Math 172 Prerequisites 1
Instructions: This is a list of some material that is essential prerequisite for this course.
That is, your instructor expects that you have a definite idea of the notions expounded
below and that you be able to solve the problems below without any review. If you find
yourself having trouble on some items below you should seek immediate help from your
instructor or from the tutors in the Learning Laboratory.
Essential prerequisites include:
(a) Graphing quadratic curves.
(b) Knowing elementary exponential, logarith-
mic, and trigonometric identities.
(c) Manipulation of algebraic fractions.
(d) Completion of the square.
(e) Knowing the basic differentiation rules.
(f) Knowing the basics of integration.
(g) Knowing integration by substitution.
You should be able to quickly answer all questions below without needing any review.
1. Make a rough sketch of
(x−2)2
9+(y+ 1)2
4= 1.
2. Graph the curve with equation |x2−y2|= 1.
3. Sketch the curve y=|x−1|and then find
Z2
−1|x−1|dx.
4. Graph y= cos xin the interval −π
2;πand
then find Zπ
−π/2
cos xdx.
5. Find dy
dx:
(a) y=x+ 1
x−1
(b) y= sec(2x)
(c) y=1
√4−x2
(d) y=x3e3x
6. Add: −2
x−2
x2+2
x−2.
7. Show that √1 + x−2/3=x−1/3√1 + x2/3.
8. Factor: 4x2−12x+ 9.
9. Find the exact value of:
(a) arccos 1
2
(b) arcsin 1
2
10. If π
2< x < π and sin x=1
2, find tan x.
11. If 0< θ < π
2and tan θ=1
x, find cos θ.
12. Evaluate the following limits:
(a) limx→2−
(x−1)(x−2)
x+ 2
(b) limx→0+ 1−cos(2x)
(c) limx→0
sin 2x
tan 3x
(d) limx→−1
x2−1
x+ 1
13. Consider the graph of y=f(x). Find the follow-
ing.
(a) f(1)
(b) f(3)
(c) limx→1f(x)
(d) limx→−1+ f(x)
(e) limx→−∞ f(x)
(f) limx→3−f(x)
(g) limx→3+ f(x)
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
Figure 1: Problem 13.
14. Find Z(sin x)ecos xdx.
15. Find constants A, a, B , b such that sin xcos 2x=
Asin ax +Bsin bx.
16. Prove that 2 sec x=cos x
1−sin x+cos x
1 + sin xand use
it to find Zsec xdx.
17. Find Zr1−x
1 + xdx.
18. Find constants A, a, , b such that x2+ 6x+ 10 =
A(x+a)2+b2. Graph the curve y=x2+6x+10 and
identify its intercepts and its minimum point.
By: Dan JACOBSON and David A. SANTOS Last Revision: January 12, 2008