Download Midterm 1, Sec 2, Spring2005

ECON1078-002 Math Tools for Economists
Midterm Examination 1
Write down your name, ID, derivation processes and answers on your bluebook or some
paper. You do not need to turn in this examination sheet.
You can use a scientific calculator. You cannot see the textbook or notebooks.
Each question counts for 5 points.
Q1 Which table specifies a function?
Table A
Table B
Table C
Domain (x)
Range(y)
Domain (x)
Range(y)
Domain (x)
Range(y)
0
–2
–2
4
0
0
1
–1
–1
1
1
–1
4
0
0
0
4
–2
1
1
9
1
2
2
2
Q2 Suppose f ( x)  2 x 2  5x . Find f (4) .
Q3 The graph of y  x 3 is shifted 4 units to the left (as the following figure shows.)
y
y?
Hint:
y  x3
Shifts
To shift the graph k units up: y  f ( x)  k
To shift the graph k units down: y  f ( x)  k
x
To shift the graph k units to the left: y  f ( x  k )
To shift the graph k units to the right: y  f ( x  k )
To reflect the graph in the x-axis: y   f ( x )  k
To expand the graph vertically by factor A: y  Af (x)
Find the function that characterizes the new graph.
Q4 Suppose you have drawn the graph of y  x .
How the graph of y  x  2  5 is obtained? (Choose one from the followings.)
A. By shifting the original graph (of y  x ) 2 units to the right and 5 units up.
B. By shifting the original graph (of y  x ) 2 units to the right and 5 units down.
C. By shifting the original graph (of y  x ) 2 units to the left and 5 units up.
D. By shifting the original graph (of y  x ) 2 units to the left and 5 units down.
Q5 Find the slope of the graph of the following equation:
5x  6 y  7
Linear Equations
1) Standard form
Ax + By = C
2) Slope-intercept form
y  mx  b
3) Point-slope form
y  y1  m( x  x1 ) ,
where ( x1 , y1 ) is a point on the line.
When m is unknown,
m
y 2  y1
x1  x 2
Q6 Write an equation of the line through the indicated point with the indicated slope m.
Then, transform the equation into the form y = mx + b.
m = 3;
(–5, 4)
Q7 Write an equation of the line through the indicated pair of points. Transform the
equation into the form y = mx + b.
(3, 2) and (6, –4)
Q8 Suppose you expand the graph of y  x 2 vertically by multiplying each ordinate by 3,
and then shift 2 units to the left and 5 units up.
Find the function that characterizes the new graph.
(Hint: Refer to the table of “shifts” given in Q3.)
Q8 Find the vertex, maximum (or minimum,) range and the intercepts of the following
function. Then graph it.
y  f ( x)  ( x  2) 2  4
The graph of y  a( x  h) 2  k
1) The vertex is ( h, k ) .
2) minimum = k if
a  0.
maximum = k if
a  0.
3) The range is y  k if
The range is y  k if
a  0.
a  0.
4) y-intercept is obtained by calculating
x-intercepts are obtained by solving
y  a ( h) 2  k .
a( x  h) 2  k  0 .
Q10 Solve (for x and y) using substitution.
4 x  3 y  1

 y  x5
Q11 Solve (for x and y) using "elimination by addition."
 2 x  9 y  16

5 x  3 y  11
Q12 Read the following, and write the corresponding system of inequalities.
(Write it mathematically. You do not need to solve or graph.)
A furniture company manufactures tables and chairs. A table requires 2 labor-hours
for assembling and 4 labor-hours for finishing. A chair requires 1 labor-hour for
assembling and 3 labor-hours for finishing. The maximum labor-hours available per day
for assembly and finishing are 30 and 80, respectively.
(Suppose x represents the number of tables and y represents the number of chairs.)
Q13 Solve the following system of inequalities graphically. Find coordinates of each
corner point.
(For this one, you need to solve and graph.)
 y  2 x  6

1
y   x  3
2

x0


y0

Q14 Suppose you are investigating the graph of some polynomial function. It has 11
turning points. What is the minimum degree of the function?
1) number of the turning points
2) number of x- intercepts
 degree.
Q15 Find any vertical and horizontal asymptotes for the graph of f ( x) 
1) Vertical Asymptotes: can be obtained by solving “numerator = 0.”
2) Horizontal Asymptotes: can be obtained as follows.
Divide the numerator and the denominator by the highest power of x.
See what happens when x approaches  or   .
Q16 Simplify.
(e 2 x 1e  x 1 ) 3
Q17 Graph the following function over the indicated interval.
y  4x
for
3 x  3
Q18 Rewrite in logarithmic form.
7 3  343
 degree – 1
2x  1
.
x4
Q19 Write in the simplest logarithmic form.
1
log b x 3 z 2
(The simplest form should not have a power or two or more factors within each log
expression.)
Q20 Solve for x and find its value using a calculator (to three decimal places.)
e x  9.2905
(Hint: Recall the definition of a logarithmic function.)
(End)
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