Problem 1 - Calculus: An Intuitive and Physical Approach

Question

In the following exercises draw the curve by working with the parametric equations. Then find the direct relation between \(x\) and \(y\) (a) \(x=2 t+1, y=1-t . \quad\) Ans. \(y=(3-x) / 2\). (b) \(x=t^{2}, y=2 t\). (c) \(x=2 \sin t, y=2 \cos t . \quad\) Ans. \(x^{2}+y^{2}=4\). (d) \(x=t^{2}, y=t^{2}-5 t\). (e) \(x=e^{t}, y=e^{-t}\) Suggestion: Use the e-table. Ans. \(y=1 / x, x>0\). (f) \(x=\cos t, y=2 \sin t\). (g) \(x=\cos ^{2} \pi t, y=\sin ^{2} \pi t . \quad\) Ans. \(x+y=1, x>0, y>0 .\) (h) \(x=\sin t, y=\sin t\). (i) \(x=e^{t}, y=e^{t}\).

Step-by-Step Solution

Answer

The direct relations are: (a) \(y = \frac{3 - x}{2}\), (b) \(y^2 = 4x\), (c) \(x^2 + y^2 = 4\), (d) \(y = x - 5\sqrt{x}\), (e) \(y = \frac{1}{x}\), (x > 0), (f) an ellipse, (g) \(x + y = 1\), (x > 0, y > 0), (h) \(y = x\), and (i) \(y = x\).

Step 1 - Graphing the Parametric Equations for (a)

To graph the parametric equations for part (a), assign various values to the parameter t and find the corresponding x and y coordinates. Plot these points on the Cartesian plane and draw a curve through them.

Step 2 - Finding the Direct Relation between x and y for (a)

To find the direct relation between x and y, solve one of the parametric equations for t and substitute that into the other equation. For (a), solving for t in the first equation gives us \(t = \frac{x - 1}{2}\). Substituting this into the second equation, we get \(y = 1 - \frac{x - 1}{2}\rightarrow y = \frac{3 - x}{2}\).

Step 3 - Graphing the Parametric Equations for (b)

Follow the same process as in Step 1 for part (b), using the given parametric equations to plot points and draw a curve.

Step 4 - Finding the Direct Relation between x and y for (b)

For part (b), since \(x = t^2\) and \(y = 2t\), we can express t in terms of x as \(t = \sqrt{x}\). Substituting this into the second equation yields \(y = 2\sqrt{x}\). However, because t can be both positive and negative, the correct relation includes the entire parabola, therefore \(y^2 = 4x\).

Step 5 - Graphing the Parametric Equations for (c)

For part (c), plot points by assigning values to t within the range \(0\leq t < 2\pi\) and draw the resulting curve.

Step 6 - Finding the Direct Relation between x and y for (c)

Use the Pythagorean identity \(\sin^2t + \cos^2t = 1\) to find the direct relation for (c). Substituting the parametric forms gives us \((\frac{x}{2})^2 + (\frac{y}{2})^2 = 1\), which simplifies to \(x^2 + y^2 = 4\), representing a circle with a radius of 2.

Step 7 - Graphing the Parametric Equations for (d)

Follow the same process as in Step 1 for part (d), using the given parametric equations.

Step 8 - Finding the Direct Relation between x and y for (d)

Since \(x = t^2\), we can express t as \(t = \sqrt{x}\). Substituting into \(y = t^2 - 5t\) using the positive root (because both t and the square root of x are positive), we find that \(y = x - 5\sqrt{x}\).

Step 9 - Graphing the Parametric Equations for (e)

For part (e), assign various values to the parameter t within a reasonable range and plot the resulting points.

Step 10 - Finding the Direct Relation between x and y for (e)

Recognizing that \(e^t\) and \(e^{-t}\) are inverses of each other, we can multiply the corresponding x and y expressions to get \(xy = e^t \times e^{-t} = 1\). Therefore, the direct relation is \(y = \frac{1}{x}\) for \(x > 0\), because \(e^t > 0\) for all t.

Step 11 - Graphing the Parametric Equations for (f)

Plot points using the parametric equations for part (f) and graph the resulting curve.

Step 12 - Finding the Direct Relation between x and y for (f)

Since \(x = \cos t\) and \(y = 2\sin t\), by squaring and adding them, we cannot use the Pythagorean identity directly due to the different coefficients. Instead, we note that \((\frac{y}{2})^2 = \sin^2 t\), which gives \(\frac{x^2}{1^2} + \frac{y^2}{2^2} = 1\), an equation of an ellipse.

Step 13 - Graphing the Parametric Equations for (g)

For part (g), find points for a range of t values and plot them.

Step 14 - Finding the Direct Relation between x and y for (g)

Using the identity that \(\cos^2\pi t + \sin^2\pi t = 1\), we can directly see that the relation between x and y is \(x + y = 1\), with the constraint that \(x > 0\) and \(y > 0\) because both \(\cos^2\pi t\) and \(\sin^2\pi t\) are positive.

Step 15 - Graphing the Parametric Equations for (h)

For part (h), the parametric equations give the same value for x and y at each t, thus the graph will be a line.

Step 16 - Finding the Direct Relation between x and y for (h)

Since \(x = \sin t\) and \(y = \sin t\), the direct relation is simply \(y = x\), representing a line with a slope of 1.

Step 17 - Graphing the Parametric Equations for (i)

Plot points for part (i) by selecting values for t.

Step 18 - Finding the Direct Relation between x and y for (i)

As \(x = e^t\) and \(y = e^t\), the direct relation is \(y = x\), the same as the previous case, representing a line with a slope of 1.

Key Concepts

Direct Relation Between x and y

When dealing with parametric equations, a fundamental goal is to express a direct relationship between the variables x and y, without the parameter t. This involves eliminating the parameter to find an equation solely in terms of x and y. For example, given the parametric equations from exercise (a), we found the direct relation by solving the first equation for t and then substituting into the second, yielding the equation y = (3 - x) / 2. This step is crucial because it provides a way to understand how x and y are related without depending on the parameter.

By applying this method to various sets of parametric equations, we can derive different types of curves, such as lines, parabolas, and circles. This understanding is essential for calculus problem solving, as it often requires manipulating equations to find a more useful or simplified form.

Graphing Parametric Equations

Graphing parametric equations is a visual representation of how the two variables x and y change concerning a third variable, typically t. This process starts by assigning different values to t, calculating the corresponding x and y, and then plotting these points on a Cartesian plane. For instance, plotting the equations from exercise (c) results in a circle. This is because as t varies, x and y trace out a circular path. The graph provides valuable insights into the behavior of the relationship between x and y and how they evolve together.

When graphing, it is important to consider the range of t and any symmetry or periodicity in the functions to get a complete picture of the curve. Exercises like these are instrumental in mastering the visualization of complex relationships in calculus.

Calculus Problem Solving

Calculus problem solving often involves working with functions and their derivatives to understand rates of change and accumulation. In the context of parametric equations, this can mean finding tangents, areas, or arc lengths. The solution steps provided show how, after graphing the parametric equations, we can find the direct relations which can be further analyzed using calculus techniques.

Note that in exercises (f) and (g), for example, incorporating trigonometric identities or recognizing the properties of exponentials aids in simplifying the expressions. These skills are central to calculus problem-solving, where recognizing patterns and applying relevant mathematical theorems pave the way to solutions.

Trigonometric Parametric Equations

Trigonometric parametric equations, such as those seen in exercises (c) and (f), involve sine and cosine functions. These types of equations often describe circular or elliptical motion. For example, equation (c) yields a circle because the sine and cosine functions are orthogonal and vary sinusoidally as t changes. With exercise (f), the differing coefficients of sine and cosine result in an ellipse.

Understanding the behavior of trigonometric functions, including their periods and amplitudes, is key when graphing these equations. Also, the use of trigonometric identities, such as the Pythagorean identity used in part (c), is a powerful tool in simplifying the relationships between x and y. By mastering trigonometric parametric equations, students gain a solid foundation for tackling more complex calculus problems involving periodic functions.