Solutions: Solving  Solutions

Use a Taylor Series approximation to compute the variable values in Problems 3 and 4. Determine to 0.1.

Problem (1)

Determine the values for a, b, and c to 0.1 for the following equations.

2a - 4b + 3c = 14.5
-a + 2b + c = -6.0
5a - b - 4c = 8.0

By substitution

Problem (2)

 A +4.0% grade is followed by a -2.0% grade on a vertical alignment. Using the information shown below, what are the station and elevation of the PVI?

Using E for elevations and S for stations,

Two equations in two unknowns, Solve using substitution.

Station is 16+00; Elev = 844.00

Use a Taylor Series approximation to compute the variable values in Problems 3 and 4. Determine to 0.1.

Problem (3)

 0.5x³ - 4x² + 2.5x = -79.2

Start with x_o= -2

Partial derivative of the function

First iteration

Second iteration

Third iteration

Since the correction is less than 0.1, we can stop.

Math check

 0.5(-3.5)³ - 4(-3.5)² + 2.5(-3.5) = -79.19 check

Answer: x = -3.5

Problem (4)

Start with x_o = 5  and y_o = -7

Partial derivatives of each function.

Set up Taylor Series and substitute in initial approximations,

There are two equations in two unknowns. Solve the equations simultaneously.

Second iteration:

Third iteration.

dx and dy small enough in the third iteration so can stop.

Answer: x=6.0, y=-9.0