TOPIC 3. Taylor polynomials. Mathematica code. Here is some basic mathematica code for plotting functions.

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1 TOPIC 3 Taylor polynomials Main ideas. Linear approximating functions: Review Approximating polynomials Key formulas: P n (x) =a 0 + a (x x )+ + a n (x x ) n P n (x + x) =a 0 + a ( x)+ + a n ( x) n where a k = f (k) (x ) k! Mathematica code. Here is some basic mathematica code for plotting functions. () Plot the function f(x) =sin(3x). Plot[Sin[3 x], {x, -Pi, Pi}] (2) Plot the function f(x) =sin(3x) in red, and the line y =3x in blue. Plot[{Sin[3 x], 3 x}, {x, -Pi, Pi}, PlotRange -> {-3, 3}, PlotStyle -> {Red, Blue}] Here is some more sophisticaled Mathematica code for exploring Taylor series. () The exponential function f(x) =e 2x Plot[{Exp[2 x], Sum[((2 x)^k)/(k!), {k, 0, n}]}, {x, -, }, PlotRange -> {0, 4}, PlotStyle -> {{Thick, Red}, {Thick, Blue, Dashed}}], {n,, 0, }] (2) The function f(x) = cos x. (Note that in this case n does not correspond to the degree of the polynomial! Plot[{Cos[x], Sum[((-)^k (x)^(2 k))/((2k)!), {k, 0,n}]}, {x, -3 Pi, 3 Pi}, PlotRange -> {-2, 2}, PlotStyle -> {{Thick, Red}, {Thick, Blue, Dashed}}], {n,, 0, }] 9

2 20 3. TAYLOR POLYNOMIALS Exercises. Exercise 3.. Write down the fourth-order Taylor polynomials, centered at x = 0, for the following functions: () e x (2) ln( + x) (3) p +x +x Can you find expressions for the general polynomials P n (x)? () e x +x + 2 x2 + 6 x x4 (2) ln ( + x) x 2 x2 + 3 x3 4 x4 (3) p +x + 2 x 4 x x x + x2 x 3 + x ! x ! x4 Optional: The general formulae for the n th order approximations are () (2) (3) e x +x + 2 x2 + 6 x x4 + + n! xn ln ( + x) x 2 x2 + 3 x3 4 x4 + +( ) n n xn p +x + 2 x 4 x ! x3 n+ ()(3)...(2n 3) +( ) 2 n n! +x x + x2 x 3 + x 4 + +( ) n n xn Exercise 3.2. () Consider f(x) =sinx. Show that f (k) (0) = 0 when k is even, and that when k = 2l + is odd we have f (2l+) (0) = ( ) l. Conclude that the Taylor polynomial, centered at x = 0, for sin x x + 3! x3 + 5! x5 + + ( )l (2l + )! x2l (2) Show that the Taylor polynomial for the cosine function, centered at x = 0, is given by + 2! x2 + 4! x4 + + ( )l (2l)! x2l +... (3) Suppose f is an even function, meaning that f(x) =f( x) for all x. Show that this implies that f (k) (0) = 0 whenever k is odd. [Hint: Take k derivatives x n

3 3. TAYLOR POLYNOMIALS 2 of the identity defining even-ness.] What can you conclude about the Taylor polynomials centered at x = 0? Exercise 3.3. Using the polynomials you found in Exercises 3. and 3.2, find the following Taylor polynomials. [Do not do any extra work!] () Find the fourth-order Taylor polynomial for e 3x, centered at x = 0. (2) Find the fourth-order Taylor polynomial for ln(x), centered at x =. (3) Find the fourth-order Taylor polynomial for cos (5x), centered at x = 0. Find the eighth-order Taylor polynomial for p +4x 2, centered at x = 0. () e 3x +3x + 2 (3x)2 + 6 (3x) (3x)4 =+3x x x3 + 8 (2) We rewrite ln (x) =ln(+[x ]). Thus (3) ln (x) (x ) p +4x (4x2 ) 24 x4 2 (x )2 + 3 (x )3 (x )4 4 cos (5x) + 2! (5x)2 + 4! (5x)4 = 25 2! x ! x4 4 (4x2 ) ! (4x2 ) ! (4x2 ) 4 = x2 4 x ! x ! Exercise 3.4. In this exercise, we explore the Taylor polynomial approximation for f(x) = x. () Compute several derivatives of f and conclude that f (k) (0) = k!. (2) Write down the formula for the Taylor polynomial P n (x) for f, centered at x = 0. (3) What is P n ()? (This will depend on n, of course.) What happens to P n () as n gets large? What is f()? In what sense does this agree with your response to the previous question? (5) Find a formula for P n ( ). (Again, your formula will depend on n.) What happens as n gets large? (6) What is f( )? In what sense does this agree with your response to the previous question? x 8

4 22 3. TAYLOR POLYNOMIALS (7) Compare P n (2) with f(2) as n gets large. Are the values close to one another? Explain. (8) Use the following code to compare the Taylor polynomials to f. Plot[{/( - x), Sum[x^k, {k, 0, n}]}, {x, -2, 3}, Exclusions -> {},PlotRange -> {-5, 5}, PlotStyle -> {{Thick,Red}, {Thick,Blue,Dashed}}], {n,, 0, }] To what extent do the polynomials do a good job approximating the function; to what extent do the polynomials do a poor job? (2) P n (x) =+x + x x n (3) P n () = =n. As n gets large, so does P n (). f() is not defined; the function has an asymptote there. This agrees with the result from part (3) in the sense that there is no number P n () approaches as n gets large. (5) We have 8 < P n ( ) = + + +( ) n 0 if n is odd = : if n is even As n gets large, P n ( ) oscillates between zero and one. (6) f( ) = 2. In some sense this agrees with the polynomial approximation, in that 2 is the average of 0 and ; however, it does not agree in the sense that the approximation never gives values close to the value of the function. (7) P n (2) = n gets very large as n gets large. On the other hand, f(2) =. These two are very di erent. (8) The polynomials seem to do a good job approximating f in the interval (, ) and otherwise do not do a good job approximation f. Exercise 3.5. In the previous problem, we saw that the Taylor polynomials P n (x) approximated the function f(x) very well for a certain range of x, but did not do a good job approximating the function for other x. One might conjecture that this is because the function f considered has an asymptote. In this exercise we see that the Taylor polynomials might not do a good job approximating a function, even if that function has no vertical asymptotes! () Consider the function g(x) =. Use Exercise 3. to show that the Taylor +x2 polynomial approximating g near x = 0 are given by P n (x) = x 2 + x 4 x

5 3. TAYLOR POLYNOMIALS 23 (2) Use the following code to explore how well the Taylor polynomials approximate g. Write a few sentences explaining your findings. Plot[{/( + x^2), Sum[(-x^2)^k, {k, 0, n}]}, {x,-2,3}, Exclusions ->{}, PlotRange ->{-, 2}, PlotStyle ->{{Thick,Red}, {Thick,Blue,Dashed}}], {n,, 0, }] () Using Exercise 3. we have +x 2 = ( x 2 ) +( x 2 )+( x 2 ) 2 +( x 2 ) 3 + +( x 2 ) n = x 2 + x 4 x 6 + +( ) n x 2n. (2) Using the Mathematica code we see that on the interval (, ) the Taylor polynomials approximate the function very well. However, for x > and x<, the Taylor polynomials do not seem to do a good job of approximating the function.