Pre Calculus Final Exam Review

Transcription

1 Pre Calculus Final Exam Review Jun 2 10:04 PM Believe It or Not!! Jun 1 7:48 PM 1

2 Jun 2 9:58 PM Jun 2 10:06 PM 2

3 Jun 2 10:27 PM Jun 1 7:55 PM 3

4 Jun 2 10:20 PM Jun 2 10:22 PM 4

5 Jun 2 10:22 PM Jun 3 9:43 PM 5

6 Jun 3 9:43 PM Jun 3 9:44 PM 6

7 Integer 1 Answer? Jun 3 9:56 PM Jun 3 9:46 PM 7

8 Believe It or Not!! Apr 24 9:33 PM 2 Answer? Jun 3 9:50 PM 8

9 Jun 3 9:52 PM 3 Answer? Jun 3 9:53 PM 9

10 nearest tenth 4 Answer? Jun 3 10:02 PM 5 Answer? nearest tenth Jun 3 10:03 PM 10

11 Law of Sines Jun 3 10:06 PM 6 Answer? nearest tenth Jun 3 10:03 PM 11

12 Jun 3 10:16 PM Jun 3 10:17 PM 12

13 998 B FINAL EXAM REVIEW.notebook Area of a triangle if you know three sides Hero's Formula Area = (s (s-a)(s-b)(s-c) ; s=(1/2)(a+b+c) Jun 3 10:19 PM Jun 3 10:27 PM 13

14 Jun 3 10:30 PM Jun 3 10:32 PM 14

15 Modeling with a Sinusoidal Function f(x) = a * sin(kθ c) + h a = amplitude k = 2π/Τ ; T = period c = φk ; φ = phase shift h = verticle shift or... f(x) = a * sin(k(θ φ)) + h Jun 3 10:36 PM F net 270 N 360 N What is the magnitude of the resultant force from the clowns onto the bull? Hint What is the direction of the force? Hint 7 Answer? nearest tenth ~450 N May 1 8:17 PM 15

16 XYZ Space 3D We use the right hand rule to determine the correct orientation between the three axis. Z X Y X Z Y Apr 26 9:50 PM Find the missing axis Z X X Y Z X Y Z Apr 26 10:02 PM 16

17 To find the length (or magnitude) of any vector in 2 D space, you need to identify the initial and terminating points, P 1 and P 2 respectively. Then use the distance formula: Three dimensional space is very similar... Apr 25 12:43 AM X 2 +Y 2 =C 2 C 2 +Z 2 =R 2 By substitution: (X 2 +Y 2 )+Z 2 =R 2 R Z X C Y Apr 27 6:13 PM 17

18 Jun 4 5:11 AM You try: u = (1, 4,2) and v = (0,8, 5) u + v u v (1/2)v 2u + 3v Apr 25 1:05 AM 18

19 Without the wind, a plane would fly due east at a rate of 150 mph. The wind is blowing southeast at a rate of 50 mph. The wind is blowing at a 45 angle from due east. What is the actual speed of the plane with the wind? nearest mph 8 Answer? tenth Jun 4 5:27 AM Forces of 18 pounds and 20 pounds act on an object at an angle of 120. Find the magnitude of the resultant force. nearest ~ 19.1 tenth 9 Answer? Jun 4 5:23 AM 19

20 Example: A $20,000 new car depreciates or loses 20% per year. Let r = 20% = 0.20 hint: equation... y=$20,000(.80) x Calculate the future value at: 2 years: $12, nearest penny 5 years: $6, years: $2, years: $2.66!!! :( 10 Answer? May 7 9:58 PM You try: Recall y = ab x ; x is # of periods b is the factor a is the initial amount Carbon 14 decays into Carbon 12 at a rate of 1/2 the sample every 5,730 years. How much of a 100 microgram sample of C 14 will remain after 11,460 years? a = 100 Hint b = (1 1/2) = 1/2 Hint x = 11,460/5730 Hint = 2 periods 25 μg 11 Answer? May 7 10:11 PM 20

21 Exponential Growth Suppose the initial population of varroa mites is 500. If the population is increasing 14% per week, what is the predicted population for 22 weeks? N = 500(1 + Hint 14%) t N = 500(1.14) Hint t N = 8,930 nearest integer 12 Answer? May 7 10:17 PM Compound Interest Use A = P(1 + r/n) nt A = final value of the investment P = initial or present value of the investment r = annual interest rate. i.e. "APR" t = number of years n = number of times interest is paid per year ex:annually; n=1 semi annually; n=2 quarterly; n=4 Suppose you set an investment goal of $20,000 for a new car after college. How much would you have to invest NOW at 8% APR compounded quarterly over the next five years? 20,000 Hint: = P(1 equation +.08/4) (4)(5) $13, nearest penny 13 Answer? May 7 10:24 PM 21

22 $$ Continuously Compounded Interest $$ Peridically Compunded Interest A = P[1+(r/n)] nt Continuously Compounded Interest A = Pe rt Compare the values of a $10,000 investment at 6.75% APR over 25 years compounded semiannually versus continuously. 14 Answer? $52, nearest penny $54, May 11 9:54 PM Since log = 2/3 25 = " log 8 2 = 1/3 2 = " 4 3 = 64 log 4 64 = " 3 3 = 1/27 log 3 (1/27) = May 11 10:44 PM 22

23 Definition of the logarithm y = b x log b y = x b = base May 8 10:26 PM Properties of Logarithms May 12 9:17 PM 23

24 Believe It or Not!! "Its curvy, with a higher bit at the end and a rather aesthetically pleasing slope downwards towards a pretty flat strait bit. The actual graph itself consists of 2 straight lines meeting at the lower left hand corner of the graph and moving away at a 90 angle each line has an arrow head on the end" Jun 1 7:49 PM Using a calculator and the properties of exponents, evaluate the following: log[(x+5)/2] = log(5/x) 1 x = or nearest x= hundredth 15 Answer? May 13 9:45 PM 24

25 Change of Base Formula log a n = log b n / log b a a,b,n are positive and neither a or b is 1. Ex: log = log1043 / log9 16 Answer? nearest thousandth You try: log 5 (x) = log(2) x = ~ May 13 10:08 PM Arithmetic means are the terms of a sequence between two other non consecutive terms. 17 Answer? You try: Write an arithmetic sequence that has six terms between 12 and 23. i.e. 12,,,, 8,??,, Hint: or a n = a 1 + (n 1)d Hint: (23 12)/7 = d May 20 5:30 AM 25

26 An arithmetic series is the sum of the terms of an arithmetic sequence. S n = (n/2)(a 1 + a n ) + Ex: Find the sum of the 1st 60 terms of Answer? You try: integer Find the sum of the first 32 terms of the series May 20 5:42 AM Geometric Sequence A sequence in which each term is the product of the preceding term and the common ratio. Example: Find the next three terms: 1, 1/2, 1/4... Step 1: Determine the common ratio between pairs of consecutive terms. 1/2 1 = 1/2 1/4 1/2 = 1/2 Step 2: Multiply the third term by he common ratio to get the fourth term, and so on. a 4 = 1/4( 1/2)=( 1/8), a 5 = ( 1/8)( 1/2)=(1/16), a 6 = (1/16)( 1/2)=( 1/32), Notice that a n = (a n 1 )r ; r is the common ratio. This is a recursive formula. Notice: a 1 = a 1 a 2 = a 1 r a 3 = a 2 r = a 1 r 2 a 4 = a 3 r = a 1 r 3... a n = (a n 1 )r = a 1 r n 1 a n = a 1 r n 1 May 19 10:32 PM 26

27 A geometric series is the sum of the terms of a geometric sequence. S n = (a 1 a 1 r n )/(1 r) Ex: Find the sum of the first ten terms of ,192 You try: Find the sum of the first eight terms of the series integer 911, Answer? May 20 5:42 AM 12.3 Infinite Sequences and Series Apr 24 9:33 PM 27

28 998 B FINAL EXAM REVIEW.notebook May 22 6:14 AM Believe It or Not!! Jun 1 7:55 PM 28

29 May 30 7:24 AM 29

30 Attachments Glencoe Pre Calc on line quizzes cash_register_x.wav