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MAT 221 Lecture Notes - Lecture 12: Dependent And Independent Variables, Scatter Plot

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12 Oct 2017

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MAT 221 Full Course Notes

MAT 221 Full Course Notes

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In a positive relationship, greater values of one variable tend to occur with greater. The relationship between house size and house price. The relationship between income and work experience. In a negative relationship, greater values of one variable tend to occur with smaller variables. The relationship between the weight of cars and fuel efficiency. The relationship between the number of classes missed and exam score. The relationship between the price of textbooks and number of pages is somewhat strong. positive and pretty weak. very weak. Sometimes one variable can explain or causes changes in another variable. in that case, Sometimes there is no need or not possible to decide which variable is independent. If there is an explanatory variable, always plot it on the horizontal axis (x-axis) of the scatterplot, and plot the response variable on the vertical axis (y-axis) Is a data value that has a very low probability of occurrence (i. e. , it is unusual or unexpected).

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Guided Project 66: Traveling waves Topics and skills: Functions of two variables, partial derivative, graphing The word wave elicits images of water waves rolling in beach. However, wave is used to describe any disturbance that propagates through a medium. So the word includes water waves, sound waves, electromagnetic waves (light waves and cell phone signals), seismic waves, and waves in traffic on a crowded highway, In this project, we examine some of the fundamental mathematical properties of waves. 1. Consider the function z= f(x, t) = 3 sin (pi x- 2 pi t). It is easy to graph this function (Figure 1) and indeed we waves. But this graph really doesn't explain how and why waves propagate. To interpret the graph suppose that x and t represent distance (measured in meters) and time (measured in seconds), respectively. Specifically, the x-axis is the direction in which the wave propagates and z represents the height of the surface of the wave at a particular location and time. The maximum values of z correspond to crests of the wave and the minimum values of z correspond to troughs of the wave. What are the minimum and maximum values of z? 2. Now imagine that you are sitting on the Jt-axis, say at x = 2, watching the wave pass. Set x = 2 in z = f(x, t) and graph z = z(2, t) on the interval 0 t 2. Explain how z varies on 0 t 2. How many times do you observe a crest of the wave pass? The time that elapses between passing crests is the period of the wave. How many wave crests pass per second? The number of wave crests that pass per second passes is the frequency of the wave, usually measured in cycles per second. Instead of freezing x as we did in Step 2, we can also freeze t. Suppose you take a snapshot of the wave at t = 1. Set t = 1 in z = f(x, t) and graph z =f(x, 1) on the interval 0 x 4. Explain how z varies on 0 x 4. What is the distance between crests? The distance between crests is called the wavelength of the wave. How far does the crest of the wave travel in one second of time? What is the speed of the wave?

Page 3 of 5 which gives us y as a function of Theoretically. the integration on the right of t= 1/ - g + alpha f(V)dv can be performed and the result could provide velocity. v,. as a function of t; Then we could perform the integration in y=y0 + V/- g + alpha f(V)dv and substitute v(t) to obtain y as a function of t. Theoretically, that is. Let's look at in example An Example Suppose the air resistance model for a particular object in free fall has where units for v area feet/second Note how f is defined in terms of the sign of v. As usual. downward has been selected as the negative direction so velocities v of an object in free fall will have negative values Thus -v will be positive and the square root makes. sense Similarly if the object is moving upward, v> 0. the value of f(v)) is negative. The object is dropped from a height: of 500 feet with no initial velocity Using g = 32, the integral models from above becomes and where we have used the definition of f(v) for negative values of v since the object is in free fall With a little hard work or the use of a CAS, the first equation can be written as Page 4 of 5 and the second as Now if we could solve the first expression for v in terms of r and substitute the result for v in the expression for y, the result would be very unwieldy to work with. Actually it is impossible to solve the first expression to get a nice closed form for v. Thanks to technology. however, these expressions can still be used to study the object's motion Choose any value of v. say v=-1. Substituting -1 for v in the equations for t and y tells us that t = 0.032 and y = 499.984 so the point (0.032, 499.984) would lie on a graph of distance versus time Similarly, setting v = -50 provides the point (1 839, 452 462) By using a wide range of values for v. we can generate the complete graph of the distance function as a function of time Fortunately, most CAS provides the capability of plotting the points (t(v), y(v)) foe a range of values of v without having to compute each point separately Such a graph appears below. One final note. It is also possible that the integrals representing t and y cannot be integrated into a closed form No problem In this case, simply use the expressions t= 1/ - g + alpha f(V)dV y=y0+ V/ - g + alpha f(V)dV Page 5 of 5 and the numerical integrators provided by a CAS to determine values of t und y Exercises In all of the exercises you will need a CAS or related tool In the example above the graph indicates the object strikes the ground some time between 6 and 7 seconds from release Determine this time to three decimal places In the case where alpha = 1/2 ,f(v) = -v, v0 = 0, y0=100 determine closed form. expressions for v and y as function. Of t. An object is thrown upward at 40 ft/sec from a height of 100 feet The air resistance can be modeled by alpha =0.001 and f(v) = - |v| v Using the trajectory model determine the speed of the object when it strike the ground, determine the time the object strikes the ground after takeoff, construct the graph of the object's height versus time.