Pre-Quiz
Elements in Multivariable Calculus and Differential Equations
MATH 211
Name:
This quiz is designed to test you on material you should have already mastered in prerequisite
courses.
This will not count toward your grade but is very important.
Please answer all questions on the scantron provided.
1. Solve the following quadratic equation.
2x2 − 5x − 3 = 0
By Factoring:
(2x + 1)(x − 3) = 0
2x + 1 = 0
2x = −1
x = − 21
x−3=0
x=3
By the Quadratic
Formula:
√
b2 −4ac
x = −b±
√ 2a
(−5)2 −4(2)(−3)
2(2)
√
x = 5± 25+24
4
√
x = 5±4 49
x = 5±7
4
5+7
x = 5−7
x= 4
4
2
x = 12
x
=
−
4
4
x = 3 x = − 21
x=
−(−5)±
By Completing the Square:
2x2 − 5x = 3
x2 − 52 x = 23
1 2 1
2
2
= 2 − 52
= − 54 = 25
2b
16
5
25
3
25
x2 −
x
+
=
+
16
2
242 16
5
5
x − 4 x − 4 = 16 + 25
16
2
49
x − 54 =
q 16
x − 54 = ± 49
16
x − 45 = ± 47
x = 45 ± 47
5
7
x= 4+4
x = 54 − 47
12
x= 4
x = − 42
x = 3 x = − 21
2. Solve the following equation for y in terms of x.
√
y x − 1 = x2
√
y x = x2 + 1
y=
x2 + 1
√
x
3. What would be the appropriate partial fraction decomposition of the expression
x+2
x(x + 1)(x2 + 1)
x is a nonrepeated linear factor
x + 1 is a nonrepeated linear factor
2
x + 1 is a nonrepeated quadratic factor
x+2
A
B
Cx + D
= +
+ 2
2
x(x + 1)(x + 1)
x
x+1
x +1
4.
R
e3x dx is equal to
u = 3x
du = 3dx
1
du = dx
3
Z
Z
Z
1
1
1
3x
u1
eu du = eu + C = e3x + C
e dx = e du =
3
3
3
3
R
5. The integral x sin xdx can be written as
u=x
dv = sin xdx
du = dx v = − cos x
Z
Z
Z
x sin xdx = (x)(− cos x) − (− cos x)dx = −x cos x + cos xdx
6. What would be the correct substitution to make for x in the integral
Z
x2
√
dx
x2 − 1
x2 − 1 = x2 − a2 where a = 1
x = a sec θ =⇒ x = sec θ
7. Find the area of the region defined below.
R : {y = x, y = 2x, x = 2}
Z
2
Z
2
2x − xdx =
A=
0
xdx =
0
8. Differentiate the following.
y=
√
2
1
1 2 1
x = (2)2 − (0)2 = 2
2 0
2
2
2x + 1 sin x
1/2
y = (2x + 1) sin x
1
(2x + 1)−1/2 (2) sin x + (2x + 1)1/2 [cos x]
y0 =
2
√
y 0 = (2x + 1)−1/2 sin x + 2x + 1 cos x