Algebra Questions

About the sequence x kezt we have the following 1 B x b for each te to b t b a MK 1 1 l b t b k20 k 1 c x kezt is uniformly Cauchy on t b t b verification of these properties is left for the reader We consider the uniform limit of the sequence x 1 8 to 8 B x b C which is given by x 1 lim x 1 1 l b to b Thus x 1 lim x f s x s ds 565 8 60 ds b Finally the validity of this integral equation has the following two implications X t x s x s ds x 0 x 1 x lim ff s x s ds x lim f s x s ds x f f s lim x s ds x ff s x s ds Thus the function x 1 satisfying the integral equation x 1 x f s x s ds for all 1 t b t b 2 dx dt d 0 ff s x s ds f 1 X t This now proves that the curve X 1 8 1 82 thus obtained is a solution of the initial value problem

MATHEMATICS Proposition 3 Let x J n y J be two solutions of the initial value problem dx f t x x t x dt then x 1 y 1 for all tJnJ Proof Recall that both x and y being solutions of the initial value problem satisfy the integral equations on their domain intervals x 1 x f s x s ds y 1 x f s y s ds Therefore x 1 y t ffs x s f s y s ds which implies x 1 y 1 0 s x s f s y s 0 SK x s y s for all 12 Applying Gronwalls result with A 0 we get 0 x 1 y 1 0 for all 1 1 This gives the desired equality x t y t for all tJJ Towards the uniqueness of the solution of the initial value problem 3 we consider all the solutions of the initial value problem 3 Let the totality of them be denoted by J XEA the solutions x being thus indexed by a suitable indexing set A Above we have verified that any two solutions say x and x are equal on the overlap of their domains Therefore we patch together all the solutions to get a maximal solution is the solution defined on the largest open interval It is obtained as follows Let J U J A clearly Jis an open sub internal of I with x J and all the solutions x patch up to get a solution x J of the initial value problem

If the point x y is equidistant from the points 3 3 and 5 5 show that x and y satisfy the linear equation x y 2 A

Usehe distance formula to show that the points 1 5 0 3 and 2 1 are collinear on a line

4 Find the values of y so that the distance from 0 2 to 12 y is 13 units

Bonus 1 The sum of the distances from x y to 1 0 and from x y to 1 0 is 4 Show x and y is related by the 3x 4y 12

a b Xx 1 x 0 5 1E t b t b k20 c x keZ is uniformly Cauchy on t b t b verification of these properties is left for the reader We consider the uniform limit of the sequence x to 8 to 8B x b C which is given by x t lim x 1 1E1 b t b x 1 B x b for each te to b t b MK k 1 Thus x t lim xo 1 2 F 8 8 89 ds f s x x lim ff s x s ds x lim f s x s ds E xXx 4 ff s lim x s ds x f f s x s ds Thus the function tx 1 satisfying the integral equation Finally the validity of this integral equation has the following two implications x dx dt x 1 x ff s f s x s ds for all 1 t b t b that is X t x f s 4 d 0 f s x s ds x 0 xo j f s x s ds f t X t This now proves that the curve X 1 8 1 82 thus obtained is a solution of the initial value problem 2 3 UNIQUENESS OF A SOLUTION 20 We prove an inequality which will lead us to the uniqueness of the solutions Let Proposition 2 Gronwall s Inequality f a b 0 g a b 000 be continuous functions and A 0 a constant satisfying ft A Chapter 2 Systmes of First Order ODE g de ffs gs ds for all te ab Then ft A e Proof First we assume A 0 and put ht A S te ab Then ht 0 for all 1 a b and h t f t g t h t g t for all te a b fsgs ds for all Salt for all 16 ab Integrating this inequality over at

Jared creates a number sequence that has a first term of 2 and a second term of 5 Each term after the second is created by subtracting the term before the previous term from twice the previous term He uses s t to denote te number t in his sequence For example s 1 2 and s 2 5 Which of the following can be used to find the value of s n for some positive integer n greater than 2 AS n S n 2 2S n 1 S n 2S n 2 S n 1 S n S n 1 2S n 2 S n 2s n 1 S n 2

8 6 4 2 y 8 6 4 2 2 4 6 8 A 4 4 2 0 2 4 2 4 6 8 X

The Perez family wants to save money to travel the world They purchase an annuity with a yearly payment of 580 that earns 4 interest compounded annually Payments will be made at the end of each year Find the total value of the annuity in 11 years Do not round any intermediate computations and round your final answer to the nearest cent If necessary refer to the list of financial formulas

UIS sm Use the equation below to answer the question y 3x 6 Which equivalent equation is correctly matched with a key feature of the graph of the function it represen y 3 x 2 highlights that the y intercept is at 2 y 3 x 2 highlights that the y intercept is at 2 y 3 x 2 highlights that the x intercept is at 2

Determine an ordered pair that satisfies the given equation by substituting the given value of the variable into the equation 2x y 7 x 3 The ordered pair is Type an ordered pair

Determine whether or not the ordered pair is a solution to the equation y 3x 7 a 5 8 c Is a Is 5 8 a solution to the given equation O No O Yes b Is 2 1 a solution to the given equation O No OYes 13 2 b 2 1 O No OYes c a solution to the given equation

t the ordered pair 0 7 in the rectangular coordinate tem Tell in which quadrant the point lies or state that point lies on the x axis or y axis t the point 0 7 on the graph point lies A in quadrant I in quadrant IV on the x axis on both coordinate axes in quadrant III in quadrant II on the v axis 2016 12 B

Identify the coordinates of each point labeled in the figure a A b B c C d D e E a A b B c C d D Type an ordered pair Type an ordered pair Type an ordered pair Type an ordered pair

Plot the following ordered pair in the rectangular coordinate system Tell which quadrant the point lies in or state that the point lies on the x axis or y axis 3 1 Plot the point 3 1 In which quadrant or on what axis does point 3 1 lie OA Quadrant I OC x axis OE Quadrant III OB y axis D Quadrant IV OF Quadrant II LLE 10 8 40 AY

Plot the points 0 8 and 4 16 in a rectangular coordinate system Then draw a line through the two points Find and interpret the slope of the line containing the points Click to enlarge graph A Select the correct choice below and if necessary fill in the answer boxes to complete your choice OA The slope of the line is The value of y will increase by unit s when x increases by unit s Type integers or simplified fractions OB The slope of the line is The value of y will increase by unit s when x decreases by unit s Type integers or simplified fractions OC The slope is undefined 20 516 12 BI 41 20 16 12 8 8 12 16 20

Find the slope and y intercept of the line whose equation is y 5x 6 What is the slope Select the correct choice below and if necessary fill in the answer box to complete your choice A The slope is Type an integer or a fraction OB The slope is undefined What is the y intercept Select the correct choice below and if necessary fill in the answer box to complete your choice A The y intercept is Type an ordered pair

Plot the given points in a rectangular coordinate system Then draw a line through the two points Find and interpret the slope of the line 4 7 and 5 7 yrapi Select the correct choice below and if necessary fill in the answer box to complete your choice OA m Type an integer or a simplified fraction OB The siope m is undefined Interpret the slope of the line Choose the correct answer below OA There is no change in x The line is vertical OB There is no change in y The line is horizontal C As x increases y decreases O x increases y increases 10 B 6 A 2 10 Ay 8 6 2 2 14 6 8 40

1 Suzette ran and biked for a total of 44 75 mi in 4 h Her average running speed was 6 5 mph and her average biking speed was 14 mph Let x total hours Suzette ran Let y total hours Suzette biked a In part b you will need to solve for x and y Write the system of equations you will need to solve this problem and explain in words what each piece of the equations represents 3 points Equation Explanation Equation Explanation b Use substitution to solve for x and y In order to receive points you must solve for x first Show ALL of your work 3 points How many hours did Suzette run How many hours did she bike

3 Graph the following system of inequalities on the coordinate plane In the top 3 boxe need to explain exactly how you used the inequality to graph the line including how you used slope what the inequality sign means and of course show work if needed The bottom 3 boxes are self explanatory The graph must include labeling write the inequality next to the graphed line and shading of all 3 lines as well as the solution to the system The top row of boxes is worth a total of 2 points bottom row total of 2 points graph total of 2 points 4x 5y 20 How did you graph the line y 3x 2 How did you graph the line Where do you shade and how do you know 5 Where do you shade and how do you know 6 8 y 4 How did you graph the line Where do you shade and how do you know 9 10

2 Solve the inequality 3x 2 6 2x For full credit you will need to show and explain every step An explanation should state what you re doing and WHY A statement is not an explanation You may add more lines if you need them Steps Work 3 points Explanation 3 points

x x y v Use the Pythagorean Theorem to calculate the distance between the points 1 3 and 4 6

Use distance formula to calculate the distance between the points 1 3 and 4 6

1 02 08 Solve the compound inequality 6b 42 or 4b 12 8 1 point Ob 6 or b 5 Ob 7 or b 1 Ob 7 or b 1 Ob 6 or b 5

Solve the equation for the unknown 2r 5 10

Solve and check 3 7m 9

Solve the following equation 5x 2 3 x 2 X x Simplify your answer

Solve the equation for the unknown 5 a 2

Solve and check 7x 7 35

Solve the equation 9x 8x 2x 5 10

The formula S P 0 60P gives the sale price S of an item whose original cost P dollars was reduced by 60 Find the sale price of a digital media player that originally cost 160 00 LEEB

For the following problem use the formula I PRT where I is interest in dollars P is the principal or loan amount R is the interest rate and T is the time in years 1 Find the interest if principal of 2600 is invested at 11 or 0 115 for four years 1 Round to the nearest dollar as needed

2x 1 13 f x 3 x 21 FIND f 5 Type a response 3 2x 1 x 1 1 x 5 x 5

2x 1 f x 3 x 21 FIND f 1 Type a response x 1 1 x 5 x 5

6 6 WA CHC 2 zoom in x K 6 The DOMAIN of the fu Select The RANGE of the fun Select

10 8 46 14 zoom in 12 10 State the DOMAIN of the given relation O O 12 13 O X 5 6zy24 6 y 4 x 4 6 x 4 NONE OF THESE

3 Evaluate each function Round to the nearest hundredth if necessary 3 n 1 3 2 3n 1 5 0 7 Find 4 4 4 Each graph represents a relation Determine if the relation is a function 4 A The relation is a function min

9 Determine if the function given is ODD EVEN or NEITHER 7x f x O O EVEN NEITHER 5

Cool Down Fabric Sale At a fabric store fabrics are sold by the yard A dressmaker spent 36 35 on 4 25 yards of silk and cotton fabrics for a dress Silk is 16 90 per yard and cotton is 4 per yard Here is a system of equations that represent the constraints in the situation 1 What does the solution to the system represent x y 4 25 16 90x 4y 36 35 2 Find the solution to the system of equations Explain or show your reasoning

Evaluate each function 1 w n 2n 3 Find w 4 B 29 D 95 A 159 C S 2 2 f n n 3n Find f 6 B 88 D 18 A 54 C 40 Evaluate each function Round to the nearest hundredth if necessary TWO DIGITS

7 Determine if the function given is ODD EVEN or NEITHER f x O NEITHER march ODD EVEN 2 3

1 2 x lim k no xo f f s lim x s ds x f f s x s ds fo Thus the function tx 1 satisfying the integral equation x 1 x dx dr 500 Finally the validity of this integral equation has the following two implications X 1 x ff s x s ds 0 d dts x f s x s ds for all 1 t b t b f s x s ds f s x s ds x 0 x 1 S lim f s x s ds 11 f s x s ds f 1 X t This now proves that the curve X 1 8 82 thus obtained is a solution of the initial value problem 2 3 UNIQUENESS OF A SOLUTION 20 We prove an inequality which will lead us to the uniqueness of the solutions fi A e Proposition 2 Let Gronwall s Inequality f a b 0 g a b 0x be continuous functions and A 0 a constant satisfying PESA Ss s gs ds for all tea b Then Chapter 2 Systmes of First Order ODE for all te a b Proof First we assume A 0 and put ht A Sf a te a b Then ht 0 for all 1 a b and ht f t g t h t g t fsgs ds for all

10 10 f x 8x 13 12 12 x x 13x 56x 81 11 11 x x 3x x 1 M 13 13 f x x 4x 3x 12 13 Select Select Select falls rises to t to t to ti to th

11 11 f x x 3x x 1 13 IN 44 13 x x 4x 3x F 13 Select Select Select rises falls neither V V to the to the le to the ri

Write the equation of the circle centered at 8 5 with diameter 16

x y 6x 16y 370 is the equation of a circle with center h k and radius for h and k and T

Write the equation of the circle centered at 4 3 with radius 11 HT

Right 3 Down 7 Right 3 Up7 Left 3 Up 7 Left 3 Down 7 Reflection Down 7 Up 7 Left 7 Right 7 V Instructions Reflection g x 5x g x 5x 7 g x 5x 7 g x 5X 7 Right 3 Down 7 Left 3 Up 7 Right 3 Up7 g x 5x 3 7 Left 3 Down 7 g x 5x 3 7 Down 7 Right 7

The parent function f x 2x is transformed into the function g x 4 2 x 3 8 Identify the transformations that occurred Vertical shift Up 8 Vertical shift Down 8 Horizontal shift Left 3 Horizontal shift Right 3 X Axis Reflection Vertical Stretch narrower Vertical Compression shrink y abx h k H Shift X reflect Left Right a 1 stretch o a 1 shrink compress N Shift Up Down Horiz Asymptote y