Taylor and Maclaurin series is called the Taylor
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Taylor and Maclaurin series. is called the Taylor series for f(x) at c. If c = 0, then the series is the Maclaurin series for f(x).
Find the Taylor series for f(x) = e– 2 x at c = 0 f(x) = e– 2 x f(0) = 1 f ’(x) = -2 e– 2 x f ’(0) = -2 f ’’(x) = 4 e– 2 x f ’’(0) = 4 f ’’’(x) = -8 e– 2 x f ’’’(0) = -8 f(4)(x) = 16 e– 2 x f(4)(0) = 16
Find the Taylor series for f(x) = sin x at c = /4 f(x) = sin x f ’(x) = cos x f ’’(x) = - sin x f ’’’(x) = - cos x f(4)(x) = sinx
Find the Taylor series for f(x) = sin x at c = /4
Find the Taylor series for f(x) = ln(x 2 + 1) at c = 0
Find the Taylor series for f(x) = ln(x 2 + 1) at c = 0
Binomial series The Maclaurin series for a function of the form (1+x)n is called the Binomial series Obtain the Maclaurin series for f(x) = (1 + x)k f(0) = 1 f’(x) = k(1 + x)k-1 f’(0) = k f’’(x) = k(k – 1)(1 + x)k-2 f’’(0) = k(k – 1) f’’’(x) = k(k – 1)(k – 2)(1 + x)k-3 f’’’(0) = k(k – 1)(k – 2) f(4)(x) = k(k – 1)(k – 2)(k – 3)(1 + x)k-4 f(4)(0) = k(k – 1)(k – 2)(k – 3) Maclauren series is:
Use the binomial series to find the Maclaurin series for (1 + x)k =
Find the Maclaurin polynomial for f(x) = x cos x We find the Maclaurin polynomial cos x and multiply by x f(x) = cos x f(0) = 1 f ’(x) = -sin x f ’(0) = 0 f ’’(x) = - cos x f ’’(0) = - 1 f ’’’(x) = sin x f ’’’(0) = 0 f(4)(x) = cos x f(4)(0) = 1
Find the Maclaurin polynomial for f(x) = sin 3 x We find the Maclaurin polynomial sin x and replace x by 3 x f(x) = sin x f(0) = 0 f ’(x) = cos x f ’(0) = 1 f ’’(x) = - sin x f ’’(0) = 0 f ’’’(x) = - cos x f ’’’(0) = -1 f(4)(x) = sin x f(4)(0) = 0
Do this on your own Find the Maclaurin polynomial for f(x) = cos 2 x Hint: Write the Maclaurin polynomial for cos x and replace x by 2 x, and then simplify. Answer:
