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Market Saturation You have just introduced a new model of DVD player. You predict that the market will saturate at \(2,000,000\) DVD players and that your total sales will be governed by the equation $$ \frac{d S}{d t}=\frac{1}{4} S(2-S) $$ where \(S\) is the total sales in millions of DVD players and \(t\) is measured in months. If you give away 1,000 DVD players when you first introduce them, what will \(S\) be? Sketch the graph of \(S\) as a function of \(t .\) About how long will it take to saturate the market? (See Exercise 39.)

Show by example that a second-order differential equation, one involving the second derivative \(y^{\prime \prime}\), usually has two arbitrary constants in its general solution.

Saving for Retirement You are saving for your retirement by investing $$\$ 700$$ per month in an annuity with a guaranteed interest rate of \(6 \%\) per year. With a continuous stream of investment and continuous compounding, how much will you have accumulated in the annuity by the time you retire in 45 years?

Saving for College When your first child is born, you begin to save for college by depositing $$\$ 400$$ per month in an account paying \(12 \%\) interest per year. You increase the amount you save by \(2 \%\) per year. With continuous investment and compounding, how much will have accumulated in the account by the time your child enters college 18 years later?Saving for College When your first child is born, you begin to save for college by depositing $$\$ 400$$ per month in an account paying \(12 \%\) interest per year. You increase the amount you save by \(2 \%\) per year. With continuous investment and compounding, how much will have accumulated in the account by the time your child enters college 18 years later?

Weekly sales of graphing calculators can be modeled by the equation $$ s(t)=10-t e^{-t / 20} $$ where \(s\) is the number of calculators sold per week after \(t\) weeks. How many graphing calculators (to the nearest unit) will be sold in the first 20 weeks?

Laplace Transforms The Laplace Transform \(F(x)\) of a function \(f(t)\) is given by the formula $$ F(x)=\int_{0}^{+\infty} f(t) e^{-x t} d t \quad(x>0) $$ a. Find \(F(x)\) for \(f(t)=1\) and for \(f(t)=t\). b. Find a formula for \(F(x)\) if \(f(t)=t^{n}(n=1,2,3, \ldots)\). c. Find a formula for \(F(x)\) if \(f(t)=e^{a t}(a\) constant).