Logs and Exponential functions e, ln, solving exponential functions, solving log and exponential equations, properties of logs

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1 Page 1 AM1 Final Exam Review Packet TOPICS Complex Numbers, Vectors, and Parametric Equations Change back and forth from and to polar and rectangular forms. Raise a term in polar form to a power (DeMoivre). Find roots of terms in polar form. Multiply terms in polar form. Find the magnitude of the resultant vector of 2 vectors. Scalar multiplication of vectors. Find a vector in component form. Find the length of a vector Give polar coordinates for a vector. Inequalities and Absolute Value Solve inequalities. Solve absolute value equations and inequalities, and graph solutions. Solve polynomial, rational and conic functions inequalities, and graph solutions. Identify the graphs of absolute value inequalities. Graphs of 2-dimensional inequalities. Limits and Rational Functions Find limits. Find intercepts, holes, vertical and horizontal asymptotes. Graph rational functions. Solve rational inequalities. Logs and Exponential functions e, ln, solving exponential functions, solving log and exponential equations, properties of logs Conic Sections Identify a circle, ellipse, hyperbola, and parabola by their equations. Find center, vertex (vertices), end points of major and minor axes, lengths of major and minor axes, foci, and equations of directrix and asymptotes for the appropriate conic section. Find point(s) of intersection of a line and a conic section or of two conic sections. Find points on the graph of a conic section. Put the equation of a conic section in standard form Statistics Mean and median (from a group of numbers and from a frequency table), mode, and standard deviation. Percentiles on the normal curve (z-scores). Use your tests and quizzes to review each of these topics as well as the review problems in this packet.

2 Page 2 The exam will have a non-calculator and a calculator part. This review is not split into calc/non-calc parts. See your tests and quizzes to know what you should be able to do without calc. Complex Numbers Write the following in simplified a+bi form. 1. 3(cos 45 + i sin 45 ) 2. [6(cos 30 + i sin30 )] 2 3. (7 + 4i) i i (cos i sin 237 ) 2(cos 60 + i sin 60 ) 6. i (3 + 2i) (6 7i) 8. i 0 9. i i Write in polar form i i 13. 5i What is the rectangular form of the complex number 2cis(135 o )? a) -2 b) 2 + i 2 c) 2 + i 2 d) 2 i i is a fourth root of which of the following? a) 8 b) 8i c) -64 d) 32

3 Page Which of these is a cube root of 27cis(210º)? a) 3cis(140º) b) 9cis(70º) c) 3cis(190º) d) 9cis(270º) 18. The rectangular coordinates of the point with polar coordinates 6, 5π 4 ( ) b. ( 3 2, 3 2 ) ( ) d. ( 6 2, 6 2 ) a. 6 2, 6 2 c. 3 2, 3 2 are 19. The polar coordinates of the point with rectangular coordinates ( 2 3, 2) are a. 2, 5π 3 c. 2, 11π 6 b. 4, 5π 3 11π d. 4, The complex number i 3 i i is equal to a i b. 1 3i c. 1 3i d. 3i The complex number (3 + 2i)(1 + 4i) is equal to a i b i c i d i 22. The rectangular form of the complex number z = 2 cos 5π 6 + isin 5π 6 a. 3 + i b i is c. 1+ 3i d i 23. The trigonometric form of the complex number 3 +3i is a. 3cis315 b. 3cis135 c. 3 2cis315 d. 3 2cis135

4 Page If z = 4 cos 3π 5 + isin 3π 5 and w = 10 cos 4π 9 + isin 4π 9 a. 14 cos 47π 47π + isin then zw equals 4π 4π b. 40 cos + isin c. 40 cos 47π 47π + isin π 4π d. 14 cos + isin Which of the following is not a cube root of 27i? 3 3 a i b. 3i 2 c i d i 26. Find (2 + 3i ) 4 in polar form and rectangular form. 27. Evaluate (3 cis 120 ) 4 in polar and rectangular. 28. Find the cube roots of z = 81 cis 180 in polar and rectangular form. Plot all roots below.

5 Page 5 Vectors and Parametrics 1. If vector u = <5, -2> and vector v = <4,6>, find 4u v. 2. Given points A (-2, 5) and B (4, 13) a. Find AB in component form y b. Find AB c. Graph the vector AB. d. Give polar coordinates for AB x 3. For the following, a) draw the resultant vector, b) find the length of the resultant, and c) find the angle between the resultant and the smaller vector A plane is flying with an airspeed of 150 mph on a bearing of 120. A wind starts blowing at 50 mph on a bearing of 70. Find the resulting speed and bearing of the plane.

6 Page 6 5. Two forces act on an object. Force G acts horizontally with a magnitude of 5 Newtons and force F acts vertically with a magnitude of 8 Newtons. The magnitude of the resultant force is a. 13 N b. 13N c. 89N d. 40N 6. (x, y) = ( 2, 4) + t<3, 1> a) Complete the table. t x y b) Where does the object start? c) What is the speed of the object? d) Where will it be located when t = 5? e) At what time will the object be located at (19, 3)? 7. If a = 2,7 and b = 3, 5 then b 2 a equals a. 9, 7 b. 7, 9 c. 3, 8 d. 8, 3 8. Let P and Q be points (1, 4) and (3, 7) respectively. Then the component form of PQ is a. <2, 3> b. < 2, 3> c. <4, 11> d. < 2, 3> Given vectors u = 5,2 and v= 2, 1 answer questions What is v u? a) 4 b) 3 2 c) 2 3 d) 18

7 Page Let w= u - 3v. If you draw with its tail at the origin and treat it as the terminal ray of an angle in standard position, approximately what is that angle? a) 15º b) 79º c) 105º d) 345º 11. A child fires a toy rocket. The rocket lifts off straight into the air with a speed of 36 mph. But, the wind is blowing the rocket directly sideways at 15 mph. Ignoring other forces like gravity, what is the resultant speed of the toy? a) 13 mph b) 24 mph c) 30 mph d) 39 mph 12. A plane is flying at 400 mph due west and encounters a headwind of 280 mph blowing northeast. Which of these is closest to the component form of the plane's resultant velocity? a) 200, 140 b) 480,200 c) 200,200 d) 140, Find the coordinates of a point ¾ of the way between (3, 1) and (15, 9). 14. When t=0, an object is at (2, 5) and at t=3, the object is at (5, -10). a) Write a vector equation for this scenario. b) Find the parametric equations for this scenario. c) Where is the object at t=10? 15. The paths of two objects are described by the following vector equations: P(x, y) = (3, 2) + t 2, 1 Q(x, y) = (-21, 10) + t 4, 1 a) Where do the paths of the objects cross? b) Do the objects collide at that point? If so, at what time? c) Find the speed of both objects.

8 Page 8 Inequalities and Absolute Value Section 1. Graph this inequality: 5-5x < Graph this inequality and give answers in interval notation (x - 2) (x + 5) (x - 4) < 0 3. Solve for x: 2 x + 2 > 3 and x < Graph this inequality: 2 < 5x-2 < 6 5. Graph this parabolic inequality and write the solution in interval notation. x 2-8x - 20 > 0 6. Which of these is the solution to 2x 3 5? a) 1 x 4 b) x 4 c)x 1or x 4 d) x 4or x 1 7. What is the solution to ( x + 2) 2 ( x 3) 0? a) x 2 b) 2 x 3 c) x = 2or x 3 d) x = 2or x = 3 8. Which of the following is the solution to x + 1 x 2 1 < 0 a) x > 1 b)x < 1,x 1 c) -1 < x < 1 d) x <-1 or x >1 9. Solve x 2 = 3

9 Page 9 Limits and Rational Functions Use the following graph for lim x 8 f ( x) = a. 1 b. 2 c. 6 d. DNE 2. lim f x x 6 ( )= a. 1 b. 1 c. d. DNE 3. lim f x x 0 ( ) = a. b. c. 2 c Draw the graph of f(x) = 3x + 1 2x.

10 Page Find the hole and the asymptotes of: f(x) = (x + 3)(x 2)(x + 4) (x + 2)(x + 3) 6. Graph f(x) = x2 x 6 x + 2 on the axes below. 7. Evaluate lim x 3 x 2 2x 3 x 3 2x Evaluate lim x x 3 9. Evaluate lim x 0 x 3 4x 2 + 8x 2x

11 Page The graph of f(x) is shown. a) lim x 2 = b) lim x = c) lim x = 11. Find the following. a) lim x 2 5x + 6 x 3 b) lim x 3 x -2 x 3 x + 2 c) lim x 2 x 2 5x + 6 x 3 d) lim x 3x 2 7x 2 + x 7x 2 e) lim x 5(x + 2)(x + 3) 2x 3 f) lim x 0 x 3 5x 5x g) lim 7x 20 h) lim x x 3 x + 2 (x 3) 2

12 Page 12 i) lim 2 x 5 j) lim x 5 x 2 2x 2 + 5x + 2 x 2 + 3x + 2 k) lim [(x 2 4x) + (3x 3 1)] l) lim 4x 11 x 2 x 4 m) lim x 1 x +1 x 2 1 n) lim x 0 x 3 4x 2x 2 4 x o) lim x 0 x Find the following a) lim f(x) = a) lim f(x) = x 3 x 1 + b) lim f(x) = b) lim f(x) = x 3 + x 1 c) lim f(x) = c) lim f(x) = x x d) lim f(x) = d) lim f(x) = x 2 + x 3 + e) lim f(x) = e) lim f(x) = x 2 x 3

13 Page 13 For #14 and #15, Graph and answer the following questions. 3x 2( x 3) 14. f(x) = 15. f(x) = x +1 x 2 6x + 9 horizontal asymptote: vertical asymptote: x-int: y-int: hole(s): horizontal asymptote: vertical asymptote: x-int: y-int: hole(s): 16. lim x x + 1 x 3 1 a) 0 b) 1 c) 3 d) 17. lim x 1 x 2 1 x 2 3x 4 a) 0 b) 2 5 c) d) D.N.E. 18. f(x) = 3x 3 has a hole at... x 2 + 2x 3 a) (-3, 3) b) 1, 3 2 c) 1, 3 4 d) There are no holes

14 Page 14 Logarithms 1. Suppose that the population of a colony of bacteria increases exponentially. If the population at the start is 300 and 4 hours later it is 1800, how long (from the start) will it take the population to reach 3000? 4 ln10 a. ln6 hrs b. 6 4 ln10 hrs c. 10ln 4 ln6 hrs d. 4(ln 10 ln 6) hrs 2. If $7500 is invested at 6% interest, compounded annually, how long will it take for the amount to reach $10,000? a years b years c years d years 3. The expression log equals a. 5 b. 11/2 c. 10/3 d. 9/2 4. The expression 4 log 5 x 6log 5 x ( )equals a. 2log 5 ( x x 2 + 1) b. x c. 4 log 5 x ( ) 6 2log 5 x ( ) 3log 5 x x 4 d. log 5 x ( ) 6 5. A principle of $10,000 is invested at an annual rate of 5.5%, compounded continuously. How many years will it take for the investment to double? a years b years c years d years 6. If ln(x + 2) 5 = ln(x 8) then x equals a. ln 12 5 b. 8e 5 3 e 5 c. 7. If log 3 2x + 1 8e e 5 1 ( ) = 2 + log 3 x then x equals a. 1/7 b. 1/5 c. 1 d. 3 d. 4 e 5 e 5

15 Page 15 Solve for x, in x = 4.1 a b c d e x = 6 a b x /4 ( ) = log(12) = c d a b c. 4 d. 4 a) log(3) + log(4) b) 3 log(4) c) ln( e ) 12 d) log(3)log(4) 12. Solve 2 3x 1 = 32. a) x = 1 b) x = 2 c) x = 4 d) x = Solve ln(x) = -1. a) x = -1 b) x = 1/e c) x = 0 d) No solution 14. What is the total value after 6 years of an initial investment of $2250 that earns 7% annual interest compounded quarterly? a) $ b) $ c) $ d) $ A single cell amoeba doubles every 4 days. About how long will it take one amoeba to produce a population of 1000? a) 10 days b) 20 days c) 30 days d) 40 days

16 Page 16 Statistics 1. Given the numbers 4, 8, 5, 5, 8, find each of the following. a) mean: b) median: c) standard deviation: 2. If the calorie contents of robin eggs are normally distributed with a mean of 25 and a standard deviation of 1.2, then 73% of all robin eggs will have a calorie content roughly greater than... a) 22.5 b) 23.5 c) 24.5 d) Percentages on the normal curve (z-scores). A set of data is normally distributed with x = 80 and the standard deviation = 5. a. Sketch the normal curve: b. 68% of the data lies between and. c. 95% of the data lies between and. d. 99% of the data lies between and. e. % of the data lies below 85. f. % of the data lies above 70. Using a standard normal table, answer the following: 4. The length of human pregnancies (in days) varies according to the distribution N(266, 16). a) What percent of pregnancies last less than 240 days? b) What percent of pregnancies last between 240 and 270 days? c) How long do the longest 20% of pregnancies last?

17 Page 17 Conics 1. What is the length of the major axis of 9x 2 + 4y 2 = 36? a) 2 b) 3 c) 4 d) 6 2. Which of these points is a focus of x2 16 y 2 4 = 1? a) ( 2 5,0) b) ( 2,2 5) 3. Which of these is the directrix of x = y 2 2 1? ( ) d) ( 20,0) c) 4,0 a) x = 2 b)x = 1 2 c)x = 3 2 d)x = How many solutions are there for the system of equations x 2 + y 2 = 25 and y = x 2 + 5? a) 0 b) 1 c) 2 d) 4 5. How many solutions are there for the system of equations x2 8 + y 2 = 1 and x 2 + y 2 8 = 1? a) 0 b) 1 c) 2 d) 4

18 Page 18 The next two questions refer to this hyperbola (assume each tick mark represents an increment of one). 6. Which of these might be the equation for the hyperbola above? a) y 2 3 x2 2 = 1 b) y 2 4 x2 9 = 1 c) y 2 9 x2 4 = 1 d) x 2 9 y 2 4 = 1 7. What are the asymptotes for the hyperbola? a) y = ± 2 3 x b)y = ± 3 2 x c)y = ± 3 2 x d) x = ± 4 9 y 8. The focus of the parabola x 2 = 8y is a. ( 2, 0) b. (0, 2) c. (0, 8) d. ( 8, 0) 9. The equation of the hyperbola with foci at ( ± 5, 0) and asymptotes of y = ± 1 2 x is a. c. x 2 1 y2 4 = 1 b. x 2 4 y2 1 = 1 x 2 4 y2 5 = 1 d. x 2 2 y2 1 = 1

19 Page The center of the ellipse x 2 + 4y 2 6x + 16y + 21 = 0 is a. (6, -16) b. ( 3, 8) c. (3, 8) d. (3, 2) 11. The foci of the hyperbola ( x 1) 2 ( y + 3)2 16 = 1 are 9 a. (-6, -3) and (4, -3) b. (-4, 3) and (6, 3) c. (-4, -3) and (6, -3) d. (4, -3) and (14, -3) 12. Write the equation in vertex form. Find the vertex. y = x 2 10x Write the following in standard form. y 2 = (x + 5) 2 For each parabola, find the: a. vertex d. coordinates of the symmetric point b. equation for the axis of symmetry e. x-intercept(s) c. y-intercept (0, ) f. sketch 14. y + 4 = (x + 1) y = x 2 4x + 3

20 Page State the radius and center of a circle with the given equation. (x 7) 2 + (y + 3) 2 = 10 r = C = 17. Write the equation of a circle with a. center (4, 2) and radius 3. b. a diameter having the endpoints ( 2, 4) and (8, 4). 18. What is the center and radius of x 2 + y 2 8x + 6y 21 = 0? 19. Sketch the following. a. (x 2) 2 + (y 1) 2 9 y < x

21 Page 21 b. (x 2) (y+1) c. y 2 16 x2 4 1 equations of asymptotes :

22 Page What is it? (line, circle, ellipse, parabola, hyperbola, or none of the above) a. 4x 2 7y 2 + y = 2 b. 4x 2 7y = 2 c. 4x 2 + 7y 2 + y = 2 d. 4x 2 + 4y 2 + y = 2 e. 4x 7y = 2 f. x 2-4y 2 + 3x - 6y + 9 = Write the equations of the following. a. b. y y x x 22. Find the foci. a) y 2 4 x2 9 = 1 b) x y2 16 =1 23. Given (x +1)2 16 (y 3)2 + 9 =1 a. When x = 2, y = b. When y = 5, x = c. When x = 7, y =

23 Page 23 ( x 1) 2 + y 2 = Find the intersection point(s) of the following: y x = The vertex of a parabola is (2, -3) and the directrix is y = -1. Find the focus and equation of the parabola.