Math 231 ADA
End-of-Semester Practice Problems
1. [Ch 7] Compute the following antiderivatives. Z Z 1 x2 + 4 √ dx (a) (e) dx x2 − 9 x2 − 2x − 3 R√ R (b) 1 − x2 dx (f) tan2 (x)dx R R √ (g) sin(2x) cos(x)dx (c) x3 1 − x2 dx R R (d) x2 cos(x)dx (h) sec3 (x) tan3 (x)dx
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Z
x+1 dx (x2 + 4)(x2 + 3x + 2)
(j)
R
ex cos(x)dx
(k)
R
tan3 (x)dx
(l)
R
x ln(x + 1)dx
(i)
2. [Ch 7] Consider f (x) = x3 on the interval [1, 4]. R4 (a) Approximate 1 x3 dx using M3 , the midpoint approximation with n = 3. (b) Using the midpoint approximation, how many rectangles do we need (i.e. what is n) in order to make the error less than 0.1? R4 (c) Approximate 1 x3 dx using T3 , the trapezoidal approximation with n = 3. (d) Using the trapezoidal approximation, how many rectangles do we need (i.e. what is n) in order to make the error less than 0.1? 3. [Ch 7] Decide if the following converge or diverge. (Hint: Don’t forget the comparison theorem for integrals.) Z ∞ Z ∞ Z 4 2 1 1 dx (c) e−x dx (a) (e) dx x ln(x) (x − 2)2 0 2 2 Z 1 Z ∞ Z ∞ 1 1 1 (f) dx (b) dx dx (d) 2 2 x(ln(x)) x +1 −1 x 2 0 4. [Ch 8 & 10] Consider the curve C with parametrization x = ln(t), y = t2 ln(t) for 1 ≤ t ≤ e (a) Find an equation for the curve in Cartesian (x, y) coordinates. (b) Sketch the curve. (c) Write an integral for the length of the curve in terms of: i. x (with differential dx)
ii. t (with differential dt)
(d) Rotate C around the x-axis. Write an integral for the surface area of the resulting surface in terms of i. x (with differential dx)
ii. t (with differential dt)
(e) Rotate C around line x = 2. Write an integral for the surface area of the resulting surface in terms of i. x (with differential dx)
ii. t (with differential dt)
5. [Ch 10] Consider the curve with parametrization x = 1 − t2 and y = t2 + 5t + 6. Let C be the part of the curve which is below the x-axis. (a) Find the area between C and the x-axis. (b) Where are the vertical tangents along the entire curve (not just C)? (c) Where are the horizontal tangents along the entire curve (not just C)?
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Math 231 ADA
End-of-Semester Practice Problems
SL050914
6. [Ch 10] Plot the following points (r, θ) on the plane and give the corresponding Cartesian coordintes. (a) (3, π/2)
√ (b) ( 2, π/4)
(c) (−1, π/3)
(d) (1, π/3)
(e) (0, π)
(f) (1, 4π/3)
7. [Ch 10] Convert each of the following polar equations to Cartesian. (a) r = 2 cos(θ)
(b) r = 4
(c) θ = π/4
8. [Ch 10] Match the curves with their equations. √
(a) r = sin(θ)
(c) r = 1 + sin(θ)
(e) r = 4
(g) r =
(b) r = sin(2θ)
(d) r = 1 + 2 sin(θ)
(f) θ = π/4
(h) r = 2 − cos(θ)
I.
II.
V.
III.
VI.
θ
IV.
VII.
VIII.
9. [Ch 10] Consider the curve C below, given by r = θ for 0 ≤ θ ≤ 3π.
(a) Find the length of C.
(c) Find the area shaded red.
(b) Find the area shaded green.
(d) What is the rate of change at θ = π/2?
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Math 231 ADA
End-of-Semester Practice Problems
10. [Ch 11] Compute the sums of the following series. ∞ ∞ X X (−5)n+1 1 1 (a) (c) − 32n n n+1 n=0 n=1 10 20 40 80 (b) + + + + ··· 3 9 27 81
(d)
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(e)
∞ X 1 n! n=0
∞ X
1 2+n n n=1
(f) 1 −
π2 π4 π6 + − + ··· 2 4! 6!
11. [Ch 11] Decide if the following series converge absolutely, converge conditionally, or diverge. Justify your answer by stating the test you use and verifying all assumptions of the test. (a)
∞ X
(−1)n arctan(n)
n=1 ∞ X
(e)
∞ X
1 ln(n) n=2
(i)
∞ X (−1)2n (2n)! n=1
(j)
∞ X (−1)2n+1 2n n=1
(b)
1 n2 n=1
(f)
∞ X (−1)n 2n n=0
(c)
∞ X 1 √ n n=1
(g)
∞ X n+1 n2 + 15 n=2
(d)
∞ X
1 n ln(n) n=2
(h)
∞ X n=3
√
(k)
n−1 n5 − 14
(l)
∞ X en nn+1 n=1 n ∞ X n+1 (n) n−1 n=2
(m)
∞ X sin(n) n2 n=1
(o)
∞ X (−1/2)n n n=1
∞ X (−1)n ln(n) n n=2
12. [Ch 11] Decide if each of the following statements it is always true or not. P∞ (a) If the series n=1 an converges, then the sequence {an } converges. P∞ (b) If limn→∞ an = 4, then the series n=1 an diverges. P∞ (c) If limn→∞ an = 0, then the series n=1 an converges. P∞ P∞ (d) If n=1 |an | converges, then n=1 an converges. P∞ (e) If n=1 an = 15, then limn→∞ an = 15. P∞ (f) If n=1 an = 15, then limN →∞ sN = 15, where sN = a1 + · · · + aN . 13. [Ch 11] For each of the following series, suppose we have to add up the first several terms to approximate the sum of the series. How many terms should we add so that our error is less than 0.01? (a)
∞ X 1 n3 n=1
(b)
∞ X (−1)n n3 n=1
14. [Ch 11] For each of the following series, determine the radius and interval of convergence. ∞ X (−1)n (x − 4)n (a) 2n + 1 n=0
∞ X (x + 1)n (b) 4n n=0
∞ X (2x)n (c) n! n=0
15. [Ch 11] Find Taylor series for the given function at the given center. 3x2 ,a=0 (1 + x3 )2 (b) f (x) = sin(4x2 ), a = 0 (a) f (x) =
(c) f (x) =
1 ,a=0 3 − x2
(d) f (x) = sin(4x), a = π/8
16. [Ch 11] Solve the following using series. Z 2 2 (a) e−x dx
(e) f (x) =
√
1 + 5x, a = 0
(f) f (x) = e3x , a = 1
cos(x3 ) − 1 x→0 x6
(b) lim
0
17. [Ch 11] Find the 4th degree Taylor polynomial for e3x centered at 1. If we use this polynomial to approximate e1.2 , find an upper bound on our error using Taylor’s inequality.
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