Modeling with non-linear functions STEM 8

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Wilfred Dixon
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1 Time Time Math Learning Center Modeling with non-linear functions STEM 8 Radicals appear in many situations in science including: Laws of Motion, Hookes Law, pendulum oscillation, Series RLC circuits (electronics), Maxwell-Boltzman distribution, and kinetic energy equations. Consider an object at rest. If we propel the object with a constant acceleration of and measure the time it takes to move in intervals of four meters, the collection of the first nine data points would create a graph that looks like: Distance Looking closely at that graph we find that initially there is a large increase in time as we increase the distance. However, as distance increases, a change in distance has less of an effect on the time. If we draw a curve through the points, we obtain the following: Distance Looking at this graph, we should be able to recognize it as being similar to a square root graph. In fact, when we mathematically compare time to both distance traveled and acceleration, we find that the points observed follow the equation:

2 Consider the Maxwell-Boltzman distribution in chemistry. One part of this relationship gives the average velocity,, of gas molecules as the following relationship: In our definitions of the variables we have used Standard International (SI) units of measure. where ( ) J = Joules (energy) K = Temperature in Kelvins g = grams mol = molecular weight These terms will be defined in chemistry. While they will be ignored here, please allow this introduction to impart the importance of these measurements and remember that they should not be ignored in chemistry! Letting and our equation now will relate average velocity, to temperature, ( ) Complete the following t-chart, graph the points and draw a smooth curve to allow for interpolation of the data. T u

3 This next problem will expand upon our first problem relating time and distance. In the sciences, the formula given is not always the formula of interest. In the first problem we used the formula: A more complete form of the same equation contains the initial position, and the initial velocity, With these two additional values, we can give the general form of the equation as: Verify that the two equations are related. Let the initial position be zero, ( be zero, ( ) and solve the resulting equation for. ) and the initial velocity At this time we will look at radicals through a reintroduction of the Quadratic Formula: The function of the quadratic formula is to find the x-intercepts of the quadratic equation which is given in generic form as: Since we are looking for the x-intercept, we let and have: If we desire to find the times that the motion formula indicates that the moving body passes through zero, we use the motion formula: And set the position equal to zero,

4 At this point we need to examine the differences between mathematics and physics. In math, students are told that the best way to write equations is to begin with the term with the highest degree first and then proceed in descending order. Thus when moving from the physics equation to the quadratic equation we make the following substitutions: At this time we recognize a major difficulty that students have when working mathematics in other fields such as physics is caused by each subject using different notation. For example, in the statement, we are using the arrow to say that one-half in physics is related to in math and not that one-half equals. In the physics equation, represents the acceleration and in the quadratic formula, is just a place holder for the leading coefficient. At this time we move from the need to understand different types of notation for variables and to the need to be able to use context to understand what each variable means. This is the same as being able to recognize the usage of words (such as to, too and two) based on context. Thus, with effort, this can be mastered. Example 1. Given a moving object that starts at an initial position, has an initial velocity of and an acceleration, find the time the object will return to the starting position. First we need to find the physics equation: Next we place these numbers into the quadratic formula (It is left to the reader to find the solutions.) Since the implied domain (feasible region) for time is usually only considered to be such that can ignore any negative answer. In the quadratic formula we us the symbol, read plusminus. This symbol implies that we need to compute two answers one with addition and one with subtraction being used at the symbol., we

5 Practice problems: 1) Given a moving object that starts at an initial position, has an initial velocity of and an acceleration, find the time the object will return to the starting position. 2) Given a moving object that starts at an initial position, has an initial velocity of and an acceleration, find the time the object will return to the starting position. 3) Given a moving object that starts at an initial position, has an initial velocity of and an acceleration, find the time the object will return to the starting position. Looking at example 1 again, we can use the quadratic formula to answer the questions: When will the position be at 100 meters? ( ). Example 1 (varied). Given a moving object that starts at an initial position, has an initial velocity of and an acceleration, find the time the object will be at 100 meters. The physics equation becomes : To use the quadratic equation, we need to set this equal to zero by subtracting 100 form each side: Complete this problem using the quadratic formula and again allow the implied domain (feasible region) to be

6 Practice problems: 4) Given a moving object that starts at an initial position, has an initial velocity of and an acceleration, find the time the object be at. 5) Given a moving object that starts at an initial position, has an initial velocity of and an acceleration, find the time the object will be at.

Topics Covered in Math 115

Topics Covered in Math 115 Basic Concepts Integer Exponents Use bases and exponents. Evaluate exponential expressions. Apply the product, quotient, and power rules. Polynomial Expressions Perform addition