Pre-Calculus Spring Final Exam Review Guide

Transcription

1 Pre-Calculus Spring Final Exam Review Guide The Final Exam will cover: Chapter 4, Chapter 5, Polar Graphing, Solving systems of equations with Matrices (part of section 7.3), Partial Fraction Decomposition (section 7.4), Chapter 8 (sections 8.1, 8.2, 8.3), Chapter 9 (9.4 and 9.5), 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, sin 2 x + cos 2 x = 1) Be able to prove identities Law of Sines Law of Cosines Chapter 7 - Section 7.3 (Example 7, 8, and 9, pg ) 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 of parabolas and ellipse from graphs or word problems Solve word problems using parabolas or ellipses

2 Chapter 9 Sections 9.4 and 9.5 Sequences Arithmetic Vs. Geometric vs. Neither Famous sequences Infinite and Finite sequences Converging and Diverging Sequences Series Sigma Notation Sums of finite Series Sums of infinite Series Limits Mystery graphs Evaluate limits algebraically Evaluate limits from a graph ***************************************************** ***************************************************** 1. Evaluate: 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!) a. sin ( ) b. cos (5 ) c. tan(7 ) d. csc ( ) 4 e. cot ( 5 ) f. sec (11 ) Describe the transformations from the basic trig functions for each equations below. a. y = -5cos(4(x - )) + 6 b. y = 12 tan (3x + 18) Identify the period of each function: a. y = 3sin(5x) b. y = -4cos(8 x) 3 c. y = cos ( 5 x) 4. Review your trig identities! 5. Simplify :

3 a. sec(x) csc(x) b. cot x tan x c. csc x cot x d. (1- cosx)(1 + cosx) e. (secx 1)(secx + 1) f. cos2 x+sin 2 x sin x g. tan2 x+1 csc 2 x 6. Sketch and solve the triangles. Use the appropriate Law.: a. A = 40, a = 14, B = 30 b. A = 40, b = 14, c = 13 c. a = 14, b= 13, c = 25 d. A = 40, b = 14, c= 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 = g + 6h 13i = 76 8g + 2h - 3i = When using Matrices, what is the meaning of: Err: Dim Mismatch 10 When trying to graph, what is the meaning of: Err: Dim Mismatch

4 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) 2 b. (y 7) (x 10)2 31 = 1 c. (y 7) (x 10)2 31 = 1 d. 5x 2 7y 2 + 8x + 9y -11 = 0 e. 3x + 5y 2 9y = 12 = 0 f. 11x 2 + 8y 2 8x 12y + 56 = Graph each equation: a. (y 7) (x 10)2 36 = 1 b. (y 7) (x 10)2 36 = 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, Which infinite series converge? If the series converges, find its sum. a. i=1 5(0.25) i b. i=1 5(2.5) i c. i=1 5i + 3 d. i=1 5i 7

5 20. Find the sun of the finite series using the correct formula. a. a n = 6n + 2, n = 8 b. a n = 4(1/2) n, n = 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. 22. I recommend doing the problems from the chapter reviews and reviewing all tests/quizzes as well.