Problem 35 Determine how the plane curves d... [FREE SOLUTION]

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Algebra and Trigonometry Real Mathematics, Real People

Ron Larson

$Math Studyset 91影视 Explanations$ Math

7 Edition

Chapter 10: Problem 35

Determine how the plane curves differ from each other. (a) $x=t$ $y=2 t+1$ (b) $x=\cos \theta$ $y=2 \cos \theta+1$ (c) $x=e^{-t}$ $y=2 e^{-t}+1$ (d) $x=e^{t}$ $y=2 e^{t}+1$

Short Answer

Expert verified

Curve (a) is a straight line with a positive linear relationship between x and y. Curve (b) is a circle centered at (0,1). Curve (c) is a declining exponential graph with y-values always larger than the x-values. Curve (d) is a rising exponential graph with both x and y increasing as t increases and y-values larger than the x-values.

Step by step solution

Plot curve (a)

We're given $x = t$ and $y = 2t + 1$. This describes a straight line. If a graph of this curve is plotted, we can see that it slopes upward, indicating a positive linear relationship between x and y.

Plot curve (b)

We're given $x = \cos\theta$ and $y = 2\cos\theta + 1$. These equations make a circle on a plane about the origin. When plotted it gives a circle of radius 1 centered at (0,1).

Plot curve (c)

The equations for curve c are $x = e^{-t}$ and $y = 2e^{-t} + 1$. This will give a declining exponential curve in a plane. The y-values will be always larger than the x-values.

Plot curve (d)

Finally, for curve d, we have $x = e^{t}$ and $y = 2e^{t} + 1$. This will represent a rising exponential curve. The y-values will always be larger than the x-values but in this case, unlike graph c, both x and y increase as t increases.

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91影视

Determine how the plane curves differ from each other. (a) \(x=t\) \(y=2 t+1\) (b) \(x=\cos \theta\) \(y=2 \cos \theta+1\) (c) \(x=e^{-t}\) \(y=2 e^{-t}+1\) (d) \(x=e^{t}\) \(y=2 e^{t}+1\)

Short Answer

Step by step solution

Plot curve (a)

Plot curve (b)

Plot curve (c)

Plot curve (d)

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