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Mathematical Models

Transcript: joules (J) but not represented as a collective unit (only done) when energy transfers to an object and the body changes in the position/motion in the SAME direction of the applied force Formula Formula Power Formula *the useful work output is never equivalent or exceeding the work input power = work/time (P = W/t) Gravitational Potential Energy Equation output force = 140 N input force = 140 N mass = 9 kg height = 15 m g = 9.8 m/s² PE = 9 kg x 15 m x 9/8 m/s² PE = 1323 J the ratio of useful work output to total work input Kinetic Energy the number that describes how much force or distance is multiplied by a machine; the ratio between the output and input force efficiency = useful work output/work input work = force x distance W = F x d Mathematical Models energy an object possesses because of its position in a gravitational field Measurement grav. PE = mass x free-fall acceleration x height PE = mgh Measurement usually expressed as a percentage (%) by multiplying the useful work output/work input by 100 Mechanical Advantage *free-fall acceleration (g) = 9.8 m/s² work, W = 800 J time, t = 40 s has no SI unit any machine with a mechanical advantage greater one multiplies a force (locate quanities measured in joules in a problem) Measurement Measurement Efficiency kinetic energy = 1/2 x mass x speed squared KE = 1/2mv² Work Formula *mechanical advantage is leverage Measurement Mechanical Advantage Equation *a rope looped through a pulley attached to a fixed spot and attached to the weight a single fixed pulley a mechanical advantage of 1, meaning no mechanical advantage (or disadvantage) however advantageous the change in direction may be. power = work/time P = W/t Formula mechanical advantage = output force/input force ignoring friction, mechanical advantage is also equivalent to input distance/output distance Formula watts (W) is not italicized, as work (W) is Measurement the energy of a moving object due to the motion, depending on mass and speed Work Equation Gravitational Potential Energy joules (newtons [force] times meters [distance]) joules (J) are also equal to 1 kg x m²/s² P = 800 J/40 s P = 20 W mechanical advantage (MA) = 140 N/140 N mechanical advantage = 1 Power Equation ex. KE = 1/2(44 kg)(31 m/s)² = 2.1 x 10⁴ kg x m²/s² simplified to where 2.1 x 10⁴ J is the answer watts [joules per second] joules (J) because it is relative to the work equation, as the it is a calculation of force times distance Works Cited force, F = 25 N distance, d = 14 m work = force x distance (W = Fd) W = 25 N x 14 m Work = 350 J 1 N x m = 1 J = 1 kg x m²/s² http://easycalculation.com/physics/classical-physics/learn-potential.php http://physicshelp.co.uk/motion.php http://www.thefreedictionary.com/ the rate at which work is done or energy is transformed in a given amount of time

Mathematical Models

Transcript: A hardware store makes a profit of $1.38 for every hasp sold. Write an equation that describes this relationship. Example: P(x)=1.38x models the profit on selling x such hasps. For example, if the data appears to be periodic, what type of model would we use? Success fitting models to given problems takes practice and some insight into how variables and data values are related. A tenet pays a non-refundable $150 deposit upon leasing an apartment and $450 per month. Rent is paid in advance. A linear model for the total amount to the landlord after x months is: Example: C(x)=150+450x A trigonometric function would likely be involved. Example: A mathematical model is an abstraction of a real world phenomenon into symbols and equations. Our models relate two variables with a function. For example C=(5/9)(F-32) models the relationship between temperature measured in degrees Fahrenheit (F) and degrees Celsius (C). Mathematical Models: An exponential function or polynomial function of a high degree. Example: What about when something grows very rapidly? Mathematical Models AP Calculus 1-2

Mathematical Models

Transcript: Cellphone Comparison Postpaid and Prepaid Specifications and Formulation of Equations Postpaid Plan: Price of iPhone 5: P 2,000 Calls per Minute: P 7.50 Calls per Month (Minutes): 210 2nd Equation: y = 1575x + 2,000 210 + 7.50 ----------- P 1575 A new model of iPhone just got out and you want to have one. You have two options, buying the phone with full payment on a retail store and get a fixed prepaid plan, or buying it bundled with a plan locked for 2 years from Smart. You need to determine if you will save money from buying the phone from a retail store or buying it from Smart. Reflections: Intersection Point: (70,000,180) Conclusion: The Postpaid and Prepaid Lines will intersect at P70,000 at 180 days or 6 months. So, we have concluded that the best plan for our new phone is the Postpaid Plan because we would save more if we take that plan. Also, we will save more with this plan. GRAPHING: Is there anything you would have done differently? A: No, because we all have agreed on the same product. Why was the use of a system of linear equations necessary for this problem? A: To make it easier for us to determine which type of cellphone cost more. EQUATIONS: 1. y = 975x + 32,000 -----> 1575x - y = 2,000 2. y = 1575x + 2,000 -----> 975x - y = 32,000 600x = 30,000 x = 50 975x - y = 32,000 975(50) - y = 32,000 x = 50 48,750 - y = 32,000 y = 16,800 -y = 31,950 - 48,750 - y = -16,800 y = 16,800 What other costs will come into play if you were really planning to buy that item? A:The taxes for the text and the calls. Also the SIM card that you are going to use. Patrick Fernandez Emmanuele Murillo Jose Angelo Sibug Marcus Cortes Situation: Prepaid Plan: Retail Price of iPhone 5: P 32,000 Calls per Minute: P 7.50 Calls per Month (Minutes): 130 1st Equation: y = 975x + 32,000 130 + 7.50 ----------- P 975 (0,2,000) Members: Mathematical Models (1, 32,975) (0, 32,000) (1, 1575) What factors would influence your decision? A: The amount of the cellphone mostly and how much do we need to pay for taxes. What is your favorite thing about your presentation? A: Making this in Prezi format.

Mathematical Models

Transcript: Geometric diagram * To display an abstract (non-tangible) idea * Allow for insight and understanding of an abstract idea * Assist in making predictions of the phenomenon My definition A mathematical model is an equation, table, geometric diagram or graph in the coordinate plane that conveys or displays a relationship among two or more variables. Types of Models Types of Models Graph in the coordinate plane Equation For example: The probability of rain Purpose of the Model 1. Visual representations make the abstract ideas easier to understand. 2. The students learn there is more than one way to represent the idea. 3. Minimizes technical terminology that may be too complicated 4. Allows for discussion that utilizes the scientific method way of thinking. Table How models promote mathematical thinking Mathematical Models By M.Michelle Madison

Mathematical Models

Transcript: Mathematical Models Comparing and Contrasting Table of Values Patterns that show Difference of linear V.S quadratic Finite differences Table of Values Patterns That Show in Table of Values Patterns To correctly determine the pattern, the x value needs to be increasing/decreasing by a consistent value. When we increase the x-value at equal steps, the y values increase by a constant amount each time. The x value is the number inputted in an equation, the y value is the number that results from the equation. Differences of Linear V.S. Quadratic Linear V.S Quadratic The equation of a linear relation has the form: The general equation of a quadratic relation has the form: The finite differences are how we determine if a table of values represents linear, quadratic, or neither. Linear relation: creates a straight line when graphed Quadratic relation: creates a parabola when graphed Finite Differences Differences One way we can check if it is a linear or quadratic equation is to calculate the finite differences. The finite differences are the differences calculated between y values in the table if x values are evenly spaced. If all first differences are equal, the table is linear! The first differences are calculated by subtracting consecutive y values. The second differences are calculated by subtracting consecutive first differences. If all first differences are not equal but the second differences are, the table is quadratic! linear Material + Equipment Shape of the graph Key features Their relation to the equation Graphs Shape of the Graph Shape Linear V.S. Quadratic A linear equation will create a straight line on a graph A quadratic equation will create a parabola on a graph Key Features of a Graph Linear V.S. Quadratic Key Features y and x are coordinates of a point on the line. (x,y) The slope of the line is the steepness. (rise/run) A parabola is symmetrical, the vertex is drawn through the axis of symmetry. The step pattern tells us how much the steps increase every time we move from the vertex. The y-intercept is the point where the line crosses the y axis, x-intercept is where the line crosses the x-axis. A graph will have a minimum or maximum: maximum opens down, minimum opens up. How Linear and Quadratic Relate to Their Equation Relation The value of M is the slope and it controls the steepness (steeper when number is higher). B is the y-intercept and controls how far up the line is on the graph. X is the x-intercept and controls how far horizontally the line is. Y is the output (a point on the graph). H is the Axis of Symmetry and it represents the point on the x-axis that is the center of the parabola. K is the Max/Min value and controls how high the parabola is on the graph. A is the multiplier and the step pattern. This controls the direction of opening (negative: down, positive: up). X and Y are points on the graph. Linear Quadratic Equations Recognizing different relations General equation What different parts of the equation mean Equations Recognizing Recognizing An equation with the form Y=MX+B, Y=MX, Y=X+b, Y=X will represent a linear equation in vertex form. An equation with the from Y=A(X-H)^2+B, Y=(X-H)^2+b, Y=X^2 will represent a quadratic equation in vertex form. General Equation General Equation Quadratic Linear Equation Broken Down Breakdown Linear Quadratic Just as linear, y and x are variables. Different Forms Equations of Linear and quadratic equations What information can be read How can we switch between forms Linear Linear Two forms of linear equations we learned are standard form and vertex form. Standard form is most commonly used when trying to find the x and y intercepts. Vertex form is most commonly used when trying to find the vertex. To turn vertex form to standard form, move MX to the left side of the equation, isolating the y-intercept on one side of the equation. Quadratic Quadratic Vertex form is most commonly used when trying to find the vertex of a parabola. Factored form is most commonly used when trying to find the roots of a parabola. To change vertex or factored form into standard form: expand and simplify To change standard form into factored form: factor the side that does not contain y To change standard form into vertex form: find vertex from x-intercepts Standard form is most commonly used when trying to find the y-intercept of a parabola and is easy to switch to different forms. Direction of opening can be found from any form using the a value Solving Strategies Methods of solving How to solve Solving Methods of Solving Methods There is 3 methods for solving a linear equation: Solve by graphing Solve by substitution Solve by elimination There is 4 methods for solving a quadratic equation: Solve by graphing Solve by factoring Solve by quadratic formula Solve by completing the square Linear Quadratic How To Solve Elimination: cancel out one of the variables, add or subtract to eliminate variable, solve with remaining, substitute the new variable to solve for

Mathematical Models

Transcript: Gr. 11 College Bound Math Mathematical Models Standard Form QUADRATICS Vertex Form Factored Form Changing to This Form factored form y= 2(x+10)(x+2) standard form 2 y=2x + 24x +40 VERTEX FORM 2 y=2(x+ 6) - 32 How can we graph using standard form? Standard form is limited into identifying key features from the equation. Graphing Sub in x values and calculate corresponding y TO FIND POINTS TO GRAPH plot points and connect the dots! Example Graph Notebook pick x values to sub in (usually pick from -2 to 2 to get range of negatives and positives as well as 0! Let x =0 , y= 2(0)^2 - 5(0) +3 y= 0 + 0 + 3 y= 3 let x=1, ..... found x-intercept when subbed in 1, y ended up being 0 found y-intercept by subbing in 0 Key Features KEY FEATURES : y-intercept ideal is to find when y-values equal 0 (your zeros) play with x values y-intercept is when x=0. 2 y= a(0) + b(0) + c y= 0 + 0 + c y= c (0,c) the c , or constant value, at the end of the equation is the y-int. try to find 2 symmetrical points in the table ideal is to find your y-values change direction (from low values to high then back to low again) why? The vertex is found when the direction changes! Key Features KEY FEATURES : y-intercept ideal is to find when y-values equal 0 (your zeros) play with x values y-intercept is when x=0. 2 y= a(0) + b(0) + c y= 0 + 0 + c y= c (0,c) the c , or constant value, at the end of the equation is the y-int. try to find 2 symmetrical points in the table ideal is to find your y-values change direction (from low values to high then back to low again) why? The vertex is found when the direction changes! Changing to This Form y= 2(x+10)(x+2) factored form a=2 and x-intercepts are x= -10 and x= -2 h = -10 + (-2) / 2 h= -12/ 2 h= -6 VERTEX FORM let x= -6 in the equation and solve for y y= 2(-6 + 10)(-6 + 2) y= 2(4)(-4) y= 32 2 standard form y=2(x+ 6) - 32 2 2 y=2x + 24x +40 y=2(x+ 6) - 32 need to change to factored first then vertex form there is a way! it is called completing the square (different course material) TRICK: h= -b/2a sub in h for k TO FIND POINTS It is called vertex form because the vertex is in the equation! (h,k) is the vertex. TO GRAPH How can we graph using vertex form? Vertex form shows the transformations done to y=x^2. You can also use the step pattern to find points! Graphing Plot the vertex the and use the steps 1a,3a,5a,.. transform y=x by using the 5 key points and move them depending on a,h,k remember symmetry! Example ex. 1. Graph Notebook ex. 2 (x,y) (x-6, -4y-8) vertex: (h,k) so (-6,-8) transformations: reflection: yes, a is neg. vertical stretch: a=-4 so stretch by factor of 4 hor. translation: left 6 ver. tranlsation: down 8 move all x values left 6, multiply each y value by 4, make it negative, then down 8 Key Features KEY FEATURES : VERTEX Step Pattern: points move from vertex in a pattern over 1, up 1, over 1, up 3, over 1, up 5 if a value, then stretches/compresses each step to 1a,3a,5a,... h value is opposite sign you see! 2 (h,k) Direction of Opening if a>0 then opens up if a<0 then opens down can find the y-intercept by subbing in x=0 and solving for y in eqn: y=a(x-h) + k so always has a minus h 2 VERTEX FORM y=2(x+ 6) - 32 2 Changing to This Form a=2 find the zeros (y=0) 0=2(x+ 6) - 32 32=2(x+6) factored form 32/2=(x+6) standard form y= 2(x+10)(x+2) y=2x + 24x +40 2 2 sq root (16) =x+6 x= -6 + 4 or x=-6-4 x=-2 and x=-10 so factors are x+2 and x+10 divide each term by 2 to factor it out! 1. common factor 2. factor the trinomial __ + __ = 12 __ x __ = 20 y= 2(x+10)(x+2) y=2(x + 12x +20) y=2(x+10)(x+2) 20 1 20 2 10 3 4 5 How can we graph from factored form The easiest form to plot from is factored form. Graphing TO FIND POINTS find your zeros and plot them find the middle and plot the AOS and then calcuate the vertex to plot sub in 0 to find y-int then can use symmetry to find last point TO GRAPH Example Notebook Graph y= (x-4)(x+10) The solutions for when y=0 (x-intercepts) are x=4 and x=-10 Key Features KEY FEATURES : ZEROS y-interept can be calculated by subbing in x=0 and solving! factored form is multiplying terms together. Anything times 0 is 0 therefore if one bracket equals 0, then y=0 and makes it an x-intercept! (0,s) and (0,t) when y=a(x-s)(x-t) the AOS is found in the middle of 2 symmetrical points and the zeros are 2 points we can use to find it! h is the AOS value. sub in x=h into equation and solve for y =k, vertex at (h,k) h= s + t / 2 What are They? EXPONENTS equations where the exponent is the number changing! The Rules Exponent Rules The Graphs y=ab x The 3 forms -connected! Growth and Decay Title plot the points from the table and connect the dots! TABLE GRAPH EQUATION The variable is in the exponent spot! plug in x values (a few positive and a few negative ) and calculate the corresponding y values Title decay example The Equations Population Example Double Time Half-Life

Mathematical Models

Transcript: This means that the image of x through the function f is y. x is the independent variable and y is the dependent variable. Domain and Range of a function The second set is called the Range of the function. The elements of the range are often thought of as the y values (outputs), they are the dependent variables. Definition: A function is the relationship between 2 sets: a first set and a second set. Each element "x" of the first set is related to one and only one element "y" of the second set. Functions Mathematical Models y=ax2 + bx + c Function notation sketching quadratic graphs The Quadratic Formula uses the "a", "b", and "c" from "ax2 + bx + c", where "a", "b", and "c" are just numbers; they are the "numerical coefficients" of the quadratic equation they've given you to solve. f(x)=y The first set is called the domain of the function. The elements of the domain often thought of as the x values are the independent variables (input). Quadratics

Mathematical models

Transcript: Low level of drug allows duplication and growth The relationship High levels increase chances of virus mutation Easier to develop new combination of drugs. Cheaper and better drugs Prescription periods are VERY important Results Application LISTEN TO THE DOCTOR! by Bang P Wide range of data ensures higher accuracy than other lab results Following prescriptions perfectly does not ensure viruses won't develop resistance (other natural factors in body) Here are some extra assets : Inverse relationship? Mutation strains of HIV virus. Drug concentration throughout body Saves money and time conducting lab experiments, removes trial-and-error process Provide accurate measurements of HIV virus' ability to develop resistance Why use model? Led by Martin Nowak, Professor of Mathematics and of Biology and Director of the Evolutionary Dynamics Program. Virus resistance model Harvard's research Commitment to doctor's prescription vs Virus' ability to develop resistance Future improvements Mathematical model to reduce chances of virus developing resistance No symptoms does not mean no viruses How model was made Output: Chances of virus developing resistance Input: Patients' characteristics Adherence level (0-100%) Drugs present in compound Concentrations of drugs present Model's precision depend on amount of data collected. Johns Hopkins University Medical School was studying HIV virus' reaction to drug doses and provided good amount of sample Output: Simulated patients Virus' growth/decay rate Chances of virus developing resistance add logo here

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