qr code vb.net open source We can take the first parametric equation x = 3t in Visual Studio .NET

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We can take the first parametric equation x = 3t
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Part Two 421
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y 6 4 t = 2 t = 1 2 2 6 4 t=0 4 6 2 2 t=1 t=2 4 6 x
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Figure 20-9
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Illustration for Question and Answer 16-2
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and multiply it through by 1/3 to get ( 1/3)x = ( 1/3)(3t) = t Deleting the middle portion, we get ( 1/3)x = t The second parametric equation tells us that t = y, so we can substitute directly in the above equation to obtain ( 1/3)x = y which is identical to the following slope-intercept equation: y = ( 1/3)x
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Question 16-4
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Consider the pair of parametric equations q = 3t
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and r = t where t is the parameter on which both q and r depend How can we sketch a polar graph of this system
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Answer 16-4
We can input various values of t, restricting ourselves to values such that we see only the part of the graph corresponding to the first full counterclockwise rotation of the direction angle, where 0 q 2p: When t = 0, we have q = 3 0 = 0 and r = 1 0 = 0 When t = p /4, we have q = 3p /4 and r = p /4 079 When t = p /2, we have q = 3p /2 and r = p /2 157 When t = 2p /3, we have q = 3 2p /3 = 2p and r = 2p /3 209
Figure 20-10 illustrates this graph, based on these four points and the intuitive knowledge that the graph must be a spiral, starting at the origin and expanding as we rotate counterclockwise The graph is a little tricky, because all of the radii are negative! Also, we should remember that the concentric circles represent radial divisions on the polar coordinate grid; the straight lines represent angular divisions
Figure 20-10
Illustration for Question and Answer 16-4 In this coordinate system, each radial division represents p /4 units
Part Two 423 Question 16-5
Consider again the pair of parametric equations q = 3t and r = t How can we use algebra to determine the equivalent equation in terms of q and r only
Answer 16-5
The equation can be derived using the same algebraic process that we used in the Cartesian situation We substitute q in place of x, and we substitute r in place of y When we do that, we get r = ( 1/3)q
Question 16-6
What are the parametric equations for a circle centered at the origin in the Cartesian xy plane
Answer 16-6
The parametric equations are x = a cos t and y = a sin t where a is the radius of the circle and t is the parameter
Question 16-7
What are the parametric equations for a circle centered at the origin in the polar coordinate plane
Answer 16-7
Let the polar direction angle be q, and let the polar radius be r The parametric equations of a circle having radius a, and centered at the origin, are q=t and r=a where t is the parameter
Review Questions and Answers Question 16-8
Why does only one of the equations in Answer 16-7 contain the parameter t Shouldn t both equations contain it
Answer 16-8
The parameter t has no effect in the second equation, because the polar radius r of a circle centered at the origin is always the same, no matter how anything else varies
Question 16-9
What are the parametric equations for an ellipse centered at the origin in the Cartesian xy plane
Answer 16-9
The parametric equations are x = a cos t and y = b sin t where a is the length of the horizontal (x-coordinate) semi-axis, b is the length of the vertical (y-coordinate) semi-axis, and t is the parameter
Question 16-10
Why is the passage of time a common parameter in science and engineering
Answer 16-10
In the physical world, many effects and phenomena depend on elapsed time If we find time acting as a mathematical variable, then that variable is almost always independent We often come across situations where two or more factors fluctuate with the passage of time An example is the variation of temperature, humidity, and barometric pressure versus time in a specific location In a situation of this sort, time can be considered as the parameter on which the other three physical variables depend
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