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Calculus is most common subject I know so far. This is one of past exam papers you can find in my uploads. Key points of the exam are: Integral is Improper, Directions, Completeness, Clarity, Assigned, Correctness, Integral is Improper, Value, Diverges, Fractional Coeficients
Typology: Exams
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Math 106 Name
Exam 2
11/2/
Read directions carefully and show all your work. Partial credit will be assigned based upon the correctness,
completeness, and clarity of your answers.
(a)
cos 12 x sin 3 x dx
(b)
x 2 sin x dx
(c)
4 x^2 − 7 x + 1
(x − 1)^2 (x + 1)
dx
(d)
4 − x^2
x^2
dx
(a) Find P 3 (x), the 3 rd order Taylor Polynomial, of f (x) centered at x = 0. Simplify your answer as much as possible, in other words, fractional coefficients must be in lowest terms.
(b) Use P 3 (x) to find estimates for ln(0.9) and ln(1.3). Show your work.
(c) Use Taylor’s Theorem to approximate the error of your estimate for ln(0.9), from part (b), on the interval [− 0. 5 , 0 .4]. Recall that error bounds for estimates using a Taylor Polynomial Pn(x) may be determined using: |f (x) − Pn(x)| ≤
Kn+
(n + 1)!
|x − x 0 | n+ .
[BONUS] Which of the approximations in part (b) would you expect to be more accurate? Briefly explain. (Note: simply evaluating ln(0.9) and ln(1.3) on your calculator to see which is closer to the estimates in (b) will not earn you any bonus points.)
dy
dx
= xe −y with y(0) = 3. Use separation of variables to find the solution
to the IVP. Be sure to use the initial condition to determine the values of any constants.
0
e^2 x^ + 2700x
dx. Use a comparison to determine if this integral converges or diverges? If it
converges find an upper bound for its value.
