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- Let f(x) = (
x − 1) 5 = (x − 1) 5 / 2 .
1a. Use the following table of derivatives of f to find the third degree Taylor polynomial approximation P 3 (x) of f(x), in
powers of x − 5, ie, x 0 = 5.
k f(k)^ (x) f(k)^ (5)
x − 1) 5 = (x − 1) 5 / 2 32
(x − 1)
3 / 2 20
x − 1
x − 1
(x − 1)
− 3 / 2 −^15
1b. Taylor’s theorem guarantees that f and P 3 will separate by no more than what value (call it ), on the interval
[2. 5 , 6]? Show all your work. (Find K 4 to two decimal places. Find the maximum guaranteed error to 4 decimal places.)
1c. Since f(x) = (
x − 1)^5 , we could use it to find (
3)^5 = 3^5 /^2 by taking x = 4. Use the same x to approximate (
3)^5
with your Taylor polynomial P 3 (x). What do you get? What’s the error in this approximation? (Write your answers to 4
decimal places).
- Find
x 3 − 4 x √ 4 + x^2
dx by introducing an appropriate trig substitution, say, x = 2 tan t. Express your answer in terms of
x and
4 + x^2 (leave no trig functions in your answer if possible)
- At some point, most calculus students learn that
d
dt
arcsin t is
1 − t^2
. But to find
arcsin x dx requires integration
by parts. Show how.
5A. Find
4 x − 5 + 7 x 2
(1 + x^2 ) (x − 3)
dx.
Show all your work, from setting up the partial fraction decomposition to solving for A, B, C, etc, to final integrations.
5B. Explain why
0
4 x − 5 + 7 x 2
(1 + x^2 ) (x − 3)
dx is an improper integral.
5C. Does the integral in 5B converge? Explain! (You do not have to make a table. Just make a good argument, knowing
standard behavior of the functions involved in the anti-derivative in 5A).
- Here is the graph of f(x) =
0 if x < − 1
0 .25(x + 1) if − 1 ≤ x < 1
λe −λx if 1 ≤ x for a certain value of λ:
6a. By definition, what is the total area of the entire (both the light and dark parts combined) shaded region required to
be in order for f to be a probability density function?
6b. It’s easy to find the area of the lightly-shaded region. What is it?
6c. Find λ so that the area of the darkly-shaded region is what it needs to be, according to your answers in 6a and 6b.
You may use without verification this fact:
a
λe −λx dx = e −λa .
6d. Suppose that X is a random variable with this f as its probability density function. What’s the probability that X
is negative?
6e. What’s the probability that X is in the interval [0, 1]?
6f. What’s the probability that X is greater than 4?
