1. No category

Math 222 Worksheet 41
1. Compute each of the following integrals.
Z
(a)
sin2 (3θ) dθ
Z
(b)
Z
(c)
Z
(d)
t2 e−t dt
ex sin(2x) dx
dz
√
z2 z2 − 4
2. Compute each of the following definite integrals.
Z
1
x arctan x dx
(a)
0
Z
2
(b)
1
Z
dt
+ 1)
t2 (t
1/2
√
(c)
0
Z
3
(d)
1
s3 + 3s − 10
ds
s3 + 2s2 + 5s
Z
3. Let In =
dz
4z 2 + 1
(ln x)n dx
(a) Find a reduction formula for In .
(b) Use the formula you found in part (a) to compute I2 .
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Math 222 Worksheet 41
4. Compute the following improper integrals.
Z ∞
arctan x
(a)
dx
1 + x2
0
Z
1
x2
√
dx
1 − x2
(b)
1/2
Z
∞
(c)
1
dx
x(ln x)2
5. Determine whether the following integrals converge or diverge.
Z
∞
(a)
2
Z
(b)
0
Z
1
sin2 x
x2 − 1
x2 + x + 1
√ dx
x4 + x
∞
3
e−x dx
(c)
0
Z
(d)
2
∞
dx
x + (ln x)4
6. Match each of the following functions with the corresponding slope field on the next
page.
(a) y 0 = tan y
(b) y 0 = xy
(c) y 0 = −y
(d) y 0 = y − x
2
Math 222 Worksheet 41
Slope fields for Question 6
I.
II.
III.
IV.
3
Math 222 Worksheet 41
7. For each of the following equations find (i) the general solution (including any equilibrium solutions) and (ii) the solution to the initial value problem.
(a)
dy
dx
+ y 2 tan2 x = 0, y(0) = 1
dy
(b) y dx
=
(c)
dy
dx
p
y 2 − 1, y(0) = −2
= y 3 ex , y(0) = 0
8. For each of the following equations find (i) the general solution and (ii) the solution to
the initial value problem.
(a) y 0 + y = e−x , y(0) = 2
(b) xy 0 + 3y = sin x, y(π) = 0
(c) (x − 1)y 0 = 1 − xy, y(0) = 0
9. Consider the initial value problem y 0 = y 2 − x, y(0) = 1. Use Euler’s method with two
steps to estimate y(1).
10. Suppose you start with no intial capital and begin investing in an account that earns
10% interest. Assume investments are made and interest is compounded continuously.
(a) Suppose you invest $1000 a year. How much money will you have after 40 years?
(b) Suppose you wish to have $1000000 after 40 years. How much will you need to
invest each year (assuming you invest the same amount each year)?
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Math 222 Worksheet 41
11. A 1000 L tank is initially full of a 4 g/L salt solution. Starting at t = 0 minutes a 2
g/L salt solution flows in at 5 L/min. The solution in the tank is kept well-mixed.
(a) Suppose the mixture in the tank flows out at 5 L/min. Find the concentation in
the tank after t minutes. What happens to the concentration as t → ∞.
(b) Suppose the mixture in the tank flows out at 10 L/min. Find the concentration
in the tank after t minutes. For which t is the solution valid?
12. Find each of the following Taylor polynomials.
π/4
(a) T2 {tan x}
(b) T31 {ln x}
13. Find each of the following Taylor polynomials without computing any derivatives.
cos x
(a) T3
1−x
2
(b) T2 {e−x (1 + x)1/2 }
14. Find each of the following Taylor series. Each can be found without any tedious or
difficult computations.
(a) T∞ {(1 + x)e−x }
(b) T∞
sin(x2 )
x
(c) T∞ arctan(x2 )
15. Find an interval of x values for which 1+x is an estimate for ex with error of magnitude
less than 3/200.
16. Give an estimate of cos(1/2) with an error of magnitude less than 0.01.
5
Math 222 Worksheet 41
Z
t
17. Use T2 {e } to estimate
error.
1
2
e−x dx and give an upper bound for the magnitude of the
0
18. Find the limit of each of the following sequences.
(a) an = ln(n2 + n − 1) − ln(2n2 − 1)
(b) an =
3n
n!
19. Find the sum of each of the following series.
∞
X
3k
(a)
22k+1
k=0
(b)
∞
X
(−1)j
j=1
2j+1
20. Determine whether each of the following series converge or diverge.
(a)
∞
X
(−1)n sin(n)
n=1
(b)
∞
X
2
ne−n
n=1
(c)
∞
X
n=2
n3
n
−1
21. Find the radius and interval of convergence of each of the following power series.
(a)
n
∞ X
x+1
n=1
(b)
∞
X
n
n!xn
n=0
∞
X
(x − 2)n
√
(c)
n
n=1
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Math 222 Worksheet 41
22. Estimate the sum
∞
X
(−1)n−1
n=1
n2
to within 1/10 of its actual value.
23. For which x does lim Rn {sin x} = 0? What does this tell us about the relationship
n→∞
between sin x and T∞ {sin x}?
24. Consider the following vectors
~a =
1
2
, ~b =
−1
1




1
0
, ~c =  3  , d~ =  −1 
1
2
Which of the following quantities are defined? Compute those that are defined.
(a) ~a + ~b
(b) ~a + ~c
(c) ~a · ~b
(d) ~a × ~b
(e) ~c × d~
(f) d~ × ~c
(g) ||~c||
25. Consider the following vectors.
~a =
4
−3
, ~b =
1
−2
Find vectors ~ak and ~a⊥ such that ~a = ~ak + ~a⊥ , ~ak k ~b, and ~a⊥ ⊥ ~b.
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Math 222 Worksheet 41
26. Give a parametrization of the line through the points A(2, 1, 3) and B(0, 1, 4).
27. Find an equation for the plane through the points A(1, 1, 0), B(2, 2, 3), and C(1, −1, −4).


1+t
28. Where does the plane x1 − x2 − x3 = 0 intersect the line ~x(t) =  2t ?
1−t
29. Find the angle between the following two lines.




1+t
s
~x(t) =  2 − t  , ~x(s) =  1 
2t
2+s
30. Consider the points A(1, 2, 1), B(0, 3, −1), and C(1, −1, 4).
(a) Find a point D such that ABCD is a parallelogram.
(b) Find the area of ABCD.
31. Find the volume of the parallelpiped spanned by the following vectors.


 


1
2
−1
~a =  1  , ~b =  0  , ~c =  2 
1
1
1
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Math 222 Worksheet 41
Answers
1. (a)
1
θ
2
1
12
−
sin(6θ) + C
(b) −t2 e−t − 2te−t − 2e−t + C
(c)
1 x
e
5
√
(d)
sin(2x) − 25 ex cos(2x) + C
z 2 −4
4z
+C
2. (a)
π
4
−
1
2
(b)
1
2
+ ln( 34 )
(c)
1
2
√
ln( 2 + 1)
(d) 2 + arctan(2) −
π
4
− 2 ln 3
3. (a) In = x(ln x)n − nIn−1
(b) x(ln x)2 − 2x ln x + 2x + C
4. (a) π 2 /8
(b)
π
6
√
+
3
8
(c) Diverges
5. (a) Converges
(b) Converges
(c) Converges
(d) Diverges
6. (a) IV
(b) II
(c) I
(d) III
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Math 222 Worksheet 41
7. (a) (i) y = 1/(tan x − x + C) or y = 0, (ii) y = 1/(tan x − x + 1)
(b) (i) y = ±
q
p
√
1 + (x + C)2 or y = ±1, (ii) y = − 1 + (x + 3)2
√
(c) (i) y = ±1/ C − 2ex or y = 0, (ii) y = 0
8. (a) (i) y = (x + C)e−x , (ii) y = (x + 2)e−x
(b) (i) y = (−x2 cos x + 2x sin x + 2 cos x + C)/x3
(ii) y = (−x2 cos x + 2x sin x + 2 cos x + 2 − π 2 )/x3
(c) (i) y = (1 + Ce−x )/(x − 1), (ii) y = (1 − e−x )/(x − 1)
9. 19/8
10. (a) 10000(e4 − 1), which is $535981.50, rounded to the nearest cent.
(b) 100000/(e4 − 1), which is $1865.74, rounded to the nearest cent.
11. (a) C(t) = 2 + 2e−t/200 , C(t) → 2 as t → ∞.
(b) C(t) = 4 − t/100, which is valid for t < 200
12. (a) 1 + 2(x − π4 ) + 2(x − π4 )2
(b) (x − 1) − 21 (x − 1)2 + 31 (x − 1)3
13. (a) 1 + x +
x2
2
+
x3
2
(b) 1 + 12 x − 89 x2
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Math 222 Worksheet 41
14. (a) 1 +
∞ X
(−1)n
n=1
(b)
∞
X
(−1)n x4n+1
n=0
(c)
n!
(−1)n−1 n
+
x
(n − 1)!
(2n + 1)!
∞
X
(−1)n x4n+2
n=0
2n + 1
15. This is guaranteed on the interval −1/10 < x < 1/10
16. 7/8
17. 23/30. The error is less than 1/14. (You may find a larger upper bound, such as 1/2,
for the error depending on the specific estimates you use).
18. (a) ln(1/2)
(b) 0
19. (a) 2
(b) −1/6
20. (a) Diverges.
(b) Converges.
(c) Converges.
21. (a) Radius: ∞, Interval: −∞ < x < ∞.
(b) Radius: 0, Interval x = 0.
(c) Radius: 1, Interval −1 ≤ x < 3.
22. 31/36
23. For all real numbers x. This implies that sin x = T∞ {sin x} for all x.
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Math 222 Worksheet 41
0
3
24. (a)
(b) Undefined
(c) 1
(d) Undefined


7
(e)  −2 
−1


−7
~ = 2 
(f) d~ × ~c = −(~c × d)
1
(g)
25. ~ak =
√
11
2
−4
, ~a⊥ =
2
1


2 − 2t
26. ~x(t) =  1 
3+t
27. x1 + 2x2 − x3 = 3
28. The line is entirely within the plane.
29. π/6
30. (a) D(2, −2, 6)
√
(b) 3 3
31. 1
12