Pre-Calculus Spring Final Exam Review Guide
The Final Exam will cover: Chapter 4, Chapter 5, Solving systems of equations with Matrices (part of section 7.3), Partial Fraction Decomposition (section 7.4), Chapter 8 (sections8.1, 8.2, 8.3),Chapter 9 (9.1, 9.2, 9.3, and the probability problems we did in class), and our review of limits, as well as basic skills, calculator skills, and vocabulary embedded in the learning throughout the year.
Basic Skills:
Order of Operations
Differentiation of types of functions
Completing the square
Factoring
Expanding/Distribution
Recognizing and using difference of squares
Solving equations – multiple methods
Calculator Skills
Chapter 4
Know the unit Circle
Use the unit circle to evaluate basic trig functions without a calculator
Solve trigonometric equations on a given interval.
Graphing Sine and cosine functions
Transformations of trig functions
Writing equations of sine and cosine functions
Solve word problems using trigonometric functions
Chapter 5
Know the basic trig identities (you may bring a notecard to class with any identity EXCEPT: quotient, reciprocal, sin2x + cos2x = 1)
Be able to prove identities
Law of Sines
Law of Cosines
Chapter7 - Section 7.3 (Example 7, 8, and 9, pg. 600-602) and Section 7.4 (examples 1 and 2 only)
Solving systems of equations using Matrices
Know how to convert a system to a matrix equation
Know how to enter matrices into your calculator and use the calculator to solve the system of equations
Partial Fraction Decomposition (with only linear factors in the denominator)
Chapter 8 - Sections 8.1, 8.2, and 8.3
Graph parabolas, ellipses, and hyperbolas
Identify the equations of parabolas, ellipses, and hyperbolas in both general and standard form
Convert between standard and general form
Convert the equation of a rotated conic into “y =” form
Write the equations ofparabolas and ellipse from graphs or word problems
Solve word problems using parabolas or ellipses
Chapter 9 – Sections 9.1, 9.2, 9.3
Know the different counting methods: tree diagrams, multiplication principle, combinations, permutations
Be able to accurately apply the counting principles
Find simple probability
Find simple probability using advanced counting methods
Find conditional probability using tree diagrams
Be able to identify binomial situations and calculate probability accordingly
Expand binomials
Limits
Mystery graphs
Evaluate limits algebraically
Evaluate limits from a graph
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Practice:
(This does not constitute a full review. It will give you an idea of what kinds of problems to review. I recommend re-doing section assignments or doing parts of the unit reviews as well!)
1. Evaluate:
a. sin ()b. cos ()c. tan()d. csc ()
e. cot ()f. sec ()
2. Describe the transformations from the basic trig functionsfor each equations below.
a. y = -5cos(4(x - )) + 6b. y = 12 tan (3x + 18) – 9
3. Identify the period of each function:
a. y = 3sin(5x)b. y = -4cos(8x) – 3c. y = cos ( x)
4. Review your trig identities!
5. Simplify :
a. sec(x) csc(x)b. c. d. (1- cosx)(1 + cosx)
e. (secx – 1)(secx + 1)f. g.
6. Sketch and solve the triangles. Use the appropriate Law.:
a. A = 40, a = 14, B = 30b. A = 40, b = 14, c = 13
c. a = 14, b= 13, c = 25d. A = 40, b = 14, c= 13
7. State how many triangles can be formed with the following measurements. Solve all possible triangles.
a. A = 30, a = 15, b = 14
b. A = 30, a = 14, b = 15
c. A = 30, a = 14, b = 7
d. A = 30, a = 14, b = 5
8. For each system of equations, write the corresponding matrix equation. Then solve the system using your matrix equation.
a. a + b + c + d + e = 15
2a + 5e = 15
3b + 2c + 4d + 3e = 29
5a + 4b + 3c + 2d + e = 55
a + 2b + 3c – 4d – 5e = 9
b. 5x – 6y = 21
7x + 11y = 36
c. 21g – 5h + 12i = 101
30g + 6h – 13i = 76
8g + 2h - 3i = 89
9. When using Matrices, what is the meaning of:
Err: Dim Mismatch
10 When trying to graph, what is the meaning of:
Err: Dim Mismatch
11. When using matrices, what is the meaning of:
Err: Singular Mat
12. For each equation below, state whether the equation represents a parabola, en ellipse, or a hyperbola.
a. (y – 6) = 18(x + 4)2b. + = 1
c. - = 1d. 5x2 – 7y2 + 8x + 9y -11 = 0
e. 3x + 5y2 – 9y = 12 = 0f. 11x2 + 8y2 – 8x – 12y + 56 = 0
13. Graph each equation:
a. - = 1b. + = 1
c. (x + 6)2 = 20(y – 11)
14. A flashlight has paraboloid reflector bowl. The bowl is 8 inches deep and 14 inches wide. How far above the bottom of eth bowl should eth filament be placed?
15. A planet revolves around its sun. Its perihelion distance is 3, 000 miles. Its aphelion distance is 12, 000 miles. What is the equation of this planet’s path?
17. What is the difference between an arithmetic and a geometric sequence?
18. Determine of each sequence is arithmetic or geometric. Then write both the explicit and the recursive formulas for each sequence.
a. 2,5,8,11,14, ……b. 2,6,18,54, ……c. 243, 81, 27, . . . .
19. Which infinite series converge? If the series converges, find its sum.
a. b. c. d.
20. Find the sun of the finite series using the correct formula.
a. an = 6n + 2, n = 8b. an = 4(1/2)n, n = 7
21. A quartic function passes through the points (1, 2.1), (2, 35), (3, 198.9), (-1, -10.3), (-2, -18.6). Use matrices to write the equation of this function.