Math 132, Test 1, 21 April 2011
Name:
Instructions. Do each of the following nine problems. Please do your best, and show all
appropriate details in your solutions; you may not use a calculator.
a
f (t) dt. Find F (x).
1. (14 pts) (a) Suppose f is a conti
Math 132, Test 2, 10 May 2011
Name:
Instructions. Do each of the following seven problems. Please do your best, and show all
appropriate details in your solutions; you may not use a calculator on questions 1. through 5.
1. (12 pts) Short answers.
(a) Let
Math 132, Test 3, 6 June 2011
Name:
Instructions. Do each of the following ve problems. Do your best, and show all appropriate
details in your solutions; you may not use a calculator so express answers in exact form.
1. (a) (5 pts) Consider a planar lamin
Math 132 Hyperbolic Functions
Hints and Answers
1. (a)
2
ex + ex
2
cosh2 sinh2 x =
ex ex
2
2
e2x + 2 + e2x e2x 2 + e2x
4
= = 1.
4
4
4
=
e2x e2x
ex ex ex + ex
(b) sinh 2x =
=2
= 2 sinh x cosh x
2
2
2
(c) tanh2 x + sech 2 x =
sinh2 x + 1
cosh2 x
=
=1
cosh2
Math 132 Inverse Trig Functions (Hints & Answers)
April 19, 2011
1. Verify f (x) = sin(arctan 2x) and g(x) =
2x
are equal, then compute their derivatives
1 + 4x2
and verify the derivatives are equal as well.
Answer. Let y = arctan 2x, then /2 < y < /2 an
Math 132, Test 1, 21 April 2011
Name:
Hints and Answers
Instructions. Do each of the following nine problems. Please do your best, and show all
appropriate details in your solutions; you may not use a calculator.
a
f (t) dt. Find F (x).
1. (14 pts) (a) Su
Math 132 Hints and Answers on Practice Questions
u
2
t2t sin t dt where u = 2x3 . According to the Chain Rule and
1. (a) Write F (u) =
4
Fundamental Theorem of Calculus,
d
d
du
2
6
F (x) =
F (u)
= u2u sin u 6x2 = 3x5 24x +2 sin(2x3 )
dx
du
dx
2x3
x2
t2
a
Math 132, Quiz 1, 5 May 2011
Name:
Instructions. Do each of the following three problems. Please do your best, and show all
appropriate details in your solutions; you may not use a calculator.
1. Solve the dierential equation
dy
= 3yx2 2yx subject to y(1)
Math 132, Test 3, 6 June 2011
Name: that] and AN
Instructions. Do each of the following ve problems. Please do your best, and Show all
appropriate details in your solutions; you may not use a calculator so express answers in exact
form.
1. (a) (5 pts
Math 132, Quiz 2 9 May 2011
Name:
Instructions. Do each of the following problems.
dy
where y = arccot u by
dx
dy
dierentiating both sides of u = cot y with respect to x and solving for
.
dx
1. Let u be a dierentiable function of x. Derive a formula for
2
Math 132 Practice Questions II
Hints and Answers
2xex
.
1 + x4
x
x
(b) g (x) = arcsin x +
= arcsin x
2
1x
1 x2
1. (a) f (x) = ex arctan(x2 ) +
2. f (x) =
1
1. Thus f (x) = 0 when |x| x2 1 = 1. Then x2 (x2 1) = 1 or
|x| x2 1
x4 x2 1 = 0. This implies x2