Download MA 252 S02 Exam 2 Review 1 The second exam will be on

MA 252 S02
Exam 2 Review
1
The second exam will be on Thursday, March 31. The exam will cover the sections we covered
from §7.2-10.2.
You may not use a notecard for this exam.
The following formulas will be given to you on the second exam. You are responsible for knowing
when and how to use them.
MA 252 Spring 2005 Formulas for Exam 2
sin(2x) = 2 sin x cos x
cos2 x = 12 (1 + cos(2x))
cos(2x) = cos2 x − sin2 x
cos x sin x =
2
cos(2x) = 1 − 2 sin x
cos(2x) = 2 cos2 x − 1
sin2 x = 21 (1 − cos(2x))
1
2
sin(2x)
sin A cos B = 12 [sin(A − B) + sin(A + B)]
sin A sin B = 12 [cos(A − B) − cos(A + B)]
cos A cos B = 12 [cos(A − B) + cos(A + B)]
The following are examples of some of the types of problems that may be on your exam. Bear in
mind that this is by no means a complete list. In addition to these problems, you may want to
look at the review exercises at the end of the chapters.
Z
Z
Z π
x+1
2
5
4
2
1.
sin x cos x dx =
dx
12.
6.
cot x dx =
x(x + 2)(2x − 1)
π
6
Z
Z
Z
3
x
x
8x3
7.
dx
√
2.
dx
2
√
13.
dx
(x
−
1)
(2x
+
3)
2−9
9 − x2
4x
Z
p
3
Z
Z π
x2 + 100 dx
8.
x
4
2x
4
6
14.
dx
3.
tan x sec x dx =
Z
(x + 1)(2x − 2)
0
9.
sin 7x sin 4x dx =
Z
Z 3
2
x
x − 2x + 3x
Z
√
15.
dx
4.
dx
8x
x2 − 1
9 − x2
√
10.
dx
4x2 − 9
Z
Z
Z
1
7
5.
tan x sec x dx =
16.
dx
11.
sin 3x cos 5x dx
3
x +x
17. Decide which of the following integrals converge, and calculate the value of those that do.
Z ∞
Z 1
3
(c)
xe−x dx
(a)
1 dx
0
3
0
Z 3
Z 1x
dx
dx
√
(d)
(b)
1
−
x
x−1
1
0
MA 252 S02
Exam 2 Review
2
18. Find the lengths of the following curves:
3
(a) 3x = 2(y − 1) 2 ,
(b) y = ln x −
2≤y≤5
x2
, 1≤x≤4
8
(c) x = a(cos t + t sin t), y = a(sin t − t cos t),
(d) x = et − t,
t
2
y = 4e ,
0 ≤ t ≤ π (a > 0)
0≤t≤2
19. Find the areas of the following regions:
(a) Region enclosed by the astroid x = a cos3 θ,
y = a sin3 θ,
0 ≤ θ ≤ 2π, (a > 0)
(b) Region bounded by the curve x = t − 1/t, y = t + 1/t and the line y = 2.5.
20. Sketch the following parametric curves, indicating the direction the curve is traced as the
parameter increases. (Also practice problems like §10.1, #24-28.)
(a) x = 1 − 2t,
y = t2 + 4,
(b) x = sin2 θ,
y = cos2 θ
(c) x = et ,
−1 ≤ t ≤ 2
y = e−t
(d) x = tan θ + sec θ,
y = tan θ − sec θ,
−
π
π
<θ<
2
2
21. Find equations of the tangent lines for the following curves:
(a) x = t − t3 ,
√
(b) x = e t ,
y = 2 − 5t at the point (6, 12)
y = t − ln t2 when t = 1
22. Consider the curve x = 2 sin 2t,
y = 2 sin t,
0 ≤ t ≤ 2π
(a) Find the equation(s) of the tangent to the curve at the point (0, 0).
(b) Find when the tangent is horizontal or vertical.
(c) Find
d2 y
. For what values of t is the curve concave up? Concave down?
dx2
23. Let f (x) = kx2 (1 − x) if 0 ≤ x ≤ 1 and f (x) = 0 if x < 0 or x > 1.
(a) For what value of k is f a probability density function?
(b) For the above value of k, find P (X ≥ 21 ).
(c) Find the mean using the above value of k.
24. Consider the p.d.f. f (x) = 19 x2 for 0 ≤ x ≤ 3 and f (x) = 0 if x < 0 or x > 3.
(a) Find the exact value of the mean and approximate it to 4 decimal places.
(b) Find the exact value of the median and approximate it to 4 decimal places.