MAT-1120 Midterm: MATH 1120 App State Fall2010 Test3 answer key
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November 10th, 2010: (24 points) taylor polynomials (a) let f (x) = x3 + 2x + 3. Find the 4th-order taylor polynomial, p4(x), for f (x) centered at x0 = 1. To nd the taylor polynomial centered at x0 = 1, we need to nd the value of the 0th, 1st, . , 4th derivatives of f (x) at x = 1. k = f (k)(x) = x3 + 2x + 3. 4 ( 1)3 + 2( 1) + 3 = 0 f (k)( 1) = P4(x) = 0 + 5(x ( 1)) + 6. Answer: p4(x) = 5(x + 1) 3(x + 1)2 + (x + 1)3. Note: since f (x) is a polynomial of degree 3, f (x) = p3(x) = p4(x) = p5(x) = . (x ( 1))2 + (x ( 1))3 + (x ( 1))4. We need k3+1 = k4 |3 cos(x)| for all 0 x 2.
