Sequences & Series Questions and Answers

1 2 Factor each completely 1 16x 36x 70 A 2 2x 7 4x 5 B 2 2x 7 4x 5 C 4x 7 2x 5 D 2 4x 7 2x 5 Simplify each expression 2 m 3 A B C D Sum 2 2m 1 m 3 2m 1 5 m 3m 4 4 2m m 3 2m 1 6m 8 m 3 2m 1 Find each product 4 4p 8 6p p 4 A 32p 24p 44p 12 B 10p 19p 46p 16 C 5p 32p 7p 2 D 24p 44p 24p 32 3 3 2 2 v1 A C 1 v v 1 2 2v 7 2v 2 v B D 8v 15 v 3 6v 15 v 2v v 3

5 Find the component form of v if v 6 and the angle it makes with the x axis is 150 O 6 3 6 O 3 2 3 2 O 3 3 3 O 3 3 3 6 6 3

6 Find u v u 6 1 v 1 7 O 8 7 O 7 7 O 8 9 O 8 8

6 Evaluate arccos without using a calculator 3 2 E 6 OF OF F M O F O 37 4

3 A television camera at ground level is filming the lift off of a space shuttle at a point a 600 meters from the launch pad see figure Let 0 be th angle of elevation to the shuttle and let s be the height of the shuttle Write as a function of s m XX Not drawn to scale

Use the method of corners to solve the linear programming problem An investor has decided to commit no more than 80 000 to the purchase of the common stocks of two companies Company A and Company 8 He has also estimated that there is a chance of at most a 1 capital loss on his investment in Company A and a chance of at most a 4 loss on his investment in Company B and he has decided that these losses should not exceed 2 000 On the other hand he expects to make a 15 profit from his investment in Company A and a 21 profit from his investment in Company B Determine how much he should invest in the stock of each company x dollars in Company A and y dollars in Company B in order to maximize his investment returns x y What is the maximum return in dollars

16 A publishing company orders black and blue ink in bulk for its 2 color printing press To keep things simple with its ink supplier each time it places an order for blue ink it buys B gallons and each time places an order for black ink it buys K gallons Over a one month period the company places m blue ink and n order of black ink a The company placed twice as many orders for black ink than for blue ink in January Create an equation to represent this situation in terms of m and n b Given an interpretation for the expression c Find the value of 72 m n in terms of the orders placed in January

Which of the following is equivalent to Select the correct answer below cos acot 3 sin a cos acot sin a tan acos sin Cosa sin atan B cos a sin atan cos a b cos B for all values of a and B for which cos a cos B is defined

Graph the image of kite KLMN 1 after a dilation with a scale factor of 5 centered at the origin 104Y N 6 4 2 8 6 4 2 2 4 6 8 1 0 2 M 4 6 8 K L X 10 ES

Find SU T 41 SU 9 2 U S Write your answer as an integer or as a decimal rounded to the nearest tenth

Find RS 49 R RS 8 S Write your answer as an integer or as a decimal rounded to the nearest tenth

7 Let V be the subset of P3 consisting of all polynomials of the form ao ax a x where ao a 0 and a2 a3 0 Find the dimension of V a 2 b 1 c 3 d 4 e 5

17 Let W be the set of all vectors in R of the form a 1 0 Which of the following statemen is true i W is not closed under additon and not closed under scalar multiplication ii W is closed under addition but not closed under scalar multiplication iii W is closed under scalar multiplication but not closed under addition iv W is a subspace a ii b iii c iv d i e none of them

4 Suppose that V1 V2 V3 is a linearly independent set of vectors Which of the foll sets are also linearly independent i V V1 V2 V V V3 ii V1 2v2 2V3 iii V1 V2 V2 V3 V1 V3 a i and ii only b all of them c ii only d i only e iii only

5 Let V be the set of all ordered pairs of real numbers a y where a and y are both nonzero Consider the following addition and scalar multiplication operations on u and v 01 02 u v 1 v ku u k u k Although V is not a vector space each vector in V does have a negative as defined in the axioms of a vector space If u is in V what is the negative of the vector u a u b u 1 u 1 c d u 1 2 1 e u 1 1

10 You deposit 8000 in an account earning 7 interest compounded continuously a Write the equation that models the amount of money in the account after t years b How much will you have in the account in 8 years c How long does it take for the money in the account to double 0

2 Find the center and the radius of the circle x y 4x 3 0

Di intercent s and y intercept s of the function f x 3r 2 7

8 Determine whether the statement is true or false Justify your answer In each of the four quadrants the signs of the cosec function and cosine function will be the same True False In each of the four quadrants the signs of the cosec function and the sine function will be the same because these functions are reciproc

2 Using the figure below if 0 21 and y 8 determine the exact value of x y D

3 Given f x and g x ln x 3 a What is the domain of g h Find the domain of f o g

1 Evaluate if possible the sine cosine and tangent of the real number

2 Evaluate the trigonometric function using its period as an aid COS 25

EX 2 Expand each logarithm 8x 5y A log c log 9x log 37 B loga D log 4

EX 1 Write each expression as a single logarithre A log 32 log 4 C 2 log 6 log 9 E

O Solve each equation 9 9 2n 2 10 8 A 3 0 10 C B 9 D 0 2 B Evaluate each using the values given 10 g m q q use m 6 and q 2 9

Prove by Insuction 1 2 3 4 2 2 4 1 n z 24 1

8 Two ocean liners leave from the same port in Puerto Rico at 10 00 a m One travels at a bearing of N 51 W at 11 miles per hour and the other travels at a bearing of S 53 W at 17 miles per hour Approximate the distance between them at noon the same day Round your answer to two decimal places O 20 28 miles O 39 93 miles O 35 75 miles O 30 25 miles

8 Determine the area of a triangle having the following measurements Round your answer to two decimal places C 73 a 12 5 and b 11 5 55 17 sq units 62 06 sq units 68 96 sq units

32 8 The range of a projectile fired at an angle with the horizontal and with an initial velocity of vo feet per second is measured in feet A golfer strikes a golf ball at 90 feet per second Ignoring the effects of air resistance at what angle must the golfer hit the ba so that it travels 150 feet Round your answer to the nearest degree 0 36 O 41 14 18 v sin 20 where r is 27

5 The mach number M of an airplane is the ratio of its speed to the speed of sound When an airplane travels faster than the speed of sound th sound waves form a cone behind the airplane see figure The mach number is related to the apex angle of the cone by sin 0 5 1 M

Systems of linear inequalities and Linear programming 12 A factory manufactures two kinds of ice skates racing skates and figure skates The racing skates require 8 work hours in the fabrication department whereas the figure skates require 4 work hours there The racing skates and the figure skates each require 2 work hours in the finishing department The fabricating department has available at most 224 work hours per day and the finishing department has no more than 72 work hours per day available The factory must also produce at least 16 racing skates If the profit on each racing skate is 15 and the profit on each figure skate is 12 how many of each should be manufactured each day to maximize profit Assume that all skates made are sold how the feasible region and the corner points

Describe the end behavior of the left side of the function f x x5 3x 2 y approaches infinity as x approaches negative infinity y approaches negative infinity as x approaches infinity Oy approaches negative infinity as x approaches negative infinity O y approaches infinity as x approaches infinity

Find the length of the side labeled x in the following triangle 8 4 62 6 93 X 30 4

What is the horizontal asymptote of the following rational function x 1 x3 6x 2 y There is no horizontal asymptote y 2 3 y 0 y 3 2 A

Solve the triangle a 4 32 m c 5 73 m B 29 6 What is the length of side b m Simplify your answer Type an integer or a decimal Round to four decimal places if needed What is the measure of angle A Simplify your answer Type an integer or a decimal Round to the nearest tenth if needed What is the measure of angle C 0 Simplify your answer Type an integer or a decimal Round to the nearest tenth if needed

K Determine the remaining sides and angles of the triangle ABC What is the measure of angle B 0 Simplify your answer Type an integer or a decimal What is the length of side a ft Simplify your answer Type an integer or a decimal Round to the nearest tenth as needed What is the length of side b ft Simplify your answer Type an integer or a decimal Round to the nearest tenth as needed B B 50 4 ft 35 69 114 4

5 5 135 10 4k 56 E 6 6 2 k 1 4k 2k 1

Graph the function y 3 sin 2x Give the period and the amplitude Choose the correct graph of the function OA Ay 3 8T OB Ay 4 0 The amplitude is Simplify your answer Type an exact answer using as needed The period is Simplify your answer Type an exact answer using as needed X 81 C Ay 4 b THU X 8x o O D 3 0 3 8

20 22 Kuta Software LLC All rights reserved Subtracting Rational Expressions 2 Simplify each expression 2 1 10m 1 10 5m 1 8 2 7 3r 9r 3r

Find the exact function value tan tan 3T 4 3T 4 Simplify your answer including any radicals Use integers or fractions for any numbers in the ex

Select the correct choice below and if necessary fill in the answer box to complete your choice O A csc 8 Simplify your answer including any radicals Use integers or fractions for any numbers in the expression OB The function is undefined

7 The measure A in degrees of an exterior angle of a regular polygon is related to the number of sides n of the polygon by the formula nA 360 If the measure of an exterior angle of a regu lar polygon is greater than 50 what is the greatest number of sides it can have

Example 1 Kathy is a repair technician for a phone company Each week she receives a batch of phones that need to be repaired The number of phones that she has left to fix at the end of each day can be estimated with the equation P 108 23d where P is the number of phones left and d is the number of days she has worked this week Q1 Define your variables below in your own words if possible P d Q2 Complete a table for this equation P 0 2 63 0 Q3 What patterns do you see in the table above Q4 How many cell phones will Kathy have left to repair after 3 days 05 How many days will it take Kathy to repair all of the cell phones

O sin u v O O 8 Find the exact value of sin u v given that sin u 25 O sin u v sin u v sin u v sin u v 204 325 204 325 253 325 36 325 36 and cos v 13 Both u and v are in Quadrant II

O 2 Simplify the expression algebraically 3 3 sin COS X 2 1 O o cosx COS X 3 O 3 cos x O 3 sin x O cosx 3

6 Write the given expression as the tangent of an angle tan 7x tan4x 1 tan 7x tan 4x tan 3x tan 5x Otan 17x Otan 11x tan 7x

1 Find the expression as the tangent of an angle O tan 60 tan 20 1 tan 60 tan 20 O O tan 40 1 O tan 80 tan 60 tan 40 tan 20

10 A student works the following problem and then stops Explain what is wrong with the problem 6 cos x 2 1 cos x cos x 3 1 V 3 1 ANSWER 1

6 Graph the expression on each side of the equals symbol to determine whether the equation might be an identity 4 7tan x sin x 4csc x 7sec x Not an Identity Identity