Exercises

1. 1. Derive the Taylor series for `cos x`.
   2. Calculate `cos 1` (radian) by substituting `x=1` in your answer to part (a) and adding up partial sums until you see no change in the sixth decimal place.
   3. Check your result in part (b) by comparing your partial sums with your computer's or calculator's value for `cos 1`.
2. 1. If `f text[(] x text[)] =sin x`, what is the coefficient of `x^13` in the Taylor series for `f text[(] x text[)] `?
   2. For any function `f text[(] x text[)] ` with infinitely many derivatives, what is the coefficient of `x^13` in the Taylor series for `f text[(] x text[)] `?
3. 1. Find the Taylor series for the function `f text[(] x text[)] =(sin x)/x`. (Hint: Don't calculate any derivatives!)
   2. Use your Taylor series to estimate `int_0^1 (sin x)/x dx`.
   3. Use the integral function of your computer or calculator to evaluate the integral in part (b). How accurate was your estimate?
4. 1. Find the Taylor series for `f text[(] x text[)] =e^(-x).`
   2. In the Taylor series for `g text[(] x text[)] =e^x`, substitute `-x` for `x`. How is the resulting series related to the one you calculated in part (a)?
5. What does the graph of an exponential function look like on a log-log graph? What does the graph of a power function look like on a semilog graph? How are these observations related to Example 3?
6. 1. Show that `2^x` grows faster than any power function. (Hint: Write `2^x` as an exponential with base `e`.)
   2. Graph `f text[(] x text[)] =x^30/2^x` on a scale that shows clearly the limiting behavior as `x rarr oo`.
7. 1. Use your computer or calculator to find the value of `sin 100`.
   2. Find an angle between `-pi`/`2` and `pi`/`2` whose sine is the same as `sin 100`.
   3. Use your small angle from part (b) in the series for `sin x` to find `sin 100`.
   4. In general, given a number `x`, how would you find a number `alpha` between `-pi`/`2` and `pi`/`2` such that `sin alpha=sin x`? Describe a procedure that could be automatic enough to be programmed into a computer or calculator and that could evaluate the sine of any number.
8. The values of `cos x` are not all found in the interval `[-pi`/`2`, `pi`/`2]`. Why not? Describe an effective automatic procedure for evaluating cosine of any number. (One possibility: Use sine of a complementary angle. Can you think of others?)
9. 1. Graph `f text[(] x text[)] =sin x` and `g text[(] x text[)] =x root(3)(cos x)` on `[-3,3]`. Observe that these functions agree closely on `[-1,1]` but less so when `x` is farther from `0`.
   2. With the help of a computer algebra system, here are the first three derivatives of `g`:

      `g' text[(] x text[)] =(3 cos x-x sin x)/(3text[(]cos xtext[)]^(2//3))`

      `g'' text[(] x text[)] =-(3 x cos^2 x+6 sin x cos x+2x sin^2 x)/(9text[(]cos xtext[)]^(5//3))`

      `g''' text[(] x text[)] =-(27 cos^3 x+9x sin x cos^2 x+18 sin^2 x cos x+10x sin^3 x)/(27text[(]cos xtext[)]^(8//3))`

      Find the Taylor polynomials for `g text[(] x text[)] ` through degree 3.
   3. Use Taylor polynomials for both functions to explain your graphical observations in part (a).

Problem adapted from Calculus Problems for a New Century, edited by Robert Fraga, MAA Notes No. 28, 1993.

10. Let `f text[(] x text[)] =1/2(e^x+e^(-x))`. Note that this is the average of `e^x` and `e^(-x)`.
    1. Graph this function on the interval `[-3,3]`. Explain the symmetry you observe.
    2. Find the Taylor polynomial of degree 6 for `f text[(] x text[)] `. You can either calculate derivatives or find a simpler way to proceed. What symmetry do you observe in this polynomial function?
    3. Add the graph of the polynomial in part (b) to your graph in part (a). Do you see two distinct graphs or only one?
    4. How big does the graphing interval have to be before you can clearly distinguish the graph of `f text[(] x text[)] ` from that of its `6`th degree Taylor approximation?
11. Let `g text[(] x text[)] =1/2(e^x-e^(-x))`. Note that this is the average of `e^x` and `-e^(-x)`.
    1. Graph this function on the interval `[-3,3]`. Explain the symmetry you observe.
    2. Find the Taylor polynomial of degree 7 for `g text[(] x text[)] `. You can either calculate derivatives or find a simpler way to proceed. What symmetry do you observe in this polynomial function?
    3. Add the graph of the polynomial in part (b) to your graph in part (a). Do you see two distinct graphs or only one?
    4. How big does the graphing interval have to be before you can clearly distinguish the graph of `g text[(] x text[)] ` from that of its `7`th degree Taylor approximation?
12. We saw in this section that

    ∫ 0 x e - t 2 d t = x - x 3 3 + x 5 2 ! ⋅ 5 - x 7 3 ! ⋅ 7 + ⋅⋅⋅ + ( - 1 ) k   x 2 k + 1 k ! ⋅ ( 2 k + 1 ) + ⋅⋅⋅ ,

    and we used this formula to calculate

    ∫ 0 1 e - t 2 d t = 1 - 1 3 + 1 10 - 1 42 + 1 216 - 1 1320 + ⋅⋅⋅ = 0.74682412

    to six-decimal-place accuracy, using only simple arithmetic.
    1. Carry out a similar calculation to find `int_0^0.8 e^(-t^2) dt` to 6-place accuracy.
    2. Check your work by using the integral function of your computer or calculator.