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Chapter 7: Problem 31
Evaluate the following integrals. $$\int \frac{t^{3}-2}{t+1} d t$$
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The heights of U.S. men are normally distributed with a mean of 69 inches and
a standard deviation of 3 inches. This means that the fraction of men with a
height between \(a\) and \(b\) (with \(a
\(A\) powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function \(f(t)\) the Laplace transform is a new function \(F(s)\) defined by $$F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t$$ where we assume that \(s\) is a positive real number. For example, to find the Laplace transform of \(f(t)=e^{-t},\) the following improper integral is evaluated: $$F(s)=\int_{0}^{\infty} e^{-s t} e^{-t} d t=\int_{0}^{\infty} e^{-(s+1) t} d t=\frac{1}{s+1}$$ Verify the following Laplace transforms, where a is a real number. $$f(t)=\cos a t \longrightarrow F(s)=\frac{s}{s^{2}+a^{2}}$$
A differential equation of the form \(y^{\prime}(t)=F(y)\) is said to be autonomous (the function \(F\) depends only on \(y\) ). The constant function \(y=y_{0}\) is an equilibrium solution of the equation provided \(F\left(y_{0}\right)=0\) (because then \(y^{\prime}(t)=0,\) and the solution remains constant for all \(t\) ). Note that equilibrium solutions correspond to horizontal line segments in the direction field. Note also that for autonomous equations, the direction field is independent of \(t\). Consider the following equations. a. Find all equilibrium solutions. b. Sketch the direction field on either side of the equilibrium solutions for \(t \geq 0\). c. Sketch the solution curve that corresponds to the initial condition \(y(0)=1\). $$y^{\prime}(t)=\sin y$$
Use the indicated methods to solve the following problems with nonuniform grids. A hot-air balloon is launched from an elevation of 5400 ft above sea level. As it rises, its vertical velocity is computed using a device (called a variometer) that measures the change in atmospheric pressure. The vertical velocities at selected times are shown in the table (with units of \(\mathrm{ft} / \mathrm{min}\) ). $$\begin{array}{|l|c|c|c|c|c|c|c|} \hline t \text { (min) } & 0 & 1 & 1.5 & 3 & 3.5 & 4 & 5 \\ \hline \begin{array}{l} \text { Velocity } \\ \text { (ft/min) } \end{array} & 0 & 100 & 120 & 150 & 110 & 90 & 80 \\ \hline \end{array}$$ a. Use the Trapezoid Rule to estimate the elevation of the balloon after five minutes. Remember that the balloon starts at an elevation of \(5400 \mathrm{ft}\) b. Use a right Riemann sum to estimate the elevation of the balloon after five minutes. c. A polynomial that fits the data reasonably well is $$g(t)=3.49 t^{3}-43.21 t^{2}+142.43 t-1.75$$ Estimate the elevation of the balloon after five minutes using this polynomial.
The region bounded by \(f(x)=(4-x)^{-1 / 3}\) and the \(x\) -axis on the interval [0,4) is revolved about the \(y\) -axis. The region bounded by \(f(x)=(x+1)^{-3 / 2}\) and the \(x\) -axis on the interval (-1,1] is revolved about the line \(y=-1\).
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