Math - Others Questions

6 Assume te a b and f n t y M show Taylor s method of order n has a local truncation error of O h 7 Computational exercise Write a script to use Euler s method to compute the approximate solution

Write the MATLAB commands to reduce the block diagrams below 1 R S 5 115 2 Write the MATLAB script and answer below 2s C s

9 Write the equation of a rational function g x that satisfies ALL of the criteria below The graph of g has an z intercept at z 2 The graph of g has a vertical asymptote at z 1 4 The graph of g has a horizontal asymptote at y 3 The graph of g has a hole at z 6

1 Suppose that an investor with a five year investment horizon is considering purchasing a seven year 7 coupon bond selling at par The investor expects that he can reinvest the coupon payments at an annual interest rate of 9 4 and that at the end of the investment horizon two year bonds will be selling to offer a yield to maturity of 11 2 What is the total return on this investment The investor has a five year investment horizon and purchases a seven year 7 coupon bond for 1 000 The yield to maturity for this bond is 7 since it is selling at par The investor expects to be able to reinvest the coupon interest payments at an annual interest rate of 9 4 and that at the end of the planned investment horizon the then two year bond will be selling to offer a yield to maturity of 11 2 Hint using semiannual interest to compute total return for a bond on p 61 C 7 1 000 2 Step 1 to apply eqn 3 7 coupon interest interest on interest C 1 r r Step 2 utilizing eqn 2 7 2 8 projected sale price PV coupon payments PV par value 1 1 r r M 1 r Step 3 sum up 1 2 as the total future dollars Step 4 to obtain the semiannual total return use the formula Step 5 Double step 4 for the total return total future dollars purchase price of bond

5 Let 0 and t 0 and let ao a1 02 n be sequence that satisfies and a 1S 1 8 a t for each i20 We will prove an inequality that is important for deriving the global error in Euler s method a Show that implies 1 1 8 a 1 t 1 1 8 b Show that and part a imply 0 1 1 s a 2 t 1 1 s 1 s c Use an inductive argument to show that a 1 1 s t 1 1 s 1 s 1 s d The term in brackets 1 1 s 1 s is a geometric series Use this fact to rewrite the inequality from c as t 9 1 1 8 0 e Finally use your answer from 04 to conclude

graph with its equation Zv 47 21 2 a b d TO 1 33 7 Free KE w CO 20 3424 Em 4 45 34345 12343 13 2 3433 J 4 H P 12 7 1 2 J 4442 NO 17 72343 2241

enex hik b Vertex 2 c Domain d Range 2 0 e axis of symmetry 2 y 2 x 4 1 3 4 1 25 5 4 1 25 a Minor Max b Vertex 4 1 C Domain d Range e axis of symmetry 3 y 2x44x 1 9 2 a Minor Max b Vertex 2 5 c Domain 00 d Range 4 e X intercepts 1 y y intercept 4 y 2x 4x 1 y 2x 4x 2 1 2 x 2 1 4 y x 2 5 V 2 5 x b 2 a 4 2 2 1

Question 12 Given the function f z 2x 4 evaluate and simplify the expressions below See special instructions on how to enter your answers f a f a h f a h f a h Instructions Simplify answers as much as possible Expressions such as 4 x 2 and z 5 should be expanded Also collect like terms so 3z z should be written as 4z

A mutual fund is currently valued at 90 per share and its value per share is increasing at a rate of 0 70 a day Let V f t be the value of the share days from now a Express the information given about the mutual fund in terms of f and f i 10 and b Assuming that the rate of growth stays constant estimate and interpret f 10

Destion 3 The plot below represents the function f z 343 ka 5 40 F Br P Evaluate f 2 f 2 Solve f z 2 PH my 4 O

Question 4 select all of the following graphs which are one to one functions up 2 75 S

010 y 001 Z This gives x 20 y 35 and z 5 Thus the firm can produce 20 units of P 35 units of P and 5 units of P to make full use of its available raw material Remark One may observe that if the manufacturer decides to manufacture according to the available raw material and not according to the purchase orders of the two clients F and F as in Example 3 he she is unable to meet these purchase orders as F demanded 6 units of P where as the manufacturer can make only 5 units of P3 Example 5 A manufacturer of medicines is preparing a production plan of medicines M and M There are sufficient raw materials available to make 20000 bottles of M and 40000 bottles of M but there are only 45000 bottles into which either of the medicines can be put Further it takes 3 hours to prepare enough material to fill 1000 bottles of M it takes 1 hour to prepare enough material to fill 1000 bottles of M and there are 66 hours available for this operation The profit is Rs 8 per bottle for M and Rs 7 per bottle for M How should the manufacturer schedule his her production in order to maximise profit and M in order to maximise the Step 2 Let x be the number of bottles of type M medicine and y be the number of bottles of type M medicine Since profit is Rs 8 per bottle for M and Rs 7 per bottle for M therefore the objective function which is to be maximised is given by Solution Step 1 To find the number of bottles profit under the given hypotheses Z Z x y 8x 7y The objective function is to be maximised subject to the constraints Refer Chapter 12 on Linear Programming not 20 35 5 x 20000 y 40000 x y 45000 0 40 000 T 3x y 66000 x20 y20 Step 3 The shaded region OPQRST is the feasible region for the constraints 1 Fig A 2 3 The co ordinates of vertices O P Q R S and T are 0 0 20000 0 20000 6000 10500 34500 5000 40000 and 0 40000 respectively Rationalised 2023 24 x 20 000 MATHEMATICAL MODELLING S 5 000 40 000 y 40 000 1 R 10 500 34 500 203

This clearly shows that to meet the purchase order of F and F the raw material required is 335 units of R 467 units of R and 147 units of R which is much more than the available raw material Since the amount of raw material required to manufacture each unit of the three products is fixed we can either ask for an increase in the available raw material or we may ask the clients to reduce their orders Remark If we replace n place A in Example 3 by A given by i e if the clients agree to reduce their purchase orders then A B 10 A 9 12 6 10 20 0 Similarly we have 9 12 6 10 20 0 3 4 0 793 5 12 7 Rationalised 2023 24 141 216 78 170 220 60 MATHEMATICAL MODELLING This requires 311 units of R 436 units of R and 138 units of R which are well below the available raw materials i e 330 units of R 455 units of R and 140 units of R Thus if the revised purchase orders of the clients are given by A then the firm can easily supply the purchase orders of the two clients Note One may further modify A so as to make full use of the available raw material R R R P 3 4 0 B P 7 Query Can we make a mathematical model with a given B and with fixed quantities of the available raw material that can help the firm owner to ask the clients to modify their orders in such a way that the firm makes the full use of its available raw material The answer to this query is given in the following example 201 Example 4 Suppose P P P3 and R R R are as in Example 2 Let the firm has 330 units of R 455 units of R and 140 units of R available with it and let the amount of raw materials R R and R required to manufacture each unit of the three products is given by a d How many units of each product is to be made so as to utilise the full available raw material Solution Step 1 The situation is easily identifiable Step 2 Suppose the firm produces x units of P y units of P and z units of P Since product P requires 3 units of R P requires 7 units of R and P requires 5 units of R observe matrix B and the total number of units of R available is 330 we have 3x 7y 5z 330 for raw material R 4x 9y 12z 455 for raw material R 3y 7z 140 for raw material R and This system of equations can be expressed in matrix form as

The above principles of mathematical modelling lead to the following steps for mathematical modelling Step 1 Identify the physical situation Step 2 Convert the physical situation into a mathematical model by introducing parameters variables and using various known physical laws and symbols Step 3 Find the solution of the mathematical problem Step 4 Interpret the result in terms of the original problem and compare the result with observations or experiments Step 5 If the result is in good agreement then accept the model Otherwise modify the hypotheses assumptions according to the physical situation and to Step 2 The above steps can also be viewed through the following diagram Let Now Physical Introduce Mathematical Situation Model Physical Laws and Symbols or Accept the Model not Modify Hypothesis Not in Good Agreement Fig A 2 1 Example 1 Find the height of a given tower using mathematical modelling Solution Step 1 Given physical situation is to find the height of a given tower Step 2 Let AB be the given tower Fig A 2 2 Let PQ be an observer measuring the height of the tower with his eye at P Let PQ h and let height of tower be H Let a be the angle of elevation from the eye of the observer to the top of the tower Solve Mathematical Interpret Solution In Good Agreement Fig A 2 2 1 Rationalised 2023 24 tan 1 PC QB AC H h PC 1 B Compare with Observation Original Problem H h H MATHEMATICAL MODELLING 199 H h 1 tan a 1 Step 3 Note that the values of the parameters h I and a using sextant are known to

tan Step 5 is not required in this situation as exact values of the are known PQ QB I 200 Example 2 Let a business firm produces three types products P P and P that uses three types of raw materials R R and R Let the firm has purchase orders from two clients F and F Considering the situation that the firm has a limited quantity of R R and R respectively prepare a model to determine the quantities of the raw material R R and R required to meet the purchase orders Solution Step 1 The physical situation is well identified in the problem Step 2 Let A be a matrix that represents purchase orders from the two clients F and F Then A is of the form MATHEMATICS A F ot to he Let B be the matrix that represents the amount of raw materials R R and R required to manufacture each unit of the products P P and P Then B is of the form A P B P or 1 h cot B AB Rationalised 2023 24 Step 3 Note that the product which in this case is well defined of matrices A and B is given by the following matrix R R R F F 10 15 6 10 20 0 AB R R R3 which in fact gives the desired quantities of the raw materials R R and R to fulfill the purchase orders of the two clients F and F Example 3 Interpret the model in Example 2 in case 3 40 neters h 1 a and B B 7 9 3 5 12 7 and the available raw materials are 330 units of R 455 units of R Solution Note that 10 15 6 10 20 0 r 793 5 12 7 lied R R3 F 165 247 87 R 140 units of R

Question 9 Question Help Message ins It is important to know the appropriate dose of ibuprofen for a child so that you are ready in the case of high fever The dosing guidelines state that children should be given 8mg kg Determine the dose of ibuprofen needed for a 2 year old weighing 28 pounds Round your final answer to the nearest whole number and make sure to state your units to earn full credit Recall 1 kg 2 2 lb Select an answer E years kg

Question 22 Round these numbers to three significant figures as shown in class 1 566 38 2643468 374 649 2 11972 0 07052

Precision and Accuracy Four samples were taken from a population with a true population percentage of p 53 The number lines below show the sample percentages that resulted Determine precision and accuracy of the sample percentages 52 52 Select an answer 53 Select an answer Question Help Ro 53 e 54 54 52 52 0000 53 Select an answer Select an 53 54 54

Four students are measuring the temperature of a solution They each measure the temperature five times The actual temperature of the solution is 24 8 degrees Celsius Student 1 20 6 20 8 20 3 20 5 21 Student 2 25 2 14 5 24 6 25 1 24 8 Student 3 27 9 22 9 27 24 3 22 a The data collected by Student 1 is b The data collected by Student 2 is c The data collected by Student 3 is d The data collected by Student 4 is Question Help Message instructor Select an answer Select an answer Select an answer Select an answer Student 4 29 9 25 7 38 26 4 36 4

1 23 67 6 62 0 4 2 5 8 Name Mark an X for each category that applies Number Real Rational Irrational 2 7 2 5 7 25 1 4 75 3 4 September 2023 Group Work The Real Number System 9 10 11 12 4 13 5 2 14 2 3 15 12 5 8 16 4982 17 17 1 18 3 9 9 0 3 0 Integer Whole Natural

cubic inches Convert 898 59 cubic meters into cubic inches Round your answer off to the nearest hundredth if necessary Measurement and Conversion Table

estion 15 What is the percent uncertainty of the given measurement 137 5776 0 0005 sec onds Question Help Merca

Question 14 More accurate More precise First determine which number x or y is more accurate then which is more precise x 5 000 y 0 0070

on 10 Use the conversion factor 1 gallon 3 785 liters Convert 21 gallons per minute to liters per hour 21 gallons per minute liters per hour Round your answer to the nearest tenth Question Help Message instructor

Question 8 miles per gallon While researching vehicles Elias finds that the fuel economy of a particular vehicle is given as 12 68 kilometers per liter Convert this into miles per gallon Round your answer off to the nearest hundredth if necessary Measurement and Conversion Table

Question 16 How many significant figures does 9 984 860 0000 have Question Help Message instructor

Question 19 Determine the accuracy i e the number of significant figures of this number 37000 Select an answer Question Help Message instructor Submit Question

Question 23 Calculate the sum below and express the result with the correct number of significant figures 69 60 0 7780 256 900 Question Help Message instructor

Core 1732 1 32 answered Question 2 Complete the following table by filling in the missing metric prefixes abbreviations or multiplication factors The multiplication factor may be entered as a decimal a fraction or a power of 10 Metric Prefix Choose no prefix Choose B Choose Choose B nano Question Help Message instructor Abbreviation Choose base unit d Choose P Choose Multiplication Factor 10 10 10 3

Use PMT nt to determine the regular payment amount rounded to the nearest dollar The price of a small cabin is 1 1 1 30 000 The bank requires a 5 down payment The buyer is offered two mortgage options 20 year fixed at 7 5 or 30 year fixed at 7 5 Calculate the amount of interest paid for each option How much does the buyer save in interest with the 20 year option Find the monthly payment for the 20 year option Round to the nearest dollar as needed

32 answer Simplify this number and write the answer in appropriate scientific notation with the mantissa rounded to 3 decimal places Question Help Message instructor Submit Question 8 640 10 1 242 10 1 2 208 10 14 Note There are no questions that combine converting units with rounding rules Please remember to complete the practice worksheet to ensure you understand how to combine those concepts There will be instances on exams where you will need to do a unit conversion and round to the correct number of significant figures

Question 29 Rewrite this number in appropriate scientific notation 0 0073 108

Question 4 Convert 233 9 feet per minute into centimeters per second Round your answer off to the nearest hundredth if necessary centimeters per second Measurement and Conversion Table

Question 31 Multiply the following and express your answer in appropriate scientific notation 6 5 x 10 26 x 8 2 10 7 product Questi

2 Let D y t la t b c y d and let fe C D Suppose there exists L 0 such that y t L y t D Use the mean value theorem from calculus to show that f is Lipschitz on D with Lipschitz constant

The farmer decided that he ordered enough planks for an arena that measured 1151 ft by 631 ft Is he correct Explain

Evan works Tuesday through Saturday at their family s convenience store Besides the free snacks they get paid 55 day to work on Tuesday Wednesday Thursday and 80 day to work on Friday Saturday they work more hours on those days How much money does Evan make in one week

Determine whether the functions y and y2 are linearly dependent You can use the definition or the Wronskian 1 y t e t y t e 4t 2 y t te t y2 t e t 3 y t e t cos 2t y2 t e t sin 2t

How long will it take Jasmine and Ralph working together at the same individual rates to peel one pound of apples Be sure to include units in your answer

Solve Several different bus routes stop at the corner of Second St and Lincoln Ave A Wilkenson bus arrives every 33 minutes a Fifth Avenue bus arrives every 21 minutes and a Harris Road bus arrives every 77 minutes If both buses arrive at the stop at 5 07 AM how many minutes will pass before both buses arrive at the same time again O231 minutes 21 minutes 77 minutes

It takes Jasmine of an hour to peel a pound of apples It takes Ralph of an hour to peel a pound of apples If each person continues to work at the same rate how many pounds of apples can Jasmine and Ralph peehin one hour 12

solution Problem 7 y y 0 Problem 8

Plastic Hanging Display Strips with hooks Product Display clips for reta

5 2 Here is one way to solve the equation 5x 45 Explain what is done in each step 5x 45 2 x 9 x 9 Original equation Step 1 Step 2 x 3 or x 3 Step 3

3 Write a statement and an expression to represent the tape diagram Then evaluate your expression 22 9 22 9 22 9 22 9 22 9 22 9 Statement Expression Value of Expression

2 y 1 2 x 4

Given the following illustration find the TIME in all locations ALSO Bases upon the location of the Sun Identify what portion of the Earth is currently DARK you can say top or bottom given given EACH position THE INFORMATION BELOW FIND THE TIME OF DAY FOR 6AM 6PM 12 Am 12 pm Am apm 3AM 3pm of EARTH 2 shade the dark side 3 SUN 4 A 2 1 1 2 3 4 7

10 A a da bir s navda soru seviyelerine g re do ru ve yanl cevaplar i in al nacak puanlar verilmi tir Tablo Soru Seviyelerine G re Do ru ya da Yanl Yap ld nda Al nacak Puanlar Soru Seviyesi Zor Orta Kolay Do ru 5 32 2 Yanl 2 2 2

stralia s Great Barrier Reef The map shows disease conditions on coral reefs at six study sites of white syndrome an infectious disease of coral was investigated in a six year study on Key Number of colonies with white syndrome per 1500 m of reef 50 O 26 50 1 25 0 Cooktown Lizard Island Cairns Townsville a i Identify the site with the highest incidence of diseased colonies 1 a ii Deduce whether there is a trend in the incidence of white syndrome over the north south range of latitude 1

0 2 2