Problem 62 - Thomas Calculus

Question

In Exercises 62 and \(63,\) you will explore graphically the behavior of the helix $$ \mathbf{r}(t)=(\cos a t) \mathbf{i}+(\sin a t) \mathbf{j}+b t \mathbf{k} $$ as you change the values of the constants \(a\) and \(b\) . Use a CAS to perform the steps in each exercise. Set \(b=1 .\) Plot the helix \(r(t)\) together with the tangent line to the curve at \(t=3 \pi / 2\) for \(a=1,2,4,\) and 6 over the interval \(0 \leq t \leq 4 \pi\) . Describe in your own words what happens to the graph of the helix and the position of the tangent line as \(a\) increases through these positive values.

Step-by-Step Solution

Answer

As \( a \) increases, the helix tightens, making more loops around the z-axis. The tangent line adjusts to the steeper curves.

Step 1: Understanding the Helix Equation

The given helix is represented by the vector function \( \mathbf{r}(t) = (\cos(at))\mathbf{i} + (\sin(at))\mathbf{j} + bt\mathbf{k} \), where \( a \) and \( b \) are constants. For this exercise, we set \( b = 1 \), making the equation \( \mathbf{r}(t) = (\cos(at))\mathbf{i} + (\sin(at))\mathbf{j} + t\mathbf{k} \). It represents a 3D curve that spirals upwards as \( t \) increases.

Step 2: Calculating the Tangent Line at \( t = \frac{3\pi}{2} \)

To find the tangent line, we first calculate the derivative of \( \mathbf{r}(t) \), which is \( \mathbf{r}'(t) = (-a\sin(at))\mathbf{i} + (a\cos(at))\mathbf{j} + \mathbf{k} \). At \( t = \frac{3\pi}{2} \), the tangent line can be expressed as \( \mathbf{r}\left(\frac{3\pi}{2}\right) + t \cdot \mathbf{r}'\left(\frac{3\pi}{2}\right) \). Compute this for each value of \( a \).

Step 3: Plotting the Helix and Tangent Line

Use a computer algebra system (CAS) to graph \( \mathbf{r}(t) \) from \( t = 0 \) to \( t = 4\pi \) along with the tangent line at \( t = \frac{3\pi}{2} \) for each given \( a \) value: \( a = 1, 2, 4, \text{ and } 6 \). Observe how the helix and tangent line change with different \( a \) values.

Step 4: Analyzing the Effect of Changing \( a \)

With increasing \( a \), the helix winds more tightly around the z-axis. Each loop of the helix becomes closer in the \( xy \)-plane, as the frequency of the sinusoidal components increases. The tangent line at \( t = \frac{3\pi}{2} \) shifts accordingly, reflecting the steeper or gentler slopes depending on \( a \), but it always remains tangent to the point on the helix.

Key Concepts

Parametric Equations

Parametric equations are a powerful tool in mathematics used to express the coordinates of points on a curve as functions of a parameter, typically denoted as \(t\). In the context of a helix, parametric equations allow us to describe a spiraling path in three dimensions. By breaking down positions into components - typically \(x\), \(y\), and \(z\) - we can easily model complex curves.

• For the helix \( \mathbf{r}(t) = (\cos(at))\mathbf{i} + (\sin(at))\mathbf{j} + bt\mathbf{k} \), the parameter \(t\) dictates the position along the length of the helix.
• The functions \(\cos(at)\) and \(\sin(at)\) describe circular motion in the \(xy\)-plane, while \( t \mathbf{k} \) results in uniform vertical movement.

This characteristic makes parametric equations ideal for describing objects with rotational symmetry or periodic behavior. They provide a seamless way to animate paths or calculate lines like the tangent line to a curve.

Tangent Lines

A tangent line to a curve at a point is a straight line that just "touches" the curve at that point and has the same direction as the curve instantaneously. To find the tangent line to a helix, we must determine the derivative of its parametric equation, \( \mathbf{r}(t) \).

• The derivative \( \mathbf{r}'(t) = (-a\sin(at))\mathbf{i} + (a\cos(at))\mathbf{j} + \mathbf{k} \) gives the direction of the tangent.
• At a specific point \( t = \frac{3\pi}{2} \), substitute \( t \) into both \( \mathbf{r}(t) \) and \( \mathbf{r}'(t) \) to form the equation of the tangent line.

The tangent line at \( t=\frac{3\pi}{2} \) will take the form \( \mathbf{r}(\frac{3\pi}{2}) + t \cdot \mathbf{r}'(\frac{3\pi}{2}) \), showing how closely related its slope and direction are to the helix at that point. By adjusting the parameter \(a\), the density of the helix's loops advances more quickly or slowly, which makes the tangent indicate how tightly the curve is wound.

Calculus with Computer Algebra Systems

Computer Algebra Systems (CAS) like Mathematica, Matlab, or GeoGebra, are invaluable when dealing with calculus problems involving parametric equations and tangent lines. They automate complex computations and allow for intuitive graphical representations.

• With CAS, graphing the helix under different values of \(a\), you can visualize how the helix structure becomes more compact as \(a\) increases.
• Tangent lines can be easily plotted by computing derivatives and evaluating them symbolically, allowing precise analysis without manual calculations.

Using CAS, students gain deeper insights by seeing these mathematical concepts in action, observing how changes in equations affect the geometry of the curve. This provides a clearer understanding of theoretical concepts and a hands-on approach to learning calculus.