Basic mathematics refresher exercises

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Exercises to review basic mathematical concepts 
 
The following exercises are useful for students to: 
- check their level of knowledge of basic mathematical concepts, which we assume all 
students are acquainted with when they start this course; 
- quickly review/brush up concepts that might be forgotten and that may be used 
throughout the semester in the classroom or in the exams. 
Any student struggling to solve the following exercises, with seemingly insurmountable 
difficulties even after checking the solutions provided, should contact the teachers as 
soon as possible and help will be provided. You should not be shy on this; you would 
do it at your own risk. 
 
Note: The review of concepts is not exhaustive and is not meant to teach you these 
concepts to those who have never seen them before. For a thorough review or a good 
introduction we recommend the following reference: 
Chiang, A. and K. Wainwright (2005), Fundamental Methods of Mathematical 
Economics, 4th edition, McGraw-Hill Education (available at the library: MQ.11.30 
CHI*Fun 4ª ed.). 
The following references present more exhaustive listings of mathematical formulas 
useful for economists. Anyone intending to pursue Economics studies at a postgraduate 
level or to be a competent economist in the future should seriously considering putting 
one of these on his/her bookshelf: 
Sydsaeter, K., A. Strom and P. Berck (2010), Economists' Mathematical Manual, 
Springer (available at the library: MQ.116 SYD*Eco). 
Luderer, B., V. Nollau and K. Vetters (2010), Mathematical Formulas for Economists, 
4th edition, Springer (3rd edition available at the library: MQ.11.30 LUD*Mat; 4th 
edition on the way). 
 
 
 
 
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1) Solve the following equations: 
a. 0743 2 =−+ xx 
b. 04 2 =− xx 
c. 0102 =−xe 
d. 072 1 =+−x 
e. 
98
132
+
=+
x
x
 (Note: 8
9
−≠x ) 
f. 
6
5
23
=+
xx
 
g. 52
3
=+
x
 
h. 3
2
1
=
+x
 
i. 5,05,0
1
−
−
= x
x
 
j. ( ) 52 212 =+x 
 
2) Solve the following systems of equations (x and y are variables; everything else 
is a parameter) 
a. 



=+
=+
542
1068
yx
yx
 
b. 



=+
=+
feydx
cbyax
 
 
3) Differentiate the following expressions with respect to x: 
a. ky 
b. kyx + 
c. kzx 57 + 
d. 27x 
e. 27 yx 
f. 25,04x 
g. 968 3 +− xx 
h. 25 −x 
i. ( )212 2+x 
j. xln 
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k. ( )x3log 2 
l. xe 
m. 4
2
2 +x 
n. xe x ln 
o. 
x
e x
ln
 
p. 
22
3
+x
x
 
q. ( ) yx
yx
ln22
3
+
 
 
4) Find the extremes of the following functions and state whether they are 
maximums or minimums 
a. 2x 
b. 2x− 
c. 3x 
d. 2ln xx − 
e. 2ln xx + 
f. 2ax 
 
5) Graphically represent the following functions: 
a. xy 210 −= 
b. xy += 3 
c. 2xy = 
d. 32 +−= xy 
e. 3xy = 
f. xey = 
g. xy ln= 
h. 0,1 >= x
x
y 
 
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Solutions: 
1) 
a. 1
3
7
=∨−= xx 
Note: Recall the formula to find the roots of quadratic equations (assuming 
042 >− acb ): 
a
acbb
xcbxax
2
40
2
2 −±−
=⇔=++
 
b. 0
4
1
=∨= xx 
c. 5ln=x 
d. 
7
2
−=x 
e. 1
8
13
−=∨−= xx 
f. 1 
g. 9 
h. 
3
5
− 
i. 1 
j. 23± 
 
2) 
a. 1
2
1
=∧= yx 
b. 
bdae
cdafy
bdae
bfce
x
−
−
=∧
−
−
= 
c. 
 
3) 
a. 0 
b. 1 
c. 7 
d. x14 
e. yx14 
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f. 
4 3
4
375,0
75,0 111
x
x
x
x ===− 
g. 624 2 −x 
h. 3
3 1010
x
x −=− − 
i. 
22 +x
x
 
j. 
x
1
 
k. 
2ln
1
x
 
l. xe 
m. 2ln2 5
2
xx + 
n. 
x
e
xe
x
x +ln
 
o. ( ) ( ) 





−=
−
22 ln
1
ln
1
ln
ln
xxx
e
x
x
e
xe
x
x
x
 
p. ( )( ) 44
6
2
223
24
24
22
322
++
+
=
+
−+
xx
xx
x
xxxx
 
q. 
y
y
xx
xx
ln44
6
24
24
×
++
+
 
 
4) 
i. 0=x is an extreme ( ) 





=⇔= 002 xx
dx
d
. It is a minimum 
( ) 





>= 0222
2
x
dx
d
. 
j. 0=x is an extreme ( ) 





=⇔= 002 xx
dx
d
. It is a maximum 
( ) 





<−= 0222
2
x
dx
d
. 
k. It has no extremes. 
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l. 
2
1
=x is an extreme ( ) 





=⇔=−
2
10ln 2 xxx
dx
d
. It is a maximum 
( ) 




 ∀<−−=− x
x
xx
dx
d 021ln 2
2
2
2
. 
m. It has no extremes. 
n. 0=x is an extreme ( ) 





=⇔= 002 xax
dx
d
. It is a minimum if 0>a . It 
is a maximum if 0<a ( ) 





= aax
dx
d 222
2
. 
 
6) 
a. xy 210 −= 
 
Note: An accurate representation of the function requires us to find the zeros, the first 
derivative (to know whether the slope is increasing or decreasing) and the second 
derivative (to know whether the function is convex or concave). 
b. xy += 3 
 
-5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-4
-2
2
4
6
8
10
12
14
16
18
20
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
-4
-2
2
4
6
8
10
x
y
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c. 2xy = 
 
d. 32 +−= xy 
 
e. 3xy = 
 
 
 
-5 -4 -3 -2 -1 1 2 3 4 5
-4
-2
2
4
6
8
10
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
-10
-8
-6
-4
-2
2
4
6
8
10
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
-10
-8
-6
-4
-2
2
4
6
8
10
x
y
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f. xey = 
 
In this case it may be useful to compute ( )xf
x −∞→
lim e ( )xf
x +∞→
lim . Note also that 10 =e . 
g. xy ln= 
 
Note also that 01ln = . 
h. 0,1 >= x
x
y 
 
-5 -4 -3 -2 -1 1 2 3 4 5
-10
-8
-6
-4
-2
2
4
6
8
10
x
y
-2 -1 1 2 3 4 5 6 7 8 9 10
-10
-8
-6
-4
-2
2
4
6
8
10
x
y
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
x
y