Callahan
    10.2: 1, 3, 5, 7, 10, 11, 12, 15, 17, 23(d)(e)

1)  Find a seventh-degree Taylor polynomial centered at for the indicated antiderivatives:

    a)  

    First we just find the taylor polynomial form of the integrand:    (p535)

    Now we can do the integral:

    b)  

    First we just find the taylor polynomial form of the integrand:    (p535)

    Now we can do the integral:

    c)  

    First we just find the taylor polynomial form of the integrand:    (p535)

    Now we can do the integral:

3)  Using the seventh-degree Taylor approximation:

    calculate the values of and .  Give only the significant digits-that is, report only those decimals of your estimates that you think are fixed:

    For :

    First we find the value for the seventh-degree Taylor approximation:

    First the value for the ninth-degree Taylor approximation:

    Now we check to see which digits are the same in both:

    For :

    First we find the value for the seventh-degree Taylor approximation:

    First the value for the ninth-degree Taylor approximation:

    Now we check to see which digits are the same in both:

5)  Find the third-degree Taylor polynomial for at .  Show that the Taylor polynomial is actually equal to g(x). :

    To find the Taylor polynomial for at we use the form centered at given on page 535:

    So we need the derivatives of the original function to find the polynomial:

    So the third-degree Taylor polynomial is:

    But our center is at , so:

    What does this imply about the fourth-degree Taylor polynomial for at :

    All of the derivatives of degree greater than three were zero, so all terms beyond the third degree will necessarily also be zero.  So, any Taylor polynomial approximation of degree three or above are the same.  Also the approximation is exact in this case because we are approximating a polynomial.

7)  In this problem you will compare computations using Taylor polynomials centered at with computations using Taylor polynomials centered at :

    a) Calculate the value of using a seventh-degree Taylor polynomial centered at .  How many decimal places of your estimate appear to be fixed?

    So, using our general form for the Taylor polynomial:

    First, we need the derivatives of the original function to find the terms in the polynomial, then we evaluate each derivative at its center:

    So the general seventh-degree Taylor polynomial is:

    But our center is at , and we substitute the derivatives evaluated at the center that we found above:

    Now we just evaluate our Taylor approximation for at to get our estimate for the value of :

    To find out how many decimal places of our estimate appear to be fixed we need to compare the above estimate to one with one more term in the approximation.  We can easily see what the next term in the series will be from the pattern of Taylor polynomial we found above:

    So, it looks like no decimal places of our estimate are fixed, because none are the same in the two approximations.

    b) Now calculate the value of using a seventh-degree Taylor polynomial centered at .  Now how many decimal places of your estimate appear to be fixed?

    So, using our general form for the Taylor polynomial:

    First, we need the derivatives of the original function to find the terms in the polynomial, then we evaluate each derivative at its center:

    So the general seventh-degree Taylor polynomial is:

    But our center is at , and we substitute the derivatives evaluated at the center that we found above:

    Now we just evaluate our Taylor approximation for at to get our estimate for the value of :

    To find out how many decimal places of our estimate appear to be fixed we need to compare the above estimate to one with one more term in the approximation.  We can easily see what the next term in the series will be from the pattern of Taylor polynomial we found above:

    So, it looks like 13 decimal places of our estimate are fixed, because they are the same in the two approximations.  Using as the center seems to make the estimate more accurate then with the center at 0. So:

    Just for comparison, here is the true value of :

10)  Use the general rule to derive the fifth-degree Taylor polynomial centered at for the function:

    Use this approximation to estimate .  How accurate is this?

    So, using our general form for the Taylor polynomial:

    First, we need the derivatives of the original function to find the terms in the polynomial, then we evaluate each derivative at its center:

    So the general fifth-degree Taylor polynomial is:

    But our center is at , and we substitute the derivatives evaluated at the center that we found above:

    Now we just evaluate our Taylor approximation for at to get our estimate for the value of :

    To find out how accurate this estimate we can compare it to the real value:

    So, it looks like 7 decimal places of our estimate match the real value.

11)  Use the general rule to derive the formula for the nth-degree Taylor polynomial centered at for the function:

    So, using our general form for the Taylor polynomial:

    First, we need the derivatives of the original function to find the terms in the polynomial, then we evaluate each derivative at its center:

    So the general nth-degree Taylor polynomial is:

    But our center is at , and we substitute the derivatives evaluated at the center that we found above:

12)  Use the result of the preceding problem to get the sixth-degree Taylor polynomial centered at for the function:

    So, if we rewrite this function we can see how it will fit into the form of the last problem:

    Now to get a sixth-degree Taylor polynomial in we can see that we only need a third-degree Taylor polynomial in :

15)

    a)  Apply the general formula for calculating Taylor polynomial centered at to the tangent function to get the fifth-degree approximation:

    So, using our general form for the Taylor polynomial:

    First, we need the derivatives of the original function to find the terms in the polynomial, then we evaluate each derivative at its center:

    The first derivative:

    The second derivative:

    The third derivative:

    The forth derivative:

    The fifth derivative:

    So the general fifth-degree Taylor polynomial is:

    But our center is at , and we substitute the derivatives evaluated at the center that we found above:

    b)  Recall that .  Multiply the fifth-degree Taylor polynomial for from part (a) by the forth-degree Taylor polynomial for and show that you get the fifth-degree polynomial for :

    So this is the fifth-degree polynomial for :

17)  Note that:

    Use these to get the nth-degree Taylor polynomial centered at for :

    So:

23)  Using a suitable formula for each of the functions involved, find the indicated limit:

    d)  

    First we use the Taylor polynomial:

    Now we substitute this into the expression, and try to simplify it so that we can find the limit:

    Now we should be able to find the limit if we note that the remainder as (where is some constant).  This is because, of all the terms included in the remainder, the terms with order greater than five go to zero faster than the 5th order term.  So, if we can manipulate the expression so that we are dividing by , then as :

    Here we also used the fact that the limit of a product, is equal to the product of the limits.

    e)  

    First we use the Taylor polynomials:

    Now we substitute this into the expression, and try to simplify it so that we can find the limit:

    Now we should be able to find the limit if we note that the remainder as , the remainder as .  So, if we can manipulate the expression so that we are dividing by , then as , and we are dividing by , then as :