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The final examination questions for a calculus 2 (ma126-6b) course. The exam covers various topics including integration, improper integrals, series, and maclaurin series. Students are required to evaluate integrals, find areas, volumes, arclengths, and determine convergence of series. Some questions involve using specific methods such as the midpoint method and cylindrical shells.
Typology: Exams
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Instruction:Use the scratch paper provided if needed. Please keep your Answer the questions in the space provided. answers neat, complete but brief, and to the point.
Question 1Question 2 Question 3Question 4 Question 5Question 6 Question 7Question 8 Question 10Question 9 Question 11Question 12 Question 13Question 14 Question 15 Total Please do not write in this box
QUESTION 1. Evaluate the integral: ∫ π/ 4 0 cos^ x^2 x dx.
QUESTION 3. The midpoint method (^) ∫M 1 n is used to approximate the following integral: 0 ex^3 dx. How large should one choose Hint: Recall that the error in the midpoint method can be estimated by: n in order to guarantee the error is less than 10−^6?
|EM | ≤ K( 24 b^ −n^2 a )^3.
QUESTION 4. Determine whether the following improper integral converges: ∫ 1 0
√x (^2) + 1 x dx.
QUESTION 6.under the curve Find the volume of the solid of revolution obtained by rotating the area y = x cos x, 0 ≤ x ≤ π/ 2 , about the y-axis:
0
y x^1
Hint: Use cylindrical shells.
QUESTION 7. Find the arclength of the curve: x = y^3 /^2 , 0 ≤ y ≤ 1.
QUESTION 9.does, find its limit. Justify your answer. Determine whether the sequence^ {(1 + (^) n^3 )^4 n}∞ n=1 converges, and if it
QUESTION 10.its sum: Determine whether the following series converges, and if it does, find∞ ∑ n=
)n
QUESTION 12.or converges conditionally: Determine whether the following series converges, converges absolutely, ∑∞ n=
( n− ln1) n n. Hint: Use the integral test.
QUESTION 13. Find the Maclaurin series for the function: f = (^) (1 −^1 x) 2. Determine the interval of convergence. Hint: 1 /(1 − x) (^2) is the derivative of 1 /(1 − x).
QUESTION 15. Check that the series ∞ ∑ n= (2^1 n)! converges, and find its sum. Hint: Find the Maclaurin series of cosh x = (ex (^) + x−x)/ 2.
