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Chapter 10: Problem 49

Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest. $$\text { Spiral } x=t \cos t, y=t \sin t ; t \geq 0$$

Short Answer

Expert verified
Question: Describe the appearance of the spiral curve represented by the parametric equations \(x(t) = t\cos t\) and \(y(t) = t\sin t\), and state any notable features. Answer: The spiral curve begins at the origin and expands in a counterclockwise direction with each loop gradually increasing in size. Notable features include the number of loops, which continues to increase as the value of \(t\) increases, and the rate of growth, which causes the distance between each loop to expand.

Step by step solution

01

Choose a graphing utility

You can use any graphing utility such as Desmos, Geogebra, or TI calculator to plot the parametric equations. In this case, let's use Desmos, a free online graphing tool.
02

Set up the parametric equations

Go to desmos.com/calculator and select "Start Graphing". In the left pane, click on the "+" button to add a new expression. Choose a "Table" and then select "Parametric" mode. You should now see two boxes, where you can enter the expressions for \(x(t)\) and \(y(t)\). Enter \(t\cos t\) for the \(x(t)\) expression, and \(t\sin t\) for the \(y(t)\) expression. Leave the domain of \(t\) as the default \(0\) to \(1\) for now.
03

Choose an appropriate interval for \(t\)

Since we want to observe all interesting features of the spiral, start by setting the domain of \(t\) from \(0\) to \(10\) and take a look at the graph. If necessary, adjust the maximum value of \(t\) until you can see the features of the spiral clearly. For this spiral, a domain of \(t\) from \(0\) to \(20\) should be a good starting point.
04

Fine-tuning the graph

To get a smoother curve, increase the number of sample points for the parameter \(t\). You can do this by clicking on the gear icon in the "Table" settings and increasing the "Step" value. For our purposes, a step value of \(0.1\) should provide a smooth curve. After this, you should see the spiral curve with all its features.
05

Analyze the graph

Observe the graph and take note of any interesting features you may see in the spiral. This includes the number of loops, the rate of growth, or any other features that may arise. Remember that you can always adjust the range of \(t\) and the step value to explore the spiral further. Once you are satisfied with the graph, you have successfully used a graphing utility to plot the given parametric equations.

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Most popular questions from this chapter

Assume a curve is given by the parametric equations \(x=f(t)\) and \(y=g(t),\) where \(f\) and \(g\) are twice differentiable. Use the Chain Rule to show that $$y^{\prime \prime}(x)=\frac{f^{\prime}(t) g^{\prime \prime}(t)-g^{\prime}(t) f^{\prime \prime}(t)}{\left(f^{\prime}(t)\right)^{3}}.$$

Consider the curve \(r=f(\theta)=\cos a^{\theta}-1.5\) where \(a=(1+12 \pi)^{1 /(2 \pi)} \approx 1.78933\) (see figure). a. Show that \(f(0)=f(2 \pi)\) and find the point on the curve that corresponds to \(\theta=0\) and \(\theta=2 \pi\) b. Is the same curve produced over the intervals \([-\pi, \pi]\) and \([0,2 \pi] ?\) c. Let \(f(\theta)=\cos a^{\theta}-b,\) where \(a=(1+2 k \pi)^{1 /(2 \pi)}, k\) is an integer, and \(b\) is a real number. Show that \(f(0)=f(2 \pi)\) and that the curve closes on itself. d. Plot the curve with various values of \(k\). How many fingers can you produce?

Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection. a. \(x=1+s, y=2 s\) and \(x=1+2 t, y=3 t\) b. \(x=2+5 s, y=1+s\) and \(x=4+10 t, y=3+2 t\) c. \(x=1+3 s, y=4+2 s\) and \(x=4-3 t, y=6+4 t\)

Consider the parametric equations $$ x=a \cos t+b \sin t, \quad y=c \cos t+d \sin t $$ where \(a, b, c,\) and \(d\) are real numbers. a. Show that (apart from a set of special cases) the equations describe an ellipse of the form \(A x^{2}+B x y+C y^{2}=K,\) where \(A, B, C,\) and \(K\) are constants. b. Show that (apart from a set of special cases), the equations describe an ellipse with its axes aligned with the \(x\) - and \(y\) -axes provided \(a b+c d=0\) c. Show that the equations describe a circle provided \(a b+c d=0\) and \(c^{2}+d^{2}=a^{2}+b^{2} \neq 0\)

Eliminate the parameter to express the following parametric equations as a single equation in \(x\) and \(y.\) $$x=\sin 8 t, y=2 \cos 8 t$$
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