MAT136H1F Calculus I (B)
Fall 2012, Challenge problems Week 9
Fridays questions
Determine whether the following sequences are convergent or divergent. If theyre convergent, nd their limit.
cfw_
n
1.
2n + 1 n=1
Solution: Convergent, and converges to 1 .
2
MAT136H1F Calculus I (B)
Fall 2012, Challenge problems Week 3
Mondays questions
Find the derivative of the following functions.
y
1. g(y) = 2 t2 sin tdt
Solution: g (y) = y 2 sin y
2. g(x) =
1/x
0
arctan xdx
Solution: g (x) =
3. (*) g(x) =
1
x2
arctan
(1
MAT136H1F Calculus I (B)
Fall 2012, Challenge problems Week 2
Mondays questions
Find the area under the following curves as a limit of sums. Do not evaluate.
1. f (x) = x for 0 x 3.
Solution: limn
n
3
i=1 n
3i
n
2. f (x) = sin x for 0 x 2.
Solution:
lim
n
MAT136H1F Calculus I (B)
Fall 2012, Challenge problems Week 4
Mondays questions
Find the area bounded by the following curves.
1. y = x2 and y = 4x x2 .
Solution:
2. (*) y =
2
0
[(4x x2 ) x2 ]dx =
x, y =
x
2
8
3
and x = 9.
Solution: In the region [0, 4],
MAT136H1F Calculus I (B)
Fall 2012, Challenge problems Week 5
Wednesdays questions
1.
(2x 3) sin(x)dx
Solution: u = 2x 3, dv = sin(x):
1
2
(2x 3) sin(x)dx =
(2x 3) cos(x) + 2 sin(x) + C
2. (*)
x tan2 xdx
Solution: Hint: What the derivative of tan x x?
is
MAT136H1F Calculus I (B)
Fall 2012, Challenge problems Week 10
Wednesdays questions
Determine whether the following sequences converge or diverge. If the converge, determine
their limit.
1. cfw_arctan(2n)
Solution: This convergent, as lim arctan(2n) = lim
MAT136H1F Calculus I (B)
Fall 2012, Challenge problems Week 8
Mondays questions
Sketch the direction elds for the following equations.
1. y = 1 + y
Solution:
10
y(t)
K
4
5
K
0
2
2
4
t
K
5
K
10
2. y = y 2x
Solution:
10
y(t)
K
4
5
K
0
2
2
4
t
K
5
K
10
Page
MAT136H1F Calculus I (B)
Fall 2012, Challenge problems Week 7
Wednesdays questions
Determine if the following are convergent or divergent, and evaluate them if convergent.
1
1.
dx
(3x + 1)2
1
Solution: Convergent (compare to
1
),
x2
and converges to
1
.