MATH 163 HOMEWORK Week 13, due Monday April 26 TOPICS. c n (x a) n then c n = f(n) (a) n!

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1 MATH 63 HOMEWORK Week 3, due Monday April 6 TOPICS 4. Taylor series Reading:.0, pages Taylor series. If a function f(x) has a power series representation f(x) = c n (x a) n then c n = f(n) (a) () Therefore, if a function f(x) has a power series representation about x = a then it must be f (n) (a) (x a) n = f(a) + f (a)(x a) + f (a) (x a) + f (3) (x a) () This series is called the Taylor series of f about x = a. If a = 0 then the series is also called the Maclaurin series of f. Whether the series converges and equals to f are two questions that still need to be determined. You should know the formula for the Taylor series by heart, and be able to use it to find at least the first few terms of the Taylor series of any function. Special Series. You should memorize the following Taylor series about x = 0 (Taylor series about 0 are also called McLaurin series) and the intervals where they convergence to f (which we havent proven yet!) x = + x + x + x 3... = x n, x < e x = + x + x! + x3 + x4 4! +... = x n, x R sin(x) = x x3 + x! x7 7! +... = ( ) n xn+ (n + )!, x R cos(x) = x! + x4 4! x6 6! +... = ( ) n xn (n)!, x R Be able to use them to find new Taylor series by substitution, addition, multiplication, differentiation and integration. Note: with what we know so far we can find the Taylor series of a function, and find its interval of convergence. But we cannot prove that the series equals the function until we use the remainder theorem (next class). Therefore we cannot yet prove that e x,cos(x),sin(x) equal their Taylor series.

2 . Taylor polynomials and the remainder theorem Reading:.0 rest (SKIP Binomial series pp middle),. (pp ) Taylor polynomials. The Taylor polynomial of degree N of a function f(x) about x = a is P N (x) = N f (n) (a) (x a) n = f(a) + f (a)(x a) + f (a) (x a) + f (3) (x a) f(n) (a) (x a) N Notes: () p N exists if f is N times differentiable at a. () If f is infinitely often differentiable, then f has a Taylor series (but is not necessarily equal to its Taylor series), and p N simply equals the first N terms of the Taylor series. (3) p N is the polynomial whose value and first N derivatives agree with those of f at x = a, ie p (j) (a) = f (j) (a),j = 0,... N, assuming f is N times differentiable at c. (4) p (x) is the linear approximation of f at x = a (whose graph = line tangent to f at a). The Taylor Remainder Theorem. Can we approximate f(x) by p N (x)? (even if f has only finitely many derivatives at a, that is, doesn t even have a Taylor series?) How good is this approximation?? The answer is given by the Taylor remainder theorem: Suppose f is N + times continuously differentiable on an open interval I of a. Then () f(x) = p N (x) + R N (x), where R N (x) = f(n+) (ξ) (x a)n+ (N + )! for some (unknown) ξ between x and a. This is particularly useful if we know an upper bound for the (N+)st derivative for all ξ I: If f (N+) (ξ) M for all ξ I, then the error R N (x) satisfies M f(x) p N (x) x a N+ (N + )! Notice that this applies even if the function is no further differentiable. Consequences (i) If derivatives of f bounded near x = a, Taylor polynomials of increasing order N approximate the function f increasingly well near x = a. (ii) If lim N R N(x) 0 in I, then Taylor series of f converges to f. We can now show that e x, sin(x), cos(x) equal their Taylor series everywhere. Big-O notation Definition. f(h) = O(h p ) in I if f(x) Mh p for some M, all h I. Note: If x a = h and f (N+) (ξ) M for all ξ I, then f(x) = p N (x) + O(h N+ ),ie, f(x) = f(a) + f (a)(x a) + f (a) f(a + h) = f(a) + f (a)h + f (a) (x a) f(n) (a) (x a) N + O( x a N+ ) h + f (a) h f(n) (a) h N + O( h N+ )

3 6. Applications (Days 38-4) Applications: - Approximating transcendental functions (including integrals) near a basepoint by polynomial. Approximation error can be estimated using Alternating series test or Taylor s theorem (or integral test) - Evaluating some special series - Solving differential equations - Approximating derivatives by finite differences using data on grid - Approximating physical quantities to determine dominant order (how to reconcile Einstein s theory of relativity with Newtonian physics? How does the electric field of a dipole change far from the dipole? Approximately what is the speed of waves in shallow water? in deep water?) Example:., example 3 Taylor series can be used to approximate quantities in physics and engineering in certain limits, for example when one parameter in the problem is very small compared to another parameter. PROBLEMS DAY 36: Taylor series.. (a) Determine f(0) and f (0) for a function f(x) with Maclaurin series T(x) = 3 + x + x + x (b) Write out the first four terms of the Maclaurin series of f(x) if f(0) =, f (0) = 3, f (0) = 4, f (0) = (c) If f(x) = b n(x ) n for all x, write a formula for b 8...0: (Given graph of f, explain why a certain series cannot be its Taylor series) 3. Using the formula c n = f(n) (a), find the first terms of the Taylor series of the given functions about the given point x = a (up to and including quartic term). (a) f(x) =, a = 0 (does your result agree with a series we already know?) x (b) f(x) =, a = (compare to your result of hw, #, which method is easier?) x (c) f(x) = sin(x), a = π/4 (d) f(x) = + x, a = 0 (e) erf(x) = x e t dt, π 0 x (, ), a = 0. (This function is called the error function and occurs for example in probability, statistics, and partial differential equations. You can plot it in MATLAB using the predefined function called erf.) 4. Find the Taylor series of the following functions about the given point x = a, and find their interval of convergence. 3

4 (a) f(x) = e x, a = 0 (b) f(x) = cos(x), a = 0 (c) f(x) = x 4 3x +, a = (d) f(x) = (x + ) 6, a = 0 (e) f(x) = (x + ) 6, a = DAY 37: Taylor polynomials and remainder theorem.. Graph the function f(x) and its first four Taylor polynomials p,p,p 3,p 4 about the given point x = a on the same graph. You may use your results of Day 36, # 3,4. (a) f(x) = sin(x), a = π/4 (b) f(x) = cos(x), a = 0 (c) f(x) = + x, a = 0. (a) For f(x) = ln( + x), find its Taylor polynomial p 3 (x) of degree 3 about a =. (b) Use Taylor s Theorem to estimate the accuracy of the approximation f(x) p 3 (x) if 0. x. (c) Use a graph to determine the actual maximal difference f(x) p 3 (x) on the interval [0.,.]. 3. How good is the approximation if x 0.? sin x x x (a) Use the known Maclaurin series for f(x) = x to find the Taylor series of f(x) = tan x and state its interval of convergence. (b) For which values of x is the approximation accurate to within an error of 0.0? tan x x x3 3 + x. Using the known Maclaurin series for /( x),e x,sin x,cos x, find the Maclaurin series for the following functions, and state their interval of convergence. (a) f(x) = xcos(x) (b) f(x) = coshx (c) f(x) = x 0 e t dt (compare to the approach taken in Day 36, #3e) (d) f(x) = ln( + x) (e) f(x) = sin x [Hint: Use sin x = ( cos x).] (f) f(x) = x 0 tan (t )dt 6. Find the Taylor polynomial p 6 of f(x) = sin(x 4 ) about a = 0. 4

5 7. In Day 33, # : you found the series of /( x) about a = by writing x = + (x ) and then using the known series for /( u) about u = 0. Use this approach to find the series of f(x) = /x about a = 3, and state its interval of convergence. DAY 38: Applications.. Use the Maclaurin series for e x to calculate e 0. correct to to within an error of : # (Use series to approximate the given integrals to within the indicated accuracy.) 3. Evaluate the following series exactly: (a) π 4 (b) (c) π π 4! π ! +... π 3! + π ! π ! +... π + π4 4! π6 6 7! +... (d) +! +! + + 4! +... (e) (Hint: see Day 37, Problem d) 4. Use series to evaluate the limits in.0:, 6, 7..9: # 3 (solutions to differential equations in series form)