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Math 1A Extra Credit Project x \M Veuxlal (5,524) - kogj0714 x / L BS UE (9 46 41 X
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2. Determine the values of x for which the quadratic approximation f(x) - P(x) in Problem 1
is accurate to within 0.1. [Hint: Graph y - P(x), y - cos x - 0.1, and y = cos x + 0.1 on
a common screen.]
3. To approximate a function f by a quadratic function P near a number a, it is best to write P
in the form
P(x) - A + B(x - a) + C(x - a)
Show that the quadratic function that satisfies conditions (i). (ii), and (iii) is
HH
P(x) = f(a) + S'la)(x - a) + 'S&quot;(a)(x - a)
4. Find the quadratic approximation to f(x) - vx + 3 near a - 1. Graph f, the quadratic
approximation, and the linear approximation from Example 2 in Section 3.10 on a common
screen. What do you conclude?
5. Instead of being satisfied with a linear or quadratic approximation to f(x) near x = a, let's
try to find better approximations with higher-degree polynomials. We look for an ath-degree
essay
polynomial
T.(x) = co + ci(x - a) + cz(x - a) + cy(x - a)' + . .. + c.(x - a)&quot;
such that 7, and its first n derivatives have the same values at x - a as f and its first n
derivatives. By differentiating repeatedly and setting x = a, show that these conditions are
satisfied if co = fla), c = f'(a), cz = if&quot;(a), and in general
C - J (a)
essay
k!
where k! = 1 . 2 . 3 . 4 . . . . . k The resulting polynomial
T.(x) - f(a) + f'a)(x - a) + @(x - a) + . ..+ (x - a)
2!
n!
is called the ath-degree Taylor polynomial of f centered at a.
6. Find the 8th-degree Taylor polynomial centered at a - 0 for the function f(x) - cos x.
etc (1).p
Graph f together with the Taylor polynomials 72. 74. To. Ts in the viewing rectangle [-5. 5]
by [-1.4. 1.4] and comment on how well they approximate f.
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