Description

1. (10 points total) Circuits
In a circuit with impressed voltage ε(t) and inductance L, Kirchhoff’s 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:
t 1.00 1.01 1.02 1.03 1.04
i 3.01 3.12 3.14 3.18 3.24
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.
(a) (6 points) Approximate the voltage ε(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.
(b) (4 points) 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. (10 points total) Approximating Derivatives
(a) (5 points) Derive a method for approximating f000(xo) whose error term is O(h2), by expanding the function f in a Taylor polynomial about xo using xo± h, xo, and xo± 2h.
(b) (5 points) The partial derivative fx(x,y) of f(x,y) with respect to x is obtained by holding y fixed and differentiating with respect to x. Similar for fy when holding x fixed. Using a Taylor series expansion of a function of two variables, determine the O(h2) numerical approximation formulas and associated truncation error for fx and fy.
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