application of calculus grade 12 pdf

The important areas which are necessary for advanced calculus are vector spaces, matrices, linear transformation. CAMI Mathematics: :: : Grade 12 12.5 Calculus12.5 Calculus 12.5 Practical application 12.5 Practical application A. Calculate the width and length of the garden that corresponds to the largest possible area that Michael can fence off. Calculus—Study and teaching (Secondary)—Manitoba. \therefore t&=-\text{0,05} \text{ or } t=\text{6,05} For example we can use algebraic formulae or graphs. \text{Hits ground: } D(t)&=0 \\ \text{and } g(x)&= \frac{8}{x}, \quad x > 0 MCV4U – Calculus and Vectors Grade 12 course builds on students’ previous experience with functions and their developing understanding of rates of change. This means that \(\frac{dS}{dt} = v\): If \(x=20\) then \(y=0\) and the product is a minimum, not a maximum. \begin{align*} Statisticianswill use calculus to evaluate survey data to help develop business plans. &\approx \text{12,0}\text{ cm} A & = \text{ area of sides } + \text{ area of base } + \text{ area of top } \\ Chapter 7. Foundations of Mathematics, Grades 11–12. A set of questions on the concepts of a function, in calculus, are presented along with their answers and solutions. 12 Exponential Form(s) of a Positive Real Number and its Logarithm(s): Pre-Requisite for Understanding Exponential and Logarithmic Functions (What must you know to learn Calculus?) This means that \(\frac{dv}{dt} = a\): \end{align*}. Calculus—Programmed instruction. 2 + 3 (10 marks) a) Determine the slope of the secant lines PR, PS, and PT to the curve, given the coordinates P(1, 1), R(4, -29), S(3, -15), T(1.1, 0.58). High marks in maths are the key to your success and future plans. When will the amount of water be at a maximum? What is the most economical speed of the car? Start by finding an expression for volume in terms of \(x\): Now take the derivative and set it equal to \(\text{0}\): Since the length can only be positive, \(x=10\), Determine the shortest vertical distance between the curves of \(f\) and \(g\) if it is given that: Given: g (x) = -2. x. The velocity after \(\text{4}\) \(\text{s}\) will be: The ball hits the ground at a speed of \(\text{20}\text{ m.s$^{-1}$}\). Grade 12 Introduction to Calculus. Let \(f'(x) = 0\) and solve for \(x\) to find the optimum point. Just because gravity is constant does not mean we should necessarily think of acceleration as a constant. The length of the block is \(y\). The diagram shows the plan for a verandah which is to be built on the corner of a cottage. 2. 5. The ends are right-angled triangles having sides \(3x\), \(4x\) and \(5x\). \end{align*}. Mathematics for Apprenticeship and Workplace, Grades 10–12. \end{align*}, \begin{align*} Those in shaded rectangles, e. Data Handling Transformations 22–32 16 Functions 33–44 17 Calculus 45 – 53 18 54 - 67 19 Linear Programming Trigonometry 3 - 21 2D Trigonometry 3D Trigonometry 68 - 74 75 - 86. Chapter 5. &= 18-6(3) \\ \text{Instantaneous velocity}&= D'(3) \\ Rearrange the formula to make \(w\) the subject of the formula: Substitute the expression for \(w\) into the formula for the area of the garden. \text{where } V&= \text{ volume in kilolitres}\\ Questions and Answers on Functions. d&= \text{ days} Lessons. Students will study theory and conduct investigations in the areas of metabolic processes, molecular genetics, homeostasis, evolution, and population dynamics. Integrals . Calculating stationary points also lends itself to the solving of problems that require some variable to be maximised or minimised. Calculate the maximum height of the ball. Application of Derivative . Determine the rate of change of the volume of the reservoir with respect to time after \(\text{8}\) days. Thomas Calculus 12th Edition Ebook free download pdf, 12th edition is the most recomended book in the Pakistani universities now days. A survey involves many different questions with a range of possible answers, calculus allows a more accurate prediction. %�쏢 \begin{align*} R�nJ�IJ��\��b�'�?¿]|}��+������.�)&+��.��K�����)��M��E�����g�Ov{�Xe��K�8-Ǧ����0�O�֧�#�T���\�*�?�i����Ϭޱ����~~vg���s�\�o=���ZX3��F�c0�ïv~�I/��bm���^�f��q~��^�����"����l'���娨�h��.�t��[�����t����Ն�i7�G�c_����_��[���_�ɘ腅eH +Rj~e���O)MW�y �������~���p)Q���pi[���D*^����^[�X7��E����v���3�>�pV.����2)�8f�MA���M��.Zt�VlN\9��0�B�P�"�=:g�}�P���0r~���d�)�ǫ�Y����)� ��h���̿L�>:��h+A�_QN:E�F�( �A^$��B��;?�6i�=�p'�w��{�L���q�^���~� �V|���@!��9PB'D@3���^|��Z��pSڍ�nݛoŁ�Tn�G:3�7�s�~��h�'Us����*鐓[��֘��O&�`���������nTE��%D� O��+]�hC 5��� ��b�r�M�r��,R�_@���8^�{J0_�����wa���xk�G�1:�����O(y�|"�פ�^�w�L�4b�$��%��6�qe4��0����O;��on�D�N,z�i)怒������b5��9*�����^ga�#A Calculus—Study and teaching (Secondary). Differential Calculus - Grade 12 Rory Adams reeF High School Science Texts Project Sarah Blyth This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License y Chapter: Di erential Calculus - Grade 12 1 Why do I have to learn this stu ? Download: ThomasCalculus12thBook. V'(d)&= 44 -6d \\ &= \frac{3000}{x}+ 3x^2 In this chapter we will cover many of the major applications of derivatives. \end{align*}. The app is well arranged in a way that it can be effectively used by learners to master the subject and better prepare for their final exam. Primary Menu. \begin{align*} > Grade 12 – Differential Calculus. Velocity after \(\text{1,5}\) \(\text{s}\): Therefore, the velocity is zero after \(\text{2}\text{ s}\), The ball hits the ground when \(H\left(t\right)=0\). Lessons. Is this correct? Click below to download the ebook free of any cost and enjoy. 1. We find the rate of change of temperature with time by differentiating: We get the following two equations: Rearranging the first equation and substituting into the second gives: Differentiating and setting to \(\text{0}\) gives: Therefore, \(x=20\) or \(x=\frac{20}{3}\). Determine the velocity of the ball when it hits the ground. Homework. Calculus Questions, Answers and Solutions Calculus questions with detailed solutions are presented. mrslawsclass@gmail.com 604-668-6478 . D'(\text{6,05})=18-5(\text{6,05})&= -\text{18,3}\text{ m.s$^{-1}$} When average rate of change is required, it will be specifically referred to as average rate of change. 3. Mathematically we can represent change in different ways. The time at which the vertical velocity is zero. \end{align*}, \begin{align*} &= 4xh + x^2 + 2x^2 \\ (16-d)(4+3d)&=0\\ The questions are about important concepts in calculus. &= 4xh + 3x^2 \\ Let the first number be \(x\) and the second number be \(y\) and let the product be \(P\). Relations and Functions Part -1 . TABLE OF CONTENTS TEACHER NOTES . V & = x^2h \\ Is the volume of the water increasing or decreasing at the end of \(\text{8}\) days. 2. Credit card companiesuse calculus to set the minimum payments due on credit card statements at the exact time the statement is processed. Determinants . Between 09:01 and 09:02 it … Chapter 2. t&=\frac{-18 \pm\sqrt{(18^{2}-4(1)(-3)}}{2(-3)} \\ (i) If the tangent at P is perpendicular to x-axis or parallel to y-axis, (ii) If the tangent at P is perpendicular to y-axis or parallel to x-axis, If we draw the graph of this function we find that the graph has a minimum. The height (in metres) of a golf ball \(t\) seconds after it has been hit into the air, is given by \(H\left(t\right)=20t-5{t}^{2}\). When we mention rate of change, the instantaneous rate of change (the derivative) is implied. Grade 12 Biology provides students with the opportunity for in-depth study of the concepts and processes associated with biological systems. The sum of two positive numbers is \(\text{10}\). Germany. Interpretation: this is the stationary point, where the derivative is zero. V'(8)&=44-6(8)\\ The interval in which the temperature is increasing is \([1;4)\). A soccer ball is kicked vertically into the air and its motion is represented by the equation: Therefore, the width of the garden is \(\text{80}\text{ m}\). Effective speeds over small intervals 1. T'(t) &= 4 - t Interpretation: the velocity is decreasing by \(\text{6}\) metres per second per second. \begin{align*} A rectangle’s width and height, when added, are 114mm. Inverse Trigonometry Functions . We will therefore be focusing on applications that can be pdf download done only with knowledge taught in this course. \end{align*}. \text{Reservoir empty: } V(d)&=0 \\ Chapter 4. 4. Explain your answer. Resources. \therefore 0 &= - \frac{3000}{x^2}+ 6x \\ A pump is connected to a water reservoir. We think you are located in Applied Mathematics 9. x^3 &= 500 \\ Related Resources. If the displacement \(s\) (in metres) of a particle at time \(t\) (in seconds) is governed by the equation \(s=\frac{1}{2}{t}^{3}-2t\), find its acceleration after \(\text{2}\) seconds. \end{align*}. D''(t)&= -\text{6}\text{ m.s$^{-2}$} &=\frac{8}{x} - (-x^{2}+2x+3) \\ (Volume = area of base \(\times\) height). &= 1 \text{ metre} Velocity is one of the most common forms of rate of change: Velocity refers to the change in distance (\(s\)) for a corresponding change in time (\(t\)). \begin{align*} If \(AB=DE=x\) and \(BC=CD=y\), and the length of the railing must be \(\text{30}\text{ m}\), find the values of \(x\) and \(y\) for which the verandah will have a maximum area. D'(\text{1,5})&=18-6(\text{1,5})^{2} \\ The container has a specially designed top that folds to close the container. 5 0 obj If \(f''(a) > 0\), then the point is a local minimum. \text{where } D &= \text{distance above the ground (in metres)} \\ Homework. E-mail *. &= \text{0}\text{ m.s$^{-1}$} &= \text{Derivative} Calculus Applications II. Sitemap. \end{align*}. Principles of Mathematics, Grades 11–12. \text{Velocity after } \text{6,05}\text{ s}&= D'(\text{6,05}) \\ Michael wants to start a vegetable garden, which he decides to fence off in the shape of a rectangle from the rest of the garden. \text{Let the distance } P(x) &= g(x) - f(x)\\ D(0)&=1 + 18(0) - 3(0)^{2} \\ Title: Grade 12_Practical application of calculus Author: teacher Created Date: 9/3/2013 8:52:12 AM Keywords () V(d)&=64+44d-3d^{2} \\ Related. These are referred to as optimisation problems. 1:22:42. Determine the initial height of the ball at the moment it is being kicked. Chapter 9 Differential calculus. We are interested in maximising the area of the garden, so we differentiate to get the following: To find the stationary point, we set \({A}'\left(l\right)=0\) and solve for the value(s) of \(l\) that maximises the area: Therefore, the length of the garden is \(\text{40}\text{ m}\). by this license. Unit 8 - Derivatives of Exponential Functions. Applications of Derivatives ... Calculus I or needing a refresher in some of the early topics in calculus. We use the expression for perimeter to eliminate the \(y\) variable so that we have an expression for area in terms of \(x\) only: To find the maximum, we need to take the derivative and set it equal to \(\text{0}\): Therefore, \(x=\text{5}\text{ m}\) and substituting this value back into the formula for perimeter gives \(y=\text{10}\text{ m}\). After how many days will the reservoir be empty? \begin{align*} Calculus Concepts Questions. View Pre-Calculus_Grade_11-12_CCSS.pdf from MATH 122 at University of Vermont. \begin{align*} \text{Instantaneous velocity } &= \text{Instantaneous rate of change } \\ 2. All Siyavula textbook content made available on this site is released under the terms of a & \\ Calculate the average velocity of the ball during the third second. 339 12.1 Introduction 339 12.2 Concept of Logarithmic 339 12.3 The Laws of Exponent 340 12… Thomas Calculus 11th Edition Ebook free download pdf. Let the two numbers be \(a\) and \(b\) and the product be \(P\). \begin{align*} Show that \(y= \frac{\text{300} - x^{2}}{x}\). v &=\frac{3}{2}t^{2} - 2 \\ The quantity that is to be minimised or maximised must be expressed in terms of only one variable. We have seen that differential calculus can be used to determine the stationary points of functions, in order to sketch their graphs. To check whether the optimum point at \(x = a\) is a local minimum or a local maximum, we find \(f''(x)\): If \(f''(a) < 0\), then the point is a local maximum. \text{Substitute } h &= \frac{750}{x^2}: \\ Notice that this formula now contains only one unknown variable. The cardboard needed to fold the top of the container is twice the cardboard needed for the base, which only needs a single layer of cardboard. MATHEMATICS . Chapter 8. During which time interval was the temperature dropping? The ball hits the ground after \(\text{4}\) \(\text{s}\). A &= 4x\left( \frac{750}{x^2} \right) + 3x^2 \\ We can check that this gives a maximum area by showing that \({A}''\left(l\right) < 0\): A width of \(\text{80}\text{ m}\) and a length of \(\text{40}\text{ m}\) will give the maximum area for the garden. The vertical velocity of the ball after \(\text{1,5}\) \(\text{s}\). \end{align*}, \begin{align*} \end{align*}. I’ve tried to make these notes as self contained as possible and so all the information needed to read through them is either from an Algebra or Trig class or contained in other sections of the notes. Substituting \(t=2\) gives \(a=\text{6}\text{ m.s$^{-2}$}\). Grade 12 Page 1 DIFFERENTIAL CALCULUS 30 JUNE 2014 Checklist Make sure you know how to: Calculate the average gradient of a curve using the formula Find the derivative by first principles using the formula Use the rules of differentiation to differentiate functions without going through the process of first principles. Aims and outcomes of tutorial: Improve marks and help you achieve 70% or more! Pre-Calculus 12. Find the numbers that make this product a maximum. The important pieces of information given are related to the area and modified perimeter of the garden. An object starts moving at 09:00 (nine o'clock sharp) from a certain point A. The ball has stopped going up and is about to begin its descent. Creative Commons Attribution License. &=18-9 \\ It can be used as a textbook or a reference book for an introductory course on one variable calculus. v &=\frac{3}{2}t^{2} - 2 GRADE 12 . Nelson Mathematics, Grades 7–8. The novels, plays, letters and life. \end{align*}. \therefore 64 + 44d -3d^{2}&=0 \\ MATHEMATICS NOTES FOR CLASS 12 DOWNLOAD PDF . Math Focus, Grades 7–9. Mathemaics Download all Formulas and Notes For Vlass 12 in pdf CBSE Board . \begin{align*} \begin{align*} A'(x) &= - \frac{3000}{x^2}+ 6x \\ \begin{align*} One of the numbers is multiplied by the square of the other. 0 &= 4 - t \\ Matrix . The fuel used by a car is defined by \(f\left(v\right)=\frac{3}{80}{v}^{2}-6v+245\), where \(v\) is the travelling speed in \(\text{km/h}\). 6x &= \frac{3000}{x^2} \\ One of the numbers is multiplied by the square of the other. We know that the area of the garden is given by the formula: The fencing is only required for \(\text{3}\) sides and the three sides must add up to \(\text{160}\text{ m}\). The vertical velocity with which the ball hits the ground. Burnett Website; BC's Curriculum; Contact Me. We set the derivative equal to \(\text{0}\): \begin{align*} O0�G�����Q�-�ƫ���N�!�`ST���`pRY:␆�A ��'y�? It contains NSC exam past papers from November 2013 - November 2016. Determine the velocity of the ball after \(\text{3}\) seconds and interpret the answer. We look at the coefficient of the \(t^{2}\) term to decide whether this is a minimum or maximum point. We have learnt how to determine the average gradient of a curve and how to determine the gradient of a curve at a given point. Mathematics for Knowledge and Employability, Grades 8–11. \[A (\text{in square centimetres}) = \frac{\text{3 000}}{x} + 3x^{2}\]. A rectangular juice container, made from cardboard, has a square base and holds \(\text{750}\text{ cm}^{3}\) of juice. The speed at the minimum would then give the most economical speed. 2. We know that velocity is the rate of change of displacement. �3֕���~�ك[=���c��/�f��:�kk%�x�B6��bG�_�O�i �����H��Z�SdJ�����g�/k"�~]���&�PR���VV�c7lx����1�m�d�����^ψ3������k����W���b(���W���P�A ^��܂Bƛ�Qfӓca�7�z0?�����M�y��Xːt�L�b�>"��مQ�O�z����)����[��o������M�&Vxtv. The total surface area of the block is \(\text{3 600}\text{ cm$^{2}$}\). Application on area, volume and perimeter A. Connect with social media. Unit 1 - Introduction to Vectors‎ > ‎ Homework Solutions. x��\��%E� �|�a`�/p�ڗ_���� �K|`|Ebf0��=��S�O�{�ńef2����ꪳ��R��דX�����?��z2֧�䵘�0jq~���~���O�� -3t^{2}+18t+1&=0\\ A railing \(ABCDE\) is to be constructed around the four edges of the verandah. \begin{align*} \therefore d = 16 \text{ or } & d = -\frac{4}{3} \\ This implies that acceleration is the second derivative of the distance. Password * During an experiment the temperature \(T\) (in degrees Celsius) varies with time \(t\) (in hours) according to the formula: \(T\left(t\right)=30+4t-\frac{1}{2}{t}^{2}, \enspace t \in \left[1;10\right]\). 36786 | 185 | 8. Sign in with your email address. \begin{align*} The rate of change is negative, so the function is decreasing. Handouts. To draw a rough sketch of the graph we need to calculate where the graph intersects with the axes and the maximum and minimum function values of the turning points: Note: the above diagram is not drawn to scale. Determine an expression for the rate of change of temperature with time. We use this information to present the correct curriculum and D(t)&=1 + 18t -3t^{2} \\ 11. Advanced Calculus includes some topics such as infinite series, power series, and so on which are all just the application of the principles of some basic calculus topics such as differentiation, derivatives, rate of change and o on. Common Core St at e St andards: Mat hemat ics - Grade 11 Mat hemat ics Grade: 11 CCSS.Math.Content.HSA PDF | The diversity of the research in the field of Calculus education makes it difficult to produce an exhaustive state-of-the-art summary. TEACHER NOTES . How long will it take for the ball to hit the ground? Handouts. MALATI materials: Introductory Calculus, Grade 12 5 3. The additional topics can be taught anywhere in the course that the instructor wishes. Calculus Concepts Questions application of calculus grade 12 pdf application of calculus grade 12 pdf Questions application of calculus grade 12 pdf and Answers on Functions. Mathematics / Grade 12 / Differential Calculus. \text{Initial velocity } &= D'(0) \\ This rate of change is described by the gradient of the graph and can therefore be determined by calculating the derivative. \end{align*}. \therefore h & = \frac{750}{x^2}\\ f(x)&= -x^{2}+2x+3 \\ \text{Velocity after } \text{1,5}\text{ s}&=D'(\text{1,5}) \\ The use of different . Acceleration is the change in velocity for a corresponding change in time. Ontario. A(x) &= \frac{3000}{x}+ 3x^2 \\ 10. Apart from whole-class teaching, teachers can utilise pair and group work to encourage peer interaction and to facilitate discussion. Calculus 12. Grade 12 Mathematics Mobile Application contains activities, practice practice problems and past NSC exam papers; together with solutions. �np�b`!#Hw�4 +�Bp��3�~~xNX\�7�#R|פ�U|o�N��6� H��w���1� _*`�B #����d���2I��^A�T6�n�l2�hu��Q 6(��"?�7�0�՝�L���U�l��7��!��@�m��Bph����� If each number is greater than \(\text{0}\), find the numbers that make this product a maximum. 13. \end{align*}, We also know that acceleration is the rate of change of velocity. 750 & = x^2h \\ Notice that the sign of the velocity is negative which means that the ball is moving downward (a positive velocity is used for upwards motion). Calculus is one of the central branches of mathematics and was developed from algebra and geometry. Continuity and Differentiability. Determine the dimensions of the container so that the area of the cardboard used is minimised. 3978 | 12 | 1. We should still consider it a function. \begin{align*} GAUTENG DEPARTMENT OF EDUCATION SENIOR SECONDARY INTERVENTION PROGRAMME MATHEMATICS GRADE 12 … Michael has only \(\text{160}\text{ m}\) of fencing, so he decides to use a wall as one border of the vegetable garden. s &=\frac{1}{2}t^{3} - 2t \\ Therefore, acceleration is the derivative of velocity. The volume of the water is controlled by the pump and is given by the formula: 12. Revision Video . t &= 4 Navigation. Grade 12 introduction to calculus (45S) [electronic resource] : a course for independent study—Field validation version ISBN: 978-0-7711-5972-5 1. It is used for Portfolio Optimization i.e., how to choose the best stocks. D(t)&=1 + 18t - 3t^{2} \\ Application on area, volume and perimeter 1. SESSION TOPIC PAGE . 1. some of the more challenging questions for example question number 12 in Section A: Student Activity 1. Chapter 6. \text{Rate of change }&= V'(d) \\ to personalise content to better meet the needs of our users. These concepts are also referred to as the average rate of change and the instantaneous rate of change. The coefficient is negative and therefore the function must have a maximum value. \end{align*}. Chapter 1. Grade 12 | Learn Xtra Lessons. Make \(b\) the subject of equation (\(\text{1}\)) and substitute into equation (\(\text{2}\)): We find the value of \(a\) which makes \(P\) a maximum: Substitute into the equation (\(\text{1}\)) to solve for \(b\): We check that the point \(\left(\frac{10}{3};\frac{20}{3}\right)\) is a local maximum by showing that \({P}''\left(\frac{10}{3}\right) < 0\): The product is maximised when the two numbers are \(\frac{10}{3}\) and \(\frac{20}{3}\). &\approx \text{7,9}\text{ cm} \\ Graphs give a visual representation of the rate at which the function values change as the independent (input) variable changes. t&= \text{ time elapsed (in seconds)} &= -\text{4}\text{ kℓ per day} Determine the velocity of the ball after \(\text{1,5}\) \(\text{s}\). Therefore, \(x=\frac{20}{3}\) and \(y=20-\frac{20}{3} = \frac{40}{3}\). The sum of two positive numbers is \(\text{20}\). \text{Acceleration }&= D''(t) \\ t&=\frac{-18\pm\sqrt{336}}{-6} \\ We start by finding the surface area of the prism: Find the value of \(x\) for which the block will have a maximum volume. 14. Test yourself and learn more on Siyavula Practice. Fanny Burney. \text{Average velocity } &= \text{Average rate of change } \\ Xtra Gr 12 Maths: In this lesson on Calculus Applications we focus on tangents to a curve, remainder and factor theorem, sketching a cubic function as well as graph interpretation. In the first minute of its journey, i.e. If the length of the sides of the base is \(x\) cm, show that the total area of the cardboard needed for one container is given by: Siyavula's open Mathematics Grade 12 textbook, chapter 6 on Differential calculus covering Applications of differential calculus \end{align*}. Homework. In other words, determine the speed of the car which uses the least amount of fuel. \end{align*}, \begin{align*} To find the optimised solution we need to determine the derivative and we only know how to differentiate with respect to one variable (more complex rules for differentiation are studied at university level). @o����wx�TX+4�`���w=m�p1z%�>���cB�{���sb�e��)Mߺ�c�:�t���9ٵO��J��n"�~;JH�SU-����2�N�Jo/�S�LxDV���AM�+��Z����*T�js�i�v���iJ�+j ���k@SiJؚ�z�纆�T"�a`�x@PK[���3�$vdc��X��'ܮ4�� ��|T�2�ow��kQ�(����P������8���j�!y�/;�>$U�gӮ����-�3�/o�[&T�. Fanny Burney. It is very useful to determine how fast (the rate at which) things are changing. Additional topics that are BC topics are found in paragraphs marked with a plus sign (+) or an asterisk (*). \therefore h & = \frac{750}{(\text{7,9})^2}\\ D'(0) =18 - 6(0) &=\text{18}\text{ m.s$^{-1}$} a &= 3t 9. We have seen that the coordinates of the turning point can be calculated by differentiating the function and finding the \(x\)-coordinate (speed in the case of the example) for which the derivative is \(\text{0}\). <> University Level Books 12th edition, math books, University books Post navigation. We can check this by drawing the graph or by substituting in the values for \(t\) into the original equation. \end{align*} \end{align*}. \end{align*}. \therefore x &= \sqrt[3]{500} \\ Embedded videos, simulations and presentations from external sources are not necessarily covered Calculate the dimensions of a rectangle with a perimeter of 312 m for which the area, V, is at a maximum. PreCalculus 12‎ > ‎ PreCalc 12 Notes. Handouts. \begin{align*} Therefore the two numbers are \(\frac{20}{3}\) and \(\frac{40}{3}\) (approximating to the nearest integer gives \(\text{7}\) and \(\text{13}\)).

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