Calculus: Early Transcendentals

James Stewart

ISBN #9781285741550

8th Edition

6,422 Questions

2,819,387 Students Helped

Homework Questions

Summary

Learning Objectives

Key Concepts

Example Problems

Explanations

Common Mistakes

Summary

This section on parametric equations highlights how curves can be described through a parameter, typically time, making it possible to analyze motion and direction along a curve. It explains methods to eliminate the parameter for transformation into Cartesian form and illustrates how different sets of parametric equations can describe the same curve. Moreover, real-world applications, such as projectile motion, cycloids, and CAD, demonstrate the versatility of parametric descriptions in both theoretical and practical settings.

Learning Objectives

Describe what parametric equations are and explain how they are used to represent curves, including those that fail the vertical line test.

Demonstrate how to eliminate the parameter to convert a set of parametric equations into a Cartesian equation.

Interpret the motion of a particle along a curve via its parametric equations, including the concepts of direction and variable speed.

Formulate parametric equations for common curves such as circles, parabolas, cycloids, and trochoids.

Apply parametric equations to real-world problems like projectile motion and computer-aided design (CAD) applications.

Key Concepts

CONCEPT

DEFINITION

Parametric Equations

A pair (or set) of equations where each coordinate (x, y, etc.) is expressed as a function of an independent parameter, usually denoted by t.

Parameter

An independent variable—typically time—that the coordinates depend on in order to define a trace or curve.

Parametric Curve

The set of points (x(t), y(t)) that are traced as the parameter t varies over a given interval.

Eliminating the Parameter

The process of removing the parameter from the parametric equations to yield a Cartesian equation in x and y.

Cycloid

The curve traced by a point on the circumference of a circle as the circle rolls along a straight line.

Trochoid

A generalization of a cycloid; it is the curve traced by a point at a fixed distance from the center of a rolling circle.

Conchoid

A family of curves generated by specific parametric equations, often resembling the shape of a conch or mussel shell.

Example Problems

Example 1

Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as $ t $ increases. $ x = 1 - t^2 $, $ \quad y = 2t - t^2 $, $ \quad -1 \leqslant t \leqslant 2 $

Example 2

Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as $ t $ increases. $ x = t^3 + t $, $ \quad y = t^2 + 2 $, $ \quad -2 \leqslant t \leqslant 2 $

Example 3

Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as $ t $ increases. $ x = t + \sin t $, $ \quad y = \cos t $, $ \quad -\pi \leqslant t \leqslant \pi $

Example 4

Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as $ t $ increases. $ x = e^{-a} + t $, $ \quad y = e^a - t $, $ \quad -2 \leqslant t \leqslant 2 $

Example 5

(a) Sketch the curve by using the parametric equations to plot points. Indicat with an arrow the direction in which the curve is traced as $ t $ increases. (b) Eliminate the parameter to find a Cartesian equation of the curve. $ x = 2t - 1 $, $ \quad y = \dfrac{1}{2}t + 1 $

Step-by-Step Explanations

QUESTION

Given the parametric equations x = t^2 - 2t and y = t + 1, how do you eliminate t to find a Cartesian equation?

STEP-BY-STEP ANSWER:

Step 1: Solve the simpler equation for t. Here, y = t + 1 gives t = y - 1.
Step 2: Substitute t = y - 1 into the x-equation: x = (y - 1)^2 - 2(y - 1).
Step 3: Expand the equation: x = (y^2 - 2y + 1) - 2y + 2.
Step 4: Combine like terms: x = y^2 - 4y + 3.
Final Answer: The Cartesian equation is x = y^2 - 4y + 3.

Eliminating the Parameter (Example 1: Parabola)

QUESTION

How can you derive parametric equations for a circle with center (h, k) and radius r?

STEP-BY-STEP ANSWER:

Step 1: Start with the standard equation of a circle: (x - h)^2 + (y - k)^2 = r^2.
Step 2: Set x = h + r cos t and y = k + r sin t, using the trigonometric identities.
Step 3: Verify by substituting these into the circle's equation, which simplifies to r^2 (cos^2 t + sin^2 t) = r^2.
Final Answer: The parametric equations for the circle are x = h + r cos t and y = k + r sin t, with 0 ≤ t < 2π.

Deriving Parametric Equations for a Circle

Common Mistakes

Assuming the parameter always represents time, even when it may be a different independent variable.
Errors in algebra when eliminating the parameter, such as mismanaging square terms or sign errors.
Forgetting that different parametric representations can yield the same curve, and not accounting for direction or speed of motion.
Neglecting domain restrictions for the parameter, which can lead to plotting the full curve versus a segment.
Overlooking the importance of the order in which points are traced, which is critical in applications like modeling motion.