. $$ f(x)=\frac{3}{26} x^{2}, \quad[1,3] $$

Find the present value \(P_{0}\) of each amount \(P\) due \(t\) years in the future and invested at interest rate \(k\), compounded continuously. $$ P=\$ 100,000, \quad t=6 \mathrm{yr}, \quad k=3 \% $$

Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. $$ \int_{0}^{\infty} 3 e^{-3 x} d x $$

\(D(x)\) is the price, in dollars per unit, that consumers will pay for \(x\) units of an item, and \(S(x)\) is the price, in dollars per unit, that producers will accept for \(x\) units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. \(D(x)=(x-8)^{2}, \quad S(x)=x^{2}\)

For each probability density function, over the given interval, find \(\mathrm{E}(x), \mathrm{E}\left(x^{2}\right),\) the mean, the variance, and the standard deviation. $$ f(x)=\frac{1}{4} x, \quad[1,3] $$

Find the general solution and three particular solutions. \(y^{\prime}=\frac{3}{x}+x^{2}-x^{4}\)

Find the volume generated by rotating the area bounded by the graphs of each set of equations around the \(x\) -axis. $$ y=e^{x}, x=-3, x=2 $$

Verify Property 2 of the definition of a probability density function over the given interval. $$ f(x)=\frac{3}{64} x^{2}, \quad[0,4] $$

Find the present value \(P_{0}\) of each amount \(P\) due \(t\) years in the future and invested at interest rate \(k\), compounded continuously. $$ P=\$ 100,000, \quad t=8 \mathrm{yr}, \quad k=4 \% $$

Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. $$ \int_{0}^{\infty} 4 e^{-4 x} d x $$

\(D(x)\) is the price, in dollars per unit, that consumers will pay for \(x\) units of an item, and \(S(x)\) is the price, in dollars per unit, that producers will accept for \(x\) units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. \(D(x)=(x-6)^{2}, \quad S(x)=x^{2}\)

For each probability density function, over the given interval, find \(\mathrm{E}(x), \mathrm{E}\left(x^{2}\right),\) the mean, the variance, and the standard deviation. $$ f(x)=\frac{3}{2} x^{2}, \quad[-1,1] $$

Show that \(y=x \ln x+3 x-2\) is a solution of \(y^{\prime \prime}-\frac{1}{x}=0\)

Find the volume generated by rotating the area bounded by the graphs of each set of equations around the \(x\) -axis. $$ y=\frac{1}{x}, x=1, x=4 $$

Find the present value \(P_{0}\) of each amount \(P\) due \(t\) years in the future and invested at interest rate \(k\), compounded continuously. $$ P=\$ 1,000,000, \quad t=25 \mathrm{yr}, \quad k=6 \% $$

Verify Property 2 of the definition of a probability density function over the given interval. $$ f(x)=\frac{1}{x}, \quad[1, e] $$

Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. $$ \int_{1}^{\infty} \frac{d x}{x^{3}} $$

\(D(x)\) is the price, in dollars per unit, that consumers will pay for \(x\) units of an item, and \(S(x)\) is the price, in dollars per unit, that producers will accept for \(x\) units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. \(D(x)=8800-30 x, \quad S(x)=7000+15 x\)

Show that \(y=x \ln x-5 x+7\) is a solution of \(y^{\prime \prime}-\frac{1}{x}=0\)

Find the volume generated by rotating the area bounded by the graphs of each set of equations around the \(x\) -axis. $$ y=\frac{1}{x}, x=1, x=3 $$

Find the present value \(P_{0}\) of each amount \(P\) due \(t\) years in the future and invested at interest rate \(k\), compounded continuously. $$ P=\$ 2,000,000, \quad t=20 \mathrm{yr}, \quad k=3.5 \% $$