WebAssign Lesson 6-3 Taylor Series (Homework)

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1 WebAssign Lesson 6-3 Taylor Series (Homework) Current Score : / 56 Due : Tuesday, August :59 AM MDT Jaimos Skriletz Math 75, section 3, Summer Instructor: Jaimos Skriletz. /4 points Consider the geometric series f(x) = x n = + x + x 2 + x 3 + x 4 +. Find the following derivatives evaluated at 0. f(0) = f ' (0) = f '' (0) = f (3) (0) = f (4) (0) = f (5) (0) = 2. Look for a pattern and find the n-th order derivative at 0. f (n) (0) =

2 2. /4 points Consider a power series centered at 0 f(x) = a n x n = a 0 + a x + a 2 x 2 + a 3 x 3 + a 4 x 4 +. Find the following derivatives evaluated at 0. f(0) = f ' (0) = f '' (0) = f (3) (0) = f (4) (0) = f (5) (0) = 2. Look for a pattern and find the n-th order derivative at 0. f (n) (0) = Note: This gives a way to write the n-th coefficient of the Taylor Series in terms of the n-th derivative of a function evaluated at x = 0.

3 3. /3 points A function is written as a Taylor series centered at 0 Viewing Saved Work Revert to Last Response f(x) = a n x n = a 0 + a x + a 2 x 2 + a 3 x 3 +. if f(0) = 3 find the constant coefficient. a 0 = 2. if f '(0) = 6 find the coefficient of x a = 3. If f ''(0) = 0 find the coefficient of x 2. a 2 = 4. If f (3) (0) = 42 find the coefficent of x 3. a 3 = 5. Write the 3rd order Taylor Polynomial for f centered at x = 0. T 3 (x) = 4. /2 points Suppose f(x) is a function such that f(0) = 5 f '(0) = 0 f ''(0) = 4 f (3) (0) = 0 f (4) (0) = 3 f (5) (0) = 0 f (6) (0) = 2 Write the 6th order Taylor Series approximate centered at x = 0 for this function T 6 (x) =

4 5. /2 points Suppose f(x) is a function such that f(0) = 2 f '(0) = 4 f ''(0) = 3 f (3) (0) = 5 f (4) (0) = 6 f (5) (0) = 8 Write the 5th order Taylor Series approximate centered at x = 0 for this function T 5 (x) = 6. /2 points Consider the function f(x) = e x. Find the 5th order Taylor series approximation centered at x = 0 for this function. T 5 (x) = 7. /2 points Consider the function f(x) = cos(x). Find the 6th order Taylor series approximation centered at x = 0 for this function. T 6 (x) = 8. /2 points Consider the function f(x) = sin(2x). Find the 7th order Taylor series approximation centered at x = 0 for this function. T 7 (x) =

5 9. /2 points Suppose the Taylor series centered at x = 0 for a function is ( ) f(x) = n 2 n x n 2 4 = x + x 2 8 x n Note: In the following you can enter in the formula (including powers and or factorials) for the answer.. What is the 2th order coefficient? a 2 = 2. What is the exact value of f (2) (0) = 0. /2 points Suppose the Taylor series centered at x = 0 for a function is f(x) = ( ) n+ 4 n x 2n + = x + 2x 3 2 x 5 + (2n)! 3 What is the exact value of f (9) (0)? Note: You can enter in the formula (including powers and/or factorials) for the answer. f (9) (0) =

6 . /2 points Consider the following Taylor Series centered at x = 0. A= x n B= C = D= E = ( ) n x 2n (2n)! x n n! ( ) n x 2n + (2n + )! ( ) n x 2n Match the following functions to their Taylor Series. x e x sin(x) cos(x)

7 2. /3 points Using the Taylor series for e x e x = x n = + x + x 2 + x 3 + x 4 + n! 2 3! 4! Find the Taylor series for the function f(x) = e x2 /2.. Find the 6th order Taylor series approximation centered at x = 0. T 6 (x) = 2. Find the Taylor Series for this function centered at x = 0. e x2 /2 = 3. /2 points Using the Taylor series for sin(x) sin(x) = ( ) n x 2n + = x x 3 + x 5 x 7 + (2n + )! 3! 5! 7! Find the Taylor series for the function f(x) = 2sin(2x).. Find the 5th order Taylor series approximation centered at x = 0. T 5 (x) = 2. Find the Taylor Series for this function centered at x = 0. 2sin(2x) =

8 4. /2 points Using the geometric series = x n = + x + x 2 + x 3 + x 4 + x Find the Taylor series for the function ln( + x). Hint: d dx ln( + x) = + x. Find the 4th order Taylor series approximation centered at x = 0. T 4 (x) = 2. Find the Taylor Series for this function centered at x = 0. ln( + x) = 5. /2 points Using the geometric series = x n = + x + x 2 + x 3 + x 4 + x Find the Taylor series for the function tan (x). Hint: d dx tan (x) = + x 2. Find the 7th order Taylor series approximation centered at x = 0. T 7 (x) = 2. Find the Taylor Series for this function centered at x = 0. tan (x) =

9 6. / points The error function is defined as 2 x erf(x) = e t2 dt π 0 Find the Taylor series centered at x = 0 for this function. erf(x) = 2 π 7. / points The sine integral is defined as x sin(t) Si(x) = dt 0 t Find the Taylor series centered at x = 0 for this function. Si(x) = 8. /2 pointsrogac alcet Find the Maclaurin series for f(x) = cos 8x f(x) = n = 0 9. /2 pointsrogac alcet Find the Maclaurin series for f(x) = e 6x f(x) = n = 0

10 20. /2 pointsrogac alcet Find the Maclaurin series for f(x) = x 3x 5 f(x) = n = 0 2. /2 pointsrogac alcet Find the terms through degree four of the Maclaurin series of substitution as necessary. f(x) = e x sin(3x) Use multiplication and 22. /2 pointsrogac alcet Find the terms through degree four of the Maclaurin series of substitution as necessary. f(x) = e x tan (5x) Use multiplication and 23. /2 pointsrogac alcet Find the terms through degree four of the Maclaurin series of substitution as necessary. f(x) = sin(x 3 3x) Use multiplication and

11 24. /2 pointsrogac alcet Find the Maclaurin series for f(x) = cos 2 (7x) Hint: Use the identity cos 2 x = + cos(2x). 2 f(x) = + n = 25. /2 points Consider the function f(x) = x 3 e x2 Find the exact value of the following (Hint: Use the Taylor Series for f.) f (9) (0) = 26. /2 points Consider the function f(x) = x 2 sin(2x) Find the exact value of the following (Hint: Use the Taylor Series for f.) f (7) (0) =

WebAssign Lesson 4-1 Partial Fractions (Homework)

WebAssign Lesson 4-1 Partial Fractions (Homework) Webssign Lesson 4-1 Partial Fractions (Homework) Current Score : / 46 Due : Wednesday, July 16 2014 11:00 M MDT Jaimos Skriletz Math 175, section 31, Summer 2 2014 Instructor: Jaimos Skriletz 1. /3 points