Review of 2nd Semester 2014-15
Exponential and Logarithmic Functions
1. Write the expression as a single log,
3ln x  4 ln y
2. Solve 3  4 ln(2 x)  15 .
x4
2 x 1
3. Solve 3  7
4. Evaluate
log 5
1
25
5. Use the exponential decay model for carbon-14, A  Ao e 0.000121t , to solve this exercise. Bones of a prehistoric
man were discovered and contained 5% of the original amount of carbon-14. How long ago did the man die?
6. Suppose you have $3000 to invest. Which investment yields the greater return over 10 years: 6.5% compounded
semiannually or 6% compounded continuously? How much more is yielded by the better investment?
7. The population of Europe was 509 million in 1990; in 2000, it was 729 million. Write the exponential growth
function that describes the population of Europe, in million, t years after 1990.
Trigonometry
8. To the nearest tenth of a square centimeter, find the area of an equilateral triangle whose sides are each 10 cm.
9. A reconnaissance plane finds that the distances from the plane to two stations (A and B) on level ground are
2925.4 ft and 3562.5 ft, respectively. If the angle between lines of sight to A and B is 118.4 , find the distance
between the two stations (to the nearest tenth of a foot).
10. I need to cut down a tree and the only direction it can fall is toward my house which is 60 feet away. I walk 70
feet from the base of the tree and measure the angle of elevation to the top of the tree is 48 . Is it safe for me to
cut down the tree? Justify your answer.
For the next three questions, use triangle ABC with C=90 , a=8, and c=10.
11. Find b
12. Find  A.
13. Find  B.
14. From two points A and B, 12m apart and in line with a tower and on the same side of the tower, the angles of
elevation to the top of a tower are 45 and 60 , respectively. How tall is the tower?
15. A wire is to be stretched from the top of a 20 m high building to a point on the ground. The angle of depression
from the top of the building to the ground is 20 . How long must the wire be?
16. A boat runs in a straight line for 7 km and then makes a 45 turn and runs for another 8 km. How far from its
starting point is the boat?
17. A road rises 110 feet per horizontal mile. What is the angle of elevation of the road?
18. What is wrong with: 2 cos =3.14?
19. A sailboat race is run on a triangular course. The first leg is 15 nautical miles in a direction of 115 . The second
is 18 nautical miles in a direction of 330 . In what direction is the third leg so that the race ends at the starting
point? Recall, directions are given in degrees clockwise from north.
20. If tan  is not defined and 0    2 , what is  ?
For the next two problems, Prove each identity.
21. sec x  cos x  sin x tan x .
2
2
22. csc x(1  cos x)  1 .
For the next three problems, find the exact value without a calculator.
sin
23.
7
4
tan
24.

6
cos

3
 cos

2
cot( 
25.
8
)
3
26. If (-2, 5) is a point on the terminal side of angle  , find the exact value of each of the six trigonometric
function of  .
27. Determine the quadrant in which  lies if cos  0 and cot   0 .
1
and tan   0 , find the exact value of each of the remaining
3
trigonometric functions of  .
28. If cos  
For the next two numbers, graph one period of each function and state the key
characteristics.
y  2 cos( x 
29.

2
)
30. y  3sin 2 x
31. Write the equation for the function graphed at right.
32. A radial arm saw has a circular cutting blade with a diameter of 10 inches. It spins at 2000 rpm. If there are 12
cutting teeth per inch on the cutting blade, how many teeth cross the cutting surface each second?
Vectors
33. Use the figure at the right
a. If c  a  b , then what are the components of c ?
c 
b. Evaluate:
c. What is the unit vector for c
34. Vector a has direction angle 120 and a  6 and vector b has
direction angle 45 and b  4 . If c  a  b , what are the direction and magnitude of vector c?
35. The magnitude and direction exerted by two tugboats towing a ship are 4200 pounds, N65E, and 3000
pounds S58E. Find the magnitude, to the nearest pound, and the direction angle in DMS, of the
resultant vector.
Polar Equations
36. Use the polar equation r = 3 sin 2θ.
a. Sketch the rectangular graph of the equation below.
b. Use the rectangular graph in part a to sketch the polar graph.
c. Identify (name) the type of curve in part b
37. Shown at the right is a circle graphed in a polar coordinate system.
a. Write the polar equation whose graph will be a circle in this position
b. Now by modifying your circle equation, find an equation whose graph has the given shape and position.
(Note: the graphs are not drawn to scale relative to the one above)
i.
ii.
iii.
Parametric Equations
38. Find a parametrization for the following curves:
a. A line segment starting at point A(-2, 7) and ending at point B(5, 0) on the interval of t [2,7].
b. An entire circle traced once with center (9, 1) and radius 5. Indicate an interval of theta and you
may have the trace start anywhere on the circle.
39. Reid hits a baseball 3 feet above the ground with an initial velocity of 145 ft/sec. The ball leaves the bat
at an angle of 20 degrees with the horizontal and heads toward a 17 foot fence that is 380 feet from
home plate.
a. Draw a picture to represent the situation, including the velocity triangle.
b. Write a set of parametric equations to represent the situation.
c. Determine whether the ball, hit the ground before the fence, hits the fence or clears the fence
algebraically
d. Using your answer from part (c) determine WHEN the ball hits the ground OR how high up the
fence the ball is when it hits the wall OR what is the distance the ball clears the fence
Matrices
40. Find (a) A - B, (b) A + B, (c) 3A, and (d) 2A - 3B.
41. Find the inverse of the matrix if it has one, or state that the inverse does not exist.
42. Happy Valley Farms produces three types of eggs: 1 (large), 2 (X-large), 3 (jumbo). The number of
dozens of type i eggs sold to grocery store j is represented by aij in the matrix. The per dozen price
Happy Valley Farms charges for egg type i is represented by bij in the matrix.
a. Find the product BTA.
b. What does the matrix BTA represent?
43. Find the reduced row echelon form of the matrix. What was the original system of equations?
44. A hospital trauma unit has determined that 30% of its patients are ambulatory and 70% are bedridden at
the time of arrival. A month after arrival, 60% of the ambulatory patients have recovered 20% remain
ambulatory and 20% have become bedridden. After the same amount of time, 20% of the bedridden
patients have recovered, 30% have become ambulatory and 50% remain bedridden. 300 patients were
admitted at the beginning of January 2011.
a. What are the three initial populations?
b. Write a square matrix modeling how the population is changing during a given time period.
c. How many patients do you expect to be ambulatory in February 2011?
d. Will all of the patients ever recover?
45. Show how to use matrix multiplication to find the number of ways to go from vertex C to vertex B using
exactly 4 edges using the diagram at the right.
46. Borda Count: Similar to the method used by writers and coaches to rank NCAA college football teams,
points are assigned in inverse order to ranking. In our example, that means 8 points for a top preference,
7 points for a second-highest preference, and so on. Use matrices to determine the winner based on this
system.
47. You are making gift baskets. Each basket will contain three different types of candles: tapers, pillars
and jar candles. Tapers cost $1 each; pillars cost $4 each, and jar candles cost $6 each. You put 8
candles costing a total of $24 in each basket, and you include as many tapers as pillars and jar candles
combined. Write this situation in an augmented matrix. Convert the first column to reduced row
echelon form. Then use an inverse matrix to solve the problem.
48. A friend sends you a secret message that was coded using the coding
5 3 
matrix C  
 and the alphabet table. The message is
3 2 
567|354|620|388. What is the coded message?
49. The diagram shows the way a group of people ranked for four candidates in an
election. For example, 10 people ranked A first, B second, C third, and D
fourth. Show a way to use matrix multiplication to find the point totals for
candidates A, B, C, and D if first-place ranking is worth 8 points, second 6
points, third 4 points and fourth 1 point.
50. Show how to use matrix multiplication to find the number of ways to go from
vertex C to vertex B using exactly 3 edges.
51. Buy-Rite Electronics has 3 locations each selling 3 different models of
Convair radios. Matrix A shows the inventory of each model at each location.
Due to holiday sales, the president of Buy-Rite Electronics expects the
inventory of X to decrease by 10%, Y to decrease by 15% and Z to decrease
by 2%. Using matrix multiplication, find the expected inventory of each type
of radio at each location after the holiday sales.
Conic Sections
52. Name the conic by finding its standard-form equation: 2x2 + 3y2 + 12c – 24y + 60 = 0.
53. Find the general equation of a hyperbola with center (3, -4), vertices (3, 1) and (3, -9), and asymptotes
3
y  4   ( x  3) .
5
54. Name the vertices of 16x2 – y2 -32x -6y -57 = 0
 x  3
2
 y  8

2
1
25
100
56. Determine the standard equation of a parabola with focus (2, -3) and directrix x = 6.
55. Find the eccentricity of
Review of 1st semester
Solve the following. Include a graph to verify your solution(s).
1. 2 x  3  x  3
2. 4 2 x  1  12  0
3.
2x
x
2


x  6x  8 x  4 x  2
2
4. 3 
2x  5
6
3
5. An architect is allowed 15 square yards of floor space to add a small bedroom to a house. Because of the room’s design
in relationship to the existing structure, the width of the rectangular floor must be 7 yards less than two times the length.
Find the length and width of the rectangular floor that the architect is permitted.
For 6-9: Use f ( x)  x 2  x  4 and g ( x)  2 x  6 , make sure to also state the domain and range if the result is a
function and know what the graph would look like.
6. Find (g – f)(x).
7. Find ( f
8. Find f 1 .
9. Find (g(f(-1)).
g )( x)
10. The annual yield per walnut tree is fairly constant at 50 pounds per tree when the number of trees per acre is 30 or
fewer. For each additional tree over 30, the annual yield per tree for all trees on the acre decreases by 1.5 pounds due to
overcrowding.
a. Express the yield per tree, Y, in pounds, as a function of the number of walnut trees per acre, x.
b. Express the total yield for an acre, T, in pounds, as a function of the number of walnut trees per acre, x.
11. The graph of f ( x)  6 x3  19 x 2  16 x  4 is shown in the figure.

Based on the graph of f, find the root of the equation that
Is an integer.

Use synthetic division to find the other two roots of
6 x3  19 x 2  16 x  4  0
12. Find the zeros: f ( x)  2 x 4  x3  13x 2  5 x  15
13. Write the rational function for the graph at the right.
14. Find the domain of the rational function and graph it. f ( x) 
x 2  3x  4
x2  x  6