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

2024
Views:

378

Pages:

2
Author:

Sincere Hanna

Final Exam Review Calculus II Sheet 2 1. True or False, and give a short reason: (a) The Ratio Test will not give a conclusive result for (b) If ∞ X an converges for some large k, then so will P ∞ X 2n+3 3n4 +2n3 +3n+5 an . n=1 n=k (c) If f is continuous on [0, ∞) and x→∞ lim f (x) = 0, then (d) If f is continuous and Z 9 f (x) dx = 4, then 0 Z 3 R∞ 0 f (x) dx converges. xf (x2 ) dx = 4. 0 2. Short Answer: (a) Suppose the series cn 3n converges. Will cn (−2)n also converge? For what P values of x will the series cn (x − 2)n converge? P P (b) If an , bn are series with positive terms, and an , bn both go to zero as n → ∞, an then what can we conclude if lim = 0? n→∞ b n (c) What is the derivative of sin−1 (x)? Of tan−1 (x)? What is the antiderivative of each? P P (d) Find the sum: P∞ n=1 e−2n 3. Suppose h(1) = −2, h0 (1) = 2, h00 (1) = 3, h(2) = 6, h0 (2) = 5, and h00 (2) = 13, and h00 R 2 00 is continuous. Evaluate 1 h (u) du. s n X 3 3i 4. Determine a definite integral representing: n→∞ lim 1+ [For extra practice, try n i=1 n writing the integral so that a = 5]. 5. Evaluate Z 5 (1 + 2x) dx by using the definition of the integral (use right endpoints). 2 6. For each function, find the Taylor series for f (x) centered at the given value of a: (a) f (x) = 1 + x + x2 at a = 2 (b) f (x) = 1 x at a = 1. 7. Find a so that half the area under the curve y = the area lies in the interval [a, 4]. 1 x2 lies in the interval [1, a] and half of sin(x) x→0 x 8. Compute the limit, by using the series for sin(x): lim 9. Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by y = x, y = 4x − x2 , about x = 7. 1 10. Evaluate each of the following: d Z sin(x) 3 (a) t dt. dx 3x d Z5 3 (b) x dx dx 1 11. Converge (absolute or conditional) or Diverge? √ ∞ ∞ X X (−1)n n n2 − 1 (a) (b) 3 2 n=1 (n + 1)(n + 2) n=1 n + 2n + 5 (c) Z 5 1 (c) d 3 x dx dx ∞ X 4k k=1 +k k! 12. Find the interval of convergence. (a) ∞ X ∞ X (x + 2)n (b) n4n n=1 n n n x n=1 (c) ∞ X 2n (x − 3)n n=1 √ n+3 13. Evaluate: Z ∞ 1 (a) dx (x + 2)(x + 3) 0 Z √ √ (b) u( u + 3 u) du (c) Z x2 dx (4 − x2 )3/2 tan−1 (x) (d) dx 1 + x2 Z 1 √ dx (e) x2 − 4x Z (f) Z x4 ln(x) dx (g) (h) Z Z 3 0 (i) e−x sin(2x) dx. Z 1 √ dx x sin2 x cos5 x dx 14. Find a power series for x2 ln(5 − x) by first finding the series for 1/(5 − x). Give the radius of convergence. 15. Find the Maclaurin series for xe−x by using a template series. 16. Prove the following by induction: 1 · 2 + 2 · 3 + 3 · 4 + · · · + n · (n + 1) = n(n + 1)(n + 2) 3 17. Find an integral for the surface area of the object obtained by rotating the curve y = e−x for 0 ≤ x ≤ 1, about the x−axis. (Do not evaluate) 18. A bucket weighing 100 lbs is filled with sand weighing 500 lbs. A crane lifts the bucket from the ground to a point 80 ft in the air at a rate of 2 ft/s, but sand is leaking out of a hole at a rate of 3 lbs/s. Neglecting friction and the weight of the cable, find an expression (integral) that will compute the amount of work being done. 2

Details

Description

Related Documents

Struggling with your assignment and deadlines?

Free up your schedule!

Take 5 seconds to unlock