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MATLAB #2 - David Thibodeaux Calculus 1 H03 MATLAB...

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David Thibodeaux Calculus 1 – H03 MATLAB Assignment #2 11/4/2010 Introduction: In this assignment, we will use the concept of the Taylor series to approximate the trends seen in complex functions. Using theory, we will first derive the formula used to approximate these curves, and then apply these approximations to various graphs. 1. Find a second-order (quadratic) approximation P 2 (x) = Ax 2 + Bx + C to the function f(x), stipulating the following for the best approximation: (a) P 2 (a) = f(a) (b) P 2 (a) = f (a) (c) P 2 ′′ (a) = f ′′ (a) Using the Definition of the Taylor Series: P 2 (x) = f(a) + f’(a)(x – a) + ½f’’(a)(x – a) 2 Substituting a for all x… P 2 (a) = f(a) + f’(a)(a – a) + ½f’’(a)(a – a) 2 P 2 (a) = f(a) For P 2 ’(a), we can say that f(a) is zero, since f(a) is a constant… P 2 ’(x) = 0 + f’(a)(1) + (x – a)(0) + ½f’’(a) x 2(x – a)(1) Now substitute a for all x… P 2 ’(a) = 0 + f’(a) + (a - a)(0) + f’’(a)(a – a)(1) P 2 ’(a) = f’(a). We can also say that for P 2 ’’(a) of f’(a) is zero, since f’(a) is a constant. P 2 ’’(a) = 0 + (0)(x – a) + ½f’’(a)(x – a) 2 . Substituting all a for x, and using the chain rule for the term ½f’’(a)(x – a) 2 P 2 ’’(a) = 0 + (0)(a – a) + ½f’’(a)(2)(1) P 2 ’’(a) = f’’(a). We have satisfied all conditions.
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2. Now that we have determined the formula for an approximation, we will now test this approximation on the function f(x) = e x at a = 0. Using the Taylor approx. formula, we derive the following approximations: P 1 (x) = x + 1 P 2 (x) = 1 + x + ½x 2 . Graph of f(x), P 1 (x) , and P 2 (x) Description: the linear approximation is only truly accurate at x = 0 due to the fact that it is linear and the function is exponential.
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