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MATH 1920.
Calculus II Fall 2007 Dwiggins HOMEWORK ASSIGNMENT # 6 Due Tuesday 13 Nov 07 Taylor Polynomials (§ 8.7)
1. Calculate the third-degree Taylor polynomial for
centered at x 0 = 1. Use T 3 to estimate the value of f (1.4), and use Taylor’s Remainder Theorem to estimate an upper bound for the error between the approximated value and the exact value. (Use a calculator to approximate the exact value of (1.4)4/3.)
2. Calculate the fourth-degree Taylor polynomial for f ( x ) = x^5
centered at x 0 = 2. Use T 4 to estimate the value of f (2.1), and explain why the remainder term (from T 5 ) gives the error exactly. Power Series (§ 8.8)
3. Find the radius of convergence and interval of convergence
(being sure to check the endpoints in each case, if any) for each of the following power series: (a) (b) (c) (d) Variations on the Geometric Series (§ 8.9)
4. From page 630 of the textbook: # 10, # 16, # 20
(4a = # 10, 4b = # 16, 4c = # 20) Series Approximation Techniques (§ 8.10) 4 f ( ) x x^3 1 1
n n n f x nx
1
n n x f x n
1
n n n x f x n n
1
n n n n n x f x
5. Use Taylor series (centered at x 0 = 0) to calculate each of the following
quantities to within 3 decimal places. (Use remainder theorems to determine how many terms you will need in each case.) (a) e 0.1^ (b) cos(1) (1 radian, not 1 degree) (c)
6. Use a binomial series to calculate the arclength along the curve
to within two decimal places. (Set up the arclength differential, use a binomial series to expand the resultant square root function using at least five terms, then integrate the series expansion term by term to get a numerical series, and finally use the alternating series remainder theorem to decide how many terms are needed.) BONUS # 1: Compare your answer in # 6 to the exact answer Obtained by using a trig substitution on the arclength integral, which involves using the antiderivative of sec^3 .
7. Find the fifth-degree Taylor polynomial (centered at x 0 = 0)
for f ( x ) = tan x using the following two different methods: (and of course verify both methods give the same result) (a) Start taking derivatives of tan x and calculate for n = 0,1,2,…,5. (b) Use the series expansions of sin x and cos x to calculate using long division. BONUS # 2. Given in the table below are six-decimal-place approximations of cos x for x between 0 and 1 (radian). Since cos(0) = 1 is known exactly, the other values are calculated using Taylor series expansions of cos x about x 0 = 0. The values for cos(0.2) in the table below have been calculated using Taylor polynomials for cos x , where TN denotes the Nth-degree polynomial. As can be seen, T2 approximates cos(0.2) to two decimal places while T4 and beyond approximate cos(0.2) to six decimal places. Fill in the rest of the table to six decimal places, and for extra extra bonus points sketch the graph of y = cos x for x < 0 < 1 along with the graphs of T2, T4, T6, and T8. x 0 0.2 0.4 0.6 0.8 1. cos(x) 1.000000 0.980067 0.921061 0.825336 0.696707 0. T2(x) 1.000000 0. T4(x) 1.000000 0. T6(x) 1.000000 0. T8(x) 1.000000 0. 1 0 sin x
^ x^ dx
1 2 C y 4 x , 0 x 1 ( ) (^) (0) ! n n f a n
sin tan cos x x x
