Practice Exam: Series and Taylor Series Time: 60 minutes
Problem 1 (15 points)
Use the fourth degree Taylor polynomial of to find the exact value of
Explain your reasoning!
Solution: Taking derivatives, if necessary, we obtain that the fourth degree Taylor polynomial for
equals
.
Using the substitution y=2x, the Taylor polynomial for
becomes
Consequently for
,
Thus
Problem 2 (15 points)
Find the third degree Taylor polynomial of
with center x_{0}=3.
Solution: We have to find the first three derivatives of f(x) at x_{0}=3.
,
so
.
,
so
.
,
so
.
Finally,
,
so
.
Thus the third degree Taylor polynomial with center x_{0}=3 is given by
Problem 3 (15 points)
Let . Find the Taylor series of f(x) with center x_{0}=0 and its radius of convergence.
Solution: This is easiest if you remember that the Taylor series with center y_{0}=0 for
has radius of convergence 1 and is given by
Using the substitution y=x^{2}, one then obtains the Taylor series for f(x):
Since
,
the Taylor series for f(x) will also have 1 as its radius of convergence.
Alternatively, observe that
,
then write down a geometric series expression for
and integrate.
Problem 4 (15 points)
Find the radius of convergence of the power series
Solution: The absolute value of the general term of the series is
Consequently
The series will converge (diverge), if this quantity is less than 1 (bigger than 1).
so the radius of convergence is
.
Problem 5 (20 points)
Find the exact value of the following series:
Solution:
Solution:
You can only do this problem if you recognize the given series as a special case (x=1/2) of the Taylor expansion
.
Problem 6 (20 points)
An antibiotic decays exponentially in the human body with a half-life of about 2.5 hours. Suppose a patient takes a 250 mg tablet of the antibiotic every 6 hours.
1.
Write an expression for Q_{2}, Q_{3}, Q_{4}, where Q_{n} is the amount (in mg) of the antibiotic in the body after the
tablet is taken. Note that Q_{1}=250 mg.
2.
Write an expression for Q_{n}, and put it in closed form.
3.
Assume the antibiotic treatment consists of a total of 28 tablets. Give a numerical estimate for the amount of antibiotic in the body immediately after the patient takes the last tablet of the treatment.
Solution: Using an exponential decay model of the form
,
where t is measured in hours, and the given information that
,
we can compute
.
Consequently