Chapter 5

Verify Property 2 of the definition of a probability density function over the given interval. $$ f(x)=\frac{1}{e-1} e^{x}, \quad[0,1] $$

Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. $$ \int_{1}^{\infty} \frac{d x}{x^{4}} $$

\(D(x)\) 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. \(D(x)=1000-10 x, \quad S(x)=250+5 x\)

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. $$ f(x)=\frac{1}{\ln 4} \cdot \frac{1}{x}, \quad[0.8,3.2] $$

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

Find the volume generated by rotating the area bounded by the graphs of each set of equations around the \(x\) -axis. $$ y=\frac{2}{\sqrt{x}}, x=4, x=9 $$

Find the accumulated future value of each continuous income stream at rate \(\mathrm{R}(t),\) for the given time \(\mathrm{T}\) and interest rate \(k\) compounded continuously. $$ R(t)=\$ 50,000, \quad T=22 \mathrm{yr}, \quad k=5 \% $$

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

Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. $$ \int_{0}^{\infty} \frac{d x}{2+x} $$

\(D(x)\) 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. \(D(x)=7-x,\) for \(0 \leq x \leq 7 ; \quad S(x)=2 \sqrt{x+1}\)

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. $$ f(x)=\frac{1}{\ln 5} \cdot \frac{1}{x}, \quad[1.5,7.5] $$

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

Find the accumulated future value of each continuous income stream at rate \(\mathrm{R}(t),\) for the given time \(\mathrm{T}\) and interest rate \(k\) compounded continuously. $$ R(t)=\$ 125,000, \quad T=20 \mathrm{yr}, \quad k=6 \% $$

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

Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. $$ \int_{0}^{\infty} \frac{4 d x}{3+x} $$

\(D(x)\) 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. \(D(x)=5-x,\) for \(0 \leq x \leq 5 ; \quad S(x)=\sqrt{x+7}\)

Let \(y^{\prime}+4 y=0\) a) Show that \(y=e^{-4 x}\) is a solution of this differential equation. b) Show that \(y=C e^{-4 x}\) is a solution, where \(C\) is a constant.

Find the accumulated future value of each continuous income stream at rate \(\mathrm{R}(t),\) for the given time \(\mathrm{T}\) and interest rate \(k\) compounded continuously. $$ R(t)=\$ 400,000, \quad T=20 \mathrm{yr}, \quad k=4 \% $$

Find the volume generated by rotating the area bounded by the graphs of each set of equations around the \(x\) -axis. $$ y=5, x=1, x=3 $$

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

Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. $$ \int_{2}^{\infty} 4 x^{-2} d x $$

\(D(x)\) 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. \(D(x)=\frac{1800}{\sqrt{x+1}}, \quad S(x)=2 \sqrt{x+1}\)

Let \(x\) be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. $$ P(0 \leq x \leq 2.13) $$

Let \(y^{\prime}-3 x^{2} y=0\) a) Show that \(y=e^{x^{3}}\) is a solution of this differential equation. b) Show that \(y=C e^{x^{3}}\) is a solution, where \(C\) is a constant.

Find the accumulated future value of each continuous income stream at rate \(\mathrm{R}(t),\) for the given time \(\mathrm{T}\) and interest rate \(k\) compounded continuously. $$ R(t)=\$ 50,000, \quad T=22 \mathrm{yr}, \quad k=2.75 \% $$

Find the volume generated by rotating the area bounded by the graphs of each set of equations around the \(x\) -axis. $$ y=4, x=1, x=3 $$

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

Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. $$ \int_{2}^{\infty} 7 x^{-2} d x $$

\(D(x)\) 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. \(D(x)=\frac{100}{\sqrt{x}}, \quad S(x)=\sqrt{x}\)

Let \(x\) be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. $$ P(-1.37 \leq x \leq 0) $$

Find the accumulated present value of each continuous income stream at rate \(R(t),\) for the given time \(T\) and interest rate \(k\) compounded continuously. $$ R(t)=\$ 250,000, \quad T=18 \mathrm{yr}, \quad k=4 \% $$

Let \(y^{\prime \prime}-y^{\prime}-30 y=0\) a) Show that \(y=e^{6 x}\) is a solution of this differential equation. b) Show that \(y=e^{-5 x}\) is a solution. c) Show that \(y=C_{1} e^{6 x}+C_{2} e^{-5 x}\) is a solution, where \(C_{1}\) and \(C_{2}\) are constants.

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

Find \(k\) such that each function is a probability density function over the given interval. Then write the probability density function. $$ f(x)=k x, \quad[1,4] $$

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

\(D(x)\) 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. \(D(x)=13-x,\) for \(0 \leq x \leq 13 ; \quad S(x)=\sqrt{x+17}\)

Let \(x\) be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. $$ P(-2.01 \leq x \leq 0) $$

Find the accumulated present value of each continuous income stream at rate \(R(t),\) for the given time \(T\) and interest rate \(k\) compounded continuously. $$R(t)=\$ 425,000, \quad T=15 \mathrm{yr}, \quad k=3.5 \%$$

Let \(y^{\prime \prime}-7 y^{\prime}-44 y=0\). a) Show that \(y=e^{11 x}\) is a solution of this differential equation. b) Show that \(y=e^{-4 x}\) is a solution. c) Show that \(y=C_{1} e^{11 x}+C_{2} e^{-4 x}\) is a solution, where \(C_{1}\) and \(C_{2}\) are constants.

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

Find \(k\) such that each function is a probability density function over the given interval. Then write the probability density function. $$ f(x)=k x, \quad[2,5] $$

\(D(x)\) 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. \(D(x)=(x-4)^{2}, \quad S(x)=x^{2}+2 x+8\)

Find the accumulated present value of each continuous income stream at rate \(R(t),\) for the given time \(T\) and interest rate \(k\) compounded continuously. $$R(t)=\$ 800,000, \quad T=20 \mathrm{yr}, \quad k=2.3 \%$$

(a) find the general solution of each differential equation, and (b) check the solution by substituting into the differential equation. \(\frac{d M}{d t}=0.05 M\)

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

Find \(k\) such that each function is a probability density function over the given interval. Then write the probability density function. $$ f(x)=k x^{2}, \quad[-1,1] $$

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

Beth enjoys skydiving and is willing to pay \(p\) dollars per jump for \(x\) jumps, where \(p=D(x)=7.5 x^{2}-60.5 x+254\) a) Find Beth's consumer surplus if she makes 2 jumps. b) Suppose the supply function for Aero Skydiving Center is given by \(p=S(x)=15 x+95 .\) Find the producer surplus if the center sells Beth 2 jumps. c) Find the equilibrium point and the consumer and producer surpluses at this point. Assume that Beth makes no more than 5 jumps. d) Explain what the equilibrium point represents to both Beth and Aero Skydiving Center.

Let \(x\) be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. $$ P(-1.89 \leq x \leq 0.45) $$

Find the accumulated present value of each continuous income stream at rate \(R(t),\) for the given time \(T\) and interest rate \(k\) compounded continuously. $$R(t)=\$ 520,000, \quad T=25 \mathrm{yr}, \quad k=6 \%$$