Single Variable Calculus by James Stewart Applied Project

[Guest]

http://64.78.63.75/samples/math/05MA100 ... CC3Ch3.pdf
goto page 11, at the bottom, or if you can't read that, goto:
http://endeavor.aspfreeserver.com/page1 ... roject.txt and http://img102.imageshack.us/my.php?image=calculus29ni.png

After doing the first problem, I got a = -0.012, b = 0.8, c = 0, and f(100) = -40ft.

In the 2nd problem, can I use these values of a, b, and c again? I think so because I developed 8 equations in 11 unknowns, but it would be 8 equations in 8 unknowns if I used the values for a, b, and c.

For 2 (b), I am supposed to use a computer algebra system, but I do not have one at the moment. Does anyone know of any free CAS programs or demos that I can download?

 

[Gene]

Suppose you are asked to design the first ascent and drop for a new roller coaster. By studying photographs of your favorite coasters, you decide to make the slope of the ascent 0.8 and the slope of the drop -1.6. You decide to connect these two straight stretches y = L₁(x) and y = L₂(x) with part of the parabola y = f(x) = ax² + bx + c, where x and f(x) are measured in feet. For the track to be smooth there can't be abrupt changes in direction, so you want the linear segments L₁ and L₂ to be tangent to the parabola at the transition points P and Q. (See the figure.) To simplify equations you decide to place the origin at P.
1. (a) Suppose the horizontal distance between P and Q is 100ft. Write equations in a, b, and c that will ensure that the track is smooth at the transition points. (b) Solve the equations in part (a) for a, b, and c to find a formula for f(x). (c) Plot L₁, f, and L₂ to verify graphically that the transitions are smooth. (d) Find the difference in elevation between P and Q.
2. The solution in Problem 1 might look smooth, but it might not feel smooth because the piecewise defined function [consisting of L₁(x) for x < 0, f(x) for 0 <= x <= 100, and L₂ for x > 100] doesn't have a continuous second derivative. So you decide to improve the design by using a quadratic function q(x) = ax² + bx + c only on the interval 10 <= x <= 90 and connecting it to the linear functions by means of two cubic functions:
g(x) = kx³ + lx² + mx + n 0 <= x < 10
h(x) = px³ + qx² + rx + s 90 < x <= 100 (a) Write a system of equations in 11 unknowns that ensure that the functions and their first two derivatives agree at the transition points. (b) Solve the equations in part (a) with a computer algebra system to find formulas for q(x), g(x), and h(x). (c) Plot L₁, g, q, h, and L₂, and compare with the plot in Problem 1(c).

 

[Guest]

oh well, i solved it all myself; and didn't use a CAS. I used a matrix for the p, q, r, and s.