Download PreCalculus Midterm Exam: Topics and Questions and more Study Guides, Projects, Research Law in PDF only on Docsity!
Topic Listing- PreCalculus Midterm Exam
What Should I Know?
I. Matrices
Determinants
o How do I find a determinant?
o What does a determinant tell me?
Inverses
o How do I find a matrix’s inverse?
Solving Systems of Equations Using Matrices
o Inverse Matrix Method
o Augmented Matrix Method
II. Family of Functions
Inverses of Functions
Transformations of Parent Functions
Polynomial Functions
o Even/Odd
o Graphing
Piece-wise Functions
o Graphing
o Evaluating
Rational Functions
o Graphing
Holes
Asymptotes
Y-Intercept/Zeroes
Domain
Range
Solving Rational Equations
o Least Common Denominator
o Extraneous Solution(s)
Solving Radical Equations
o Extraneous Solution(s)
III. Trigonometry
Right Triangle Trigonometry
o Sin/cos/tan/sec/csc/cot
o System of Tangents
Reference Angles
Coterminal Angles
Degrees, Minutes, Seconds
o Degrees
o Degrees
Law of Sines
o Ambiguous Case (SSA)
Law of Cosines
Area of triangles (Heron’s Formula/Sines)
Area of regular polygons
Radian Measure
o Arc Length
o Sector Area
o Linear/Angular Velocity/Speed
IV. Unit Circle
Exact Trigonometric Coordinates
Scaling the Unit Circle
Expressing sine and cosine as vertical and
horizontal movement
Identifying quadrants where ratios are
positive/negative
Attached are problems that serve as good examples of what you can expect to find on the exam. Tests and quizzes from
the semester also contain good review problems. Skills quiz problems (there have been 4 sets thus far) may also appear on
the exam.
Precalculus Formula Chart
This is the formula chart you will have available on Midterm Exam
Trigonometric Ratios and functions:
sideopposite
sideadjacent
cot
sideadjacent
sideopposite
tan
adjacentside
hypotenuse
sec
hypotenuse
sideadjacent
cos
oppositeside
hypotenuse
csc
hypotenuse
sideopposite
sin
y
x
x
y
x
r
r
x
y
r
r
y
tan cot
cos sec
sin csc
Area:
Triangle Area:
𝑏ℎ
2
Heron: √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐) 𝑤ℎ𝑒𝑟𝑒 𝑠 =
1
2
𝑝𝑒𝑟𝑚𝑖𝑡𝑒𝑟
Regular Polygon:
𝑎𝑝
2
Circle: 𝜋𝑟
2
Sector:
𝜃𝑟
2
2
Length of an Arc: s r
Angular speed/velocity:
t
w
Linear speed/velocity: v rw
t
v r or
Angular Acceleration: 𝛼 =
𝜃
𝑡
2
Law of sines:
sin A sin B sin C
a b c
Law of cosines:
2 2 2
2 2 2
2 2 2
a b c bc CosA
b a c ac CosB
c a b ab CosC
Conversions:
5,280 feet = 1 mile 𝐻𝑧 =
2 𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
1 𝑠𝑒𝑐𝑜𝑛𝑑
1 revolution = 2 𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 180
0
- Solve the system of equations using any method.
4 x + 4 y + 3 z = 6
4 x + 7 y + 5 z = 5
5 x + 3 y + 3 z = 7
- Find the value (determinant) of this matrix.
a. c.
b. d.
- If in Q1, find. Find 𝜃 in DMS.
- A pulley of radius 10 cm turns at 6 revolutions per second. What is the linear velocity of the belt driving the
pulley in meters per second?
- A roller of radius 12.5 cm turns at 5 revolutions per second. What is the linear velocity of the roller in
meters per second?
- Given and Find
[Hint: ]
- What is the exact value ratio of 𝑠𝑖𝑛
𝜋
3
- Find the inverse of the matrix , if it exists.
- Solve the system by using a matrix equation.
- Give the angle measure represented by rotations counterclockwise.
- Solve the triangle:
- Find the values of the six trigonometric ratios for.
18. Find cos if is an angle in standard position and the point with coordinates (3, – 4) lies on the terminal
side of the angle.
- Find the values of the six trigonometric functions of an angle in standard
position if the point with coordinates (–2, 0) lies on its terminal side.
- If c = and , find a. Round to the nearest tenth.
- , find all possible values of x.
- , find all possible values of x
- Two gears are adjusted so the smaller gear with a radius of 3 inches drives the larger one with a radius
of 8.5 in. If the smaller gear rotates through
11 𝜋
6
𝑟𝑎𝑑𝑖𝑎𝑛𝑠 ( 330 °), through how many degrees will the
larger gear rotate?
- Find the area of the triangle with A = , b = feet, and c = feet. Round to the nearest tenth.
- A gear of radius 5.8 cm turns at 4 revolutions per second. What is the linear velocity of the gear in
centimeters per second?
Use the Graphs below to answer questions 33- 35
33. Evaluate q(-4) 34. Evaluate 𝑞(𝑓
35. Find f ( x ) = 3 36. Evaluate f (-4): 3 √𝑥 + 8 − 2 𝑥
2
37. Let
= 9 + 𝑥 and 𝑔
2
Find g( f ( x )). Simplify to lowest terms.
38. If an intercepted arc's length is 15 cm and the circle's radius is 3 cm, what is the angle formed by the
arc? What is the sector area formed by the arc in terms?
39. Let 𝑔
2
Find the values for: g (3):_____ g(-3):______ g(5):____ g(g(2)):__
40. Find the inverse function: 𝑗
− 1
(𝑘). Show all steps in simplified radical form: 𝑗(𝑘) = 3 √
- lim
𝑥→− 1
𝑓(𝑥) 2. lim
𝑥→±∞
𝑓(𝑥) 3. lim
𝑥→− 2
42. Given
csc 60
, list two other functions using co-functions and reciprocal functions that would
also have a ratio of exactly
(4 points)?
43. Find the area of a regular pentagon with a side length of 10 cm. Round your final answer to the nearest
tenth of a square cm. (4 points)
44. Ben can paint a house in 5 days. Mary can paint a house in 4 days. Charlotte can paint a house in 7 days.
If they all work together, how long will it take the team to paint 2 houses?
45. A bicyclist travels 15 km in the same time it takes a jogger to travel 4 km. The speed of the bicyclist is
12 km/hour faster than the speed of the jogger. How fast does each travel in km/hr? Show all work.
46. Label the following functions as Even, Odd, or Neither.
a. 𝑓(𝑥) = 3 𝑥 b. 𝑔(𝑥) = 3 𝑥
2
− 3 c. 2 𝑥
4
2
______________ ______________ ____________
47. When will the function, 𝑓
3 𝑥
2
𝑥− 1
be equal to zero?
a) never b) when x = 0 c) when x = 1 d) when x =
48. Identify the end behavior of the function: 𝑓(𝑥) = − 4 𝑥
3
a. As x g ( x )
As x g ( x )
c. As x g ( x )
As x g ( x )
b. As x g ( x )
As x g ( x )
d. As x g ( x )
As x g ( x )
49. The volume of a block of ice with dimensions of 5 feet by 6 feet by 8 feet will be reduced by shaving the
same amount from each dimension. If your goal is to reduce the volume of the block to 2/5 of its
original volume, how many feet must be shaven off from each dimension, rounded to the nearest tenth
of a foot? Show all work.
Shaven from each dimension? ___________ Dimensions of new block? _______________
50. The cost of manufacturing a textbook is $35 for each book. The fixed costs associated with producing
the books are $350,000.
a. Write the function, f ( b ), modeling the average cost for producing b books.
b. Find the average cost per book for:: f (500) and f (5,000)
c. Find the horizontal asymptote for the average cost. What does it represent for this problem?
d. If the average cost was $45 per book, how many books were produced?
10cm
