[Solved] MA3457 Homework 4

1. Circuits

In a circuit with impressed voltage (t) and inductance L, Kirchhoffs first law gives the relationship

where R is the resistance in the circuit and i is the current. Suppose we measure the current i for several values of t and obtain:

where t is measured in seconds, i is in amperes, the inductance L is a constant 0.98 henries, and the resistance is 0.142 ohms.

• Approximate the voltage (t) when t = 1.00,1.01,1.02,1.03, and 1.04 using the derivative approximations derived in class and/or 4.1 in Burden & Faires textbook.
• Specify the order of accuracy of each approximation and whether it is a forward,backward, or centered approximation.

You can complete this by hand or with a Matlab code.

2. Approximating Derivatives
   • Derive a method for approximating f⁰⁰⁰(x_o) whose error term is O(h²), by expanding the function f in a Taylor polynomial about x_o using x_o h, x_o, and x_o 2h.
   • The partial derivative f_x(x,y) of f(x,y) with respect to x is obtained by holding y fixed and differentiating with respect to x. Similar for f_y when holding x Using a Taylor series expansion of a function of two variables, determine the O(h²) numerical approximation formulas and associated truncation error for f_x and f_y.