University:
Whitman College
Course:
MATH-126 | Calculus II
Academic year:

2021
Views:

229

Pages:

2
Author:

tristepeon32s

Final Exam Review Calculus II Sheet 1 1. Prove by induction: n X i2 = i=1 2. State the definition of Z b n(n + 1)(2n + 1) 6 f (x) dx. a 3. True or False, and give a short reason: (a) The Alternating Series Test is sufficient to show that a series is conditionally convergent. (b) You can use the Integral Test to show that a series is absolutely convergent. (c) Consider P an . If lim an = 0, then the sum is said to converge. n→∞ (d) The sequence an = 0.1n converges to 1 1−0.1 4. Set up an √ integral for the volume of the solid obtained by rotating the region defined by y = x − 1, y = 0 and x = 5 about the y−axis. √ 5. Write the area under y = 3 1 + x, 1 ≤ x ≤ 4 as the limit of a Riemann sum (use right endpoints). For the same function, write an integral representing the arc length (do not evaluate the integral). √ 6. Find the Taylor series for f (x) = x centered at a = 4 (write the first four terms of the series). 7. Find dy , dx if y = R 5x cos(x) cos(t2 ) dt 8. Let f (x) = ex on the interval [0, 2]. (a) Find the average value of f . (b) Find c that is guaranteed by the Mean Value Theorem for Integrals. 9. Use a template series to find the Maclaurin series for R cos(x2 ) dx. 10. Does the series converge (absolute or conditional), or diverge? (a) ∞ X cos(n/2) 2 n=1 n + 4n (b) ∞ X n2 + 1 5n n=1 (c) ∞ X 3n n2 n=1 n! 11. Find the interval of convergence: (a) ∞ X n 2 xn n n=1 10 (b) ∞ X (3x − 2)n n3n n=1 1 (c) ∞ X (−1)n x2n−1 n=1 (2n − 1)! 12. Evaluate the integral. You should be able to do these without the table of integrals. 1 (a) dy 2 y − 4y − 12 Z 2 2x + 3 (b) + 2 dx 3x + 1 x + 9 Z (c) (d) Z x2 cos(3x) dx Z 2 |x − 1| dx −2 dx x ln(x) Z √ (f) x x − 1 dx (e) Z 13. The velocity function is v(t) = 3t − 5, 0 ≤ t ≤ 3 (a) Find the displacement. (b) Find the distance traveled. 14. Use a template series to find the Maclaurin series for the indefinite integral: Z ex − 1 dx x 15. Evaluate the indefinite integral as a power series. What is the radius of convergence? Z t dt 1 + t3 16. Find the length of the curve y= Z x q√ t − 1 dt, 1 ≤ x ≤ 16 1 17. Find an integral that gives the surface area, if the parabola y = 1 − x2 , 0 ≤ x ≤ 1, is rotated about the line y = 2. (Do not evaluate it). 18. A leaky 10 kg bucket is lifted from the ground to a height of 12 m at a constant speed with a rope that weighs 0.8 kg/m. Initially, the bucket contains 36 kg of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the 12 m level. Set up the computations you would make to determine how much work is done. You do not have to actually compute the work, just set up the computations and/or integrals needed. You may keep gravity as g m/s2 , and use the constant σ if you need the density of water (in kg/m3 ). 2

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