Exercises

1. Evaluate each of the following expressions. Exact answers are preferable to decimal approximations.

   a.   `sum_(k=0)^oo (2/3)^k`	b.   `sum_(k=0)^oo (0.9)^k`
   c.   `int_0^oo 1/(1+t^2) dt`	d.   `int_0^oo 1/(1+t^2)^2 dt`
   e.   `int_3^oo x e^(-2x) dx`	f.   `1+3+3^2/2+3^3/(3!)+3^4/(4!)+cdots`
   g.   `int_1^oo 1/(x sqrt(1+x^2)) dx`	h.   `1-3+3^2/2-3^3/(3!)+3^4/(4!)-cdots`

2. 1. Find the Taylor polynomials of degrees `0` through `5` for the function `f text[(] t text[)] =1/(1+t)` . How do these polynomials compare to the geometric polynomials `1+x+cdots+x^n` for `x=-t`?
   2. In a single graphing window, graph `f text[(] t text[)]` and the six Taylor polynomials from part (a).
   3. How do the values of the Taylor polynomials compare with `f text[(] t text[)]` at `t=1`? at `t=-1`?
   4. For what range of `t` values do these polynomials converge to `f text[(] t text[)]`?
3. 1. If `f text[(] x text[)]=ln text[(] 1+x text[)]`, what is the coefficient of `x^5` in the seventh-degree polynomial approximation `P_7 text[(] x text[)]` of `f text[(] x text[)]`?
   2. What is the seventh-degree Taylor polynomial approximation to `f text[(] x text[)] =ln text[(] 1+x text[)]`?
4. 1. If `g text[(] x text[)] =tan^(-1) text[(] 1+x text[)]`, what is the coefficient of `x^5` in the seventh-degree polynomial approximation `P_7 text[(] x text[)]` of `g text[(] x text[)]`?
   2. What is the fourth-degree Taylor polynomial approximation to `g text[(] x text[)] =tan^(-1) text[(] 1+x text[)]`?
5. 1. What is the sixth-degree Taylor polynomial approximation to `h text[(] x text[)] =1/(1-x)`?
   2. What is the sixth-degree Taylor polynomial approximation to `g text[(] x text[)] =1/(1+x^2)`?
6. Suppose that 80 cents of every dollar spent in the United States is spent again in the United States. (Economists call this the multiplier effect.) If the federal government pumps an extra billion dollars into the economy, how much total spending in the United States occurs as a result?
7. For what value or values of `x` is

   `x-1/2x^2+1/3x^3-1/4x^4+cdots=0.5`?

8. 1. Calculate `ln 1.5`  by substituting `x=0.5` in the formula

      `ln text[(] 1+x text[)] =sum_(k=0)^oo text[(]-1text[)]^k x^(k+1)/(k+1)`

      and adding up partial sums until you see no change in the third decimal place.
   2. Check your result in part (a) by comparing your partial sums with your computer's or calculator's value for `ln 1.5`.
9. Find the Taylor series for the function `f text[(] x text[)] = text[(]1+xtext[)]^4`.
10. In the preceding exercise you found a Taylor series for a function of the form `f text[(] x text[)] =text[(]1+xtext[)]^m`, where `m` happened to be a positive integer. However, the calculation of Taylor coefficients can be carried out even if `m` is not a positive integer.
    1. Calculate the first three or four derivatives of `f text[(] x text[)] `. When you see the pattern, write down a formula for the `k`th derivative.
    2. Evaluate each of your derivatives at `x=0`, and divide by the appropriate factorial to find the coefficients in a Taylor expansion for `f text[(] x text[)]`. The usual notation for the `k`th coefficient is `((m),(k))`.
    The numbers `((m),(k))` are called binomial coefficients, and the Taylor series `sum_(k=0)^oo ((m),(k)) x^k` is called a binomial series. [We will see later that, for any number `m`, the binomial series converges to `f text[(] x text[)] =text[(]1+xtext[)]^m` if `text[|] x text[|]<1`.]
    
    
    3. Suppose `m` is a positive integer. Show that the binomial coefficients `((m),(k))` are zero for `k>m`. (The nonzero coefficients are the binomial coefficients you learned about in high school.) Explain why, in this case, the series is an `m`th degree polynomial that equals `f text[(] x text[)]` for all real numbers `x`.
11. 1. For `f text[(] x text[)] =sqrt(1+x)` (i.e., for `m=0.5`), calculate the first five coefficients in the binomial series defined in the preceding exercise. Notice that the first two coefficients are positive, but thereafter they alternate in sign.
    2. Use your five-term polynomial to estimate `sqrt(1.2)`. Use your computer or calculator to check your answer. How accurate is the polynomial estimate?
    3. Graph `f text[(] x text[)] =sqrt(1+x)` and your polynomial from part (b) in the same window. Describe in your own words what you see.
12. 1. Use the result of Exercise 11 to find the first five terms of the Taylor series for `g text[(] x text[)] =sqrt(1+x^3)`.
    2. Use the result of part (a) to estimate `int_0^0.5 sqrt(1+x^3) dx`.
    3. Use the integral function of your computer or calculator to evaluate the integral in part (b). How accurate is your polynomial estimate?
13. Here, courtesy of a computer algebra system, are the first six derivatives of the function `f text[(] x text[)] =lntext[(]1+x^2text[)]`:

    `f' text[(] x text[)]`	`=(2x)/(1+x^2)`	`f^(4) text[(] x text[)]`	`=-(12text[(]1-6x^2+x^4text[)])/(text[(]1+x^2text[)]^4)`
    `f'' text[(] x text[)]`	`=(2text[(]1-x^2text[)])/(text[(]1+x^2text[)]^2)`	`f^(5) text[(] x text[)]`	`=(48xtext[(]5-10x^2+x^4text[)])/(text[(]1+x^2text[)]^5)`
    `f''' text[(] x text[)]`	`=-(4xtext[(]3-x^2text[)])/(text[(]1+x^2text[)]^3)`	`f^(6) text[(] x text[)]`	`=(240text[(]1-15x^2+15x^4-x^6text[)])/(text[(]1+x^2text[)]^6)`

    1. Find the sixth-degree Taylor approximation `P_6 text[(] x text[)]` to `f text[(] x text[)]`.
    2. Use your graphing tool to find an interval over which the graph of `P_6 text[(] x text[)]` appears to fit the graph of `f text[(] x text[)]` well.
    3. Using an appropriate `y`-scale, graph `f text[(] x text[)] -P_6 text[(] x text[)]` on the interval you determined in part (b), and determine the maximum error in the approximation on that interval.
14. Confirm the result in part (a) of the preceding exercise — without any help from a computer algebra system — by completing the following steps.
    1. Recall the approximating polynomials for `1text[/(]1+t^2text[)]` calculated in Activity 2(a); multiply each by `2t` to find approximating polynomials for `f' text[(] t text[)] =2t text[/(]1+t^2text[)]`.
    2. Integrate term by term from `0` to `x` to find approximating polynomials for `lntext[(]1+x^2text[)]`. In particular, verify that the sixth-degree approximating polynomial is the one you computed in the preceding exercise.
15. Graph `f text[(] x text[)] =ln text[(] 1+x text[)]` and `P text[(] x text[)] =x` for `-2<=x<=2` and `-5<=x<=5`. One term at a time, change `P text[(] x text[)]` to higher degree Taylor polynomial approximations for `ln text[(] 1+x text[)]`, and regraph. That is, first subtract `x^2/2`, then add `x^3/3`, and so on. Continue up to at least degree `6`. Describe in your own words the characteristics that distinguish all the Taylor polynomials from `ln text[(] 1+x text[)]`.
16. As in the preceding exercise, start with `f text[(] x text[)]=ln text[(] 1+x text[)]` and `P text[(] x text[)] =x`. Define `N text[(] x text[)] =-x^2`/`2`. The error in the approximation of `f text[(] x text[)]` by `P text[(] x text[)]` is `f text[(] x text[)] -P text[(] x text[)] `, and `N text[(] x text[)]` is the first term left out of the approximation. Your task is to compare graphically the error with the next term. Set the graphing window at `-0.5<=x<=0.5` with tick marks at spacing of `0.05` and `-0.2<=y<=0.2` with tick marks at spacing of `0.02.` If your graphing tool has a grid option, turn it on.
    1. What is the approximate shape of the error curve?
    2. For what values of `x` is the error smaller (in absolute value) than the next term?
    3. For what values of `x` is the error larger (in absolute value) than the next term?
    4. What is the largest error (in absolute value) on the interval `[-0.5,0.5]`, and where does this largest error occur?
    5. In what interval on the `x`-axis is |error| no larger than the distance between `y` tick marks?
    6. Now make `P text[(] x text[)] =x-x^2`/`2` and `N text[(] x text[)] =x^3`/`3`. Change the `y`-range to `[-0.1,0.1]` with tick marks on the `y`-axis at a spacing of `0.01`. Answer the same questions as in parts (a)–(e). Record your numerical results in a table like the one shown in Table E1.

       Table E1   Errors in polynomial approximations to ln(1+x)

       Degree	Max y	y tick spacing	Max |error| on [-0.5,0.5]	Interval for |error| < tick spacing
       1	0.2	0.02	0.19	[-0.182,0.206]
       2	0.1	0.01
       3	0.025	0.0025
       4	0.001	0.0001
       5	0.0005	0.00005
       6	0.0002	0.00002

    7. Continue adding terms to `P text[(] x text[)]` and changing the "next term" `N text[(] x text[)]`, answering the questions of parts (a)–(e), and filling in each line of Table E1. Use the ranges and tick spacing indicated in the table.
17. 1. What is the largest interval on which Taylor polynomials could possibly approximate `sin^(-1)x`?
       
       
    Here, thanks to a computer algebra system, are the first seven derivatives of the function `f text[(] x text[)] =sin^(-1)x`:

    `f' text[(] x text[)] `	`=1/(text[(]1-x^2text[)]^(1//2))`	`f^(5) text[(] x text[)] `	`=(3text[(]8x^4+24x^2+3text[)])/(text[(]1-x^2text[)]^(9//2))`
    `f'' text[(] x text[)] `	`=x/(text[(]1-x^2text[)]^(3//2))`	`f^(6) text[(] x text[)] `	`=(15xtext[(]8x^4+40x^2+15text[)])/(text[(]1-x^2text[)]^(11//2))`
    `f^(3) text[(] x text[)] `	`=(2x^2+1)/(text[(]1-x^2text[)]^(5//2))`	`f^(7) text[(] x text[)] `	`=(45text[(]16x^6+120x^4+90x^2+5text[)])/(text[(]1-x^2text[)]^(13//2))`
    `f^(4) text[(] x text[)] `	`=(3xtext[(]2x^2+3text[)])/(text[(]1-x^2text[)]^(7//2))`

    2. Find the Taylor polynomials of degrees `0` through `7` for `sin^(-1)x`.
    3. Let `E_7 text[(] x text[)] =sin^(-1)x-P_7 text[(] x text[)]`, where `P_7 text[(] x text[)]` is the seventh-degree polynomial you found in part (b). Use your computer or calculator to make a table of `x`, `sin^(-1)x`, `P_7 text[(] x text[)]`, and `E_7 text[(] x text[)]` for `x=0.1, 0.3, 0.5, 0.7, 0.9`.
    4. Without any additional calculation, make a table of `x`, `sin^(-1)x`, `P_7 text[(] x text[)]`, and `E_7 text[(] x text[)]`for `x=-0.1, -0.3, -0.5, -0.7, -0.9`.
    5. Complete the following formula: For `-1<x<1`,

       `sin^(-1)x=tan^(-1)`____________ .

       (Hint: Draw a triangle in which one angle is `sin^(-1)x`.)
    6. Some computer algebra systems convert all arcsines to arctangents for evaluation. Can you think of a reason why this conversion might be built into the design of the system?
The Taylor polynomials for a function `f text[(] x text[)]` at the reference point `x=a` are the polynomials of the form

P n ( x ) = f ( a ) + f ′ ( a ) ( x - a ) + f ′ ′ ( a ) 2 ! ( x - a ) 2 + ⋅⋅⋅ + f ( n ) ( a ) n ! ( x - a ) n .

In sigma notation, we may write the defining formula as

P n ( x ) = ∑ k = 0 n f ( k ) ( a ) k ! ( x - a ) k .

The Taylor series for a function `f text[(] x text[)]` at the reference point `x=a` is the infinite series

f ( a ) + f ′ ( a ) ( x - a ) + f ′ ′ ( a ) 2 ! ( x - a ) 2 + ⋅⋅⋅ + f ( n ) ( a ) n ! ( x - a ) n + ⋅⋅⋅

            = ∑ k = 0 ∞ f ( k ) ( a ) k ! ( x - a ) k .

18. Find the Taylor polynomials of degrees `0` through `6` for each of the following functions and reference points.
    1.   `ln x`, `a=1`
    2.   `sin x`, `a=pi//2`
    3.   `cos x`, `a=pi//2`
    4.   `e^x`, `a=1`
    5.   `e^(-x)`, `a=1`
    6.   `tan^(-1)x`, `a=2`
19. For each of the following functions and reference points, find the interval on which the Taylor polynomial of degree `6` approximates the function with an error no greater than `0.01`.
    1.   `ln x`, `a=1`
    2.   `sin x`, `a=pi//2`
    3.   `cos x`, `a=pi//2`
    4.   `e^x`, `a=1`
    5.   `e^(-x)`, `a=1`
    6.   `tan^(-1)x`, `a=2`
20. Find the Taylor series for each of the following functions and reference points.
    1.   `ln x`, `a=1`
    2.   `sin x`, `a=pi//2`
    3.   `cos x`, `a=pi//2`
    4.   `e^x`, `a=1`
    5.   `e^(-x)`, `a=1`
    6.   `tan^(-1)x`, `a=2`
21. As best you can, estimate the interval of convergence of the Taylor series for each of the following functions and reference points.
    1.   `ln x`, `a=1`
    2.   `sin x`, `a=pi//2`
    3.   `cos x`, `a=pi//2`
    4.   `e^x`, `a=1`
    5.   `e^(-x)`, `a=1`
    6.   `tan^(-1)x`, `a=2`