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Chapter 3: Q 3.6-11E (page 130)
Use the improved Euler’s method with tolerance to approximate the solution to ,at t= 1. For a tolerance of , use a stopping procedure based on the absolute error.
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A cup of hot coffee initially at 95°C cools to 80°C in 5 min while sitting in a room of temperature 21°C. Using just Newton’s law of cooling, determine when the temperature of the coffee will be a nice 50°C.
The Taylor method of order 2 can be used to approximate the solution to the initial value problem\({\bf{y' = y,y(0) = 1}}\) , at x= 1. Show that the approximation \({{\bf{y}}_{\bf{n}}}\)obtained by using the Taylor method of order 2 with the step size \(\frac{{\bf{1}}}{{\bf{n}}}\) is given by the formula\({{\bf{y}}_{\bf{n}}}{\bf{ = }}{\left( {{\bf{1 + }}\frac{{\bf{1}}}{{\bf{n}}}{\bf{ + }}\frac{{\bf{1}}}{{{\bf{2}}{{\bf{n}}^{\bf{2}}}}}} \right)^{\bf{n}}}\). The solution to the initial value problem is\({\bf{y = }}{{\bf{e}}^{\bf{x}}}\), so \({{\bf{y}}_{\bf{n}}}\)is an approximation to the constant e.
Falling Body.In Example 1 of Section 3.4, page 110, we modeled the velocity of a falling body by the initial value problem \({\bf{m}}\frac{{{\bf{dv}}}}{{{\bf{dt}}}}{\bf{ = mg - bv,v(0) = }}{{\bf{v}}_{\bf{o}}}{\bf{ = 0}}\)under the assumption that the force due to air resistance is –bv. However, in certain cases the force due to air resistance behaves more like\({\bf{ - b}}{{\bf{v}}^{\bf{r}}}\), where \({\bf{(r > 1)}}\) is some constant. This leads to the model \({\bf{m}}\frac{{{\bf{dv}}}}{{{\bf{dt}}}}{\bf{ = mg - b}}{{\bf{v}}^{\bf{r}}}{\bf{,v(0) = }}{{\bf{v}}_{\bf{o}}}\) (14).To study the effect of changing the parameter rin (14),take \({\bf{m = 1,}}\,\,{\bf{g = 9}}{\bf{.81,}}\,\,{\bf{b = 2}}\) and \({{\bf{v}}_{\bf{o}}}{\bf{ = 0}}\).Then use the improved Euler’s method subroutine with \({\bf{h = 0}}{\bf{.2}}\) to approximate the solution to (14) on the interval \(0 \le {\bf{t}} \le 5\)for \({\bf{r = 1}}{\bf{.0,}}\,\,{\bf{1}}{\bf{.5}}\) and 2.0. What is the relationship between these solutions and the constant solution\({\bf{v(t) = }}{\left( {\frac{{{\bf{9}}{\bf{.81}}}}{{\bf{2}}}} \right)^{\frac{{\bf{1}}}{{\bf{r}}}}}\)?
If the object in Problem 2 is released from rest 30ftabove the ground instead of 500ft, when will it strike the ground? [ Hint:Use Newton’s method to solve for t ]
In 1990 the Department of Natural Resources released 1000 splake (a crossbreed of fish) into a lake. In 1997 the population of splake in the lake was estimated to be 3000. Using the Malthusian law for population growth, estimate the population of splake in the lake in the year 2020.
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