Book cover for Thomas Calculus

Thomas Calculus

George B. Thomas, Jr.

ISBN #9780321878960

13th Edition

6,812 Questions

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127,035 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter introduces an alternative framework for defining curves using parametric equations, where both x- and y-coordinates are functions of an independent parameter, typically time. It covers techniques for eliminating the parameter to obtain Cartesian equations, methods for determining tangents and applying calculus, and special curves like cycloids which have unique physical properties. A strong understanding of these concepts is crucial for modeling real-world phenomena and solving problems in physics and engineering.

Learning Objectives

1

Describe the concept of a parametric curve and explain how it generalizes the graph of a function.

2

Eliminate the parameter to convert parametric equations into a Cartesian equation when possible.

3

Apply parametric equations to model real-world phenomena such as projectile motion, planetary or satellite motion, and cycloidal motion.

4

Calculate derivatives for parametrized curves including dy/dx and d²y/dx² using the chain rule.

5

Recognize and differentiate between various parametrizations of the same geometric object (e.g., a line, parabola, circle, or hyperbola).

Key Concepts

CONCEPT

DEFINITION

Parametric Curve

A curve whose points are defined by a pair of functions, x = f(t) and y = g(t), where t is a parameter over a specified interval.

Parameter Interval

The set of t-values over which the parametric equations are defined; it determines the starting and terminal points of the curve if the interval is closed.

Elimination of the Parameter

A process of solving one of the parametric equations for the parameter t and substituting into the other, resulting in an algebraic equation in x and y.

Cycloid

A curve traced by a fixed point on the circumference of a circle as it rolls along a straight line, described by the equations x = a(t − sin t) and y = a(1 − cos t).

Brachistochrone

The curve along which a frictionless particle will slide under gravity from one point to another in the shortest time; the cycloid is a unique solution.

Tautochrone

A curve with the property that the time taken for an object sliding without friction to its lowest point is independent of the starting point; the cycloid has this property.

Parametric Derivative

The derivative of y with respect to x for a parametric curve, calculated as (dy/dt) divided by (dx/dt) when dx/dt ≠ 0.

Example Problems

Example 1

Give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. $$x=3 t, \quad y=9 t^{2}, \quad-\infty< t <\infty$$

Example 2

Give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. $$x=-\sqrt{t}, \quad y=t, \quad t \geqslant 0$$

Example 3

Give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. $$x=2 t-5, \quad y=4 t-7, \quad-\infty< t <\infty$$

Example 4

Give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. $$x=3-3 t, \quad y=2 t, \quad 0 \leq t \leq 1$$

Example 5

Give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. $$x=\cos 2 t, \quad y=\sin 2 t, \quad 0 \leq t \leq \pi$$
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Step-by-Step Explanations

QUESTION

Given the parametric equations x = t² and y = t + 1, eliminate the parameter to find the Cartesian equation of the curve.

STEP-BY-STEP ANSWER:

Step 1: Solve for t in the simpler equation, y = t + 1, which gives t = y − 1.
Step 2: Substitute t = y − 1 into the equation for x: x = (y − 1)².
Step 3: Expand the expression if needed: x = y² − 2y + 1.
Final Answer: The Cartesian equation of the curve is x = y² − 2y + 1.

Eliminating the Parameter

QUESTION

Find the tangent line to the curve defined by x = sec t and y = tan t at t = π/4.

STEP-BY-STEP ANSWER:

Step 1: Differentiate x = sec t with respect to t: dx/dt = sec t tan t, and differentiate y = tan t: dy/dt = sec² t.
Step 2: At t = π/4, compute dx/dt = sec(π/4) tan(π/4) = √2·1 = √2 and dy/dt = sec²(π/4) = 2.
Step 3: Compute the slope of the tangent: dy/dx = (dy/dt)/(dx/dt) = 2/√2 = √2.
Step 4: Find the coordinates at t = π/4: x = sec(π/4) = √2, y = tan(π/4) = 1.
Step 5: Write the equation of the tangent line using point-slope form: y − 1 = √2 (x − √2).
Final Answer: The tangent line is given by y − 1 = √2 (x − √2).

Finding the Tangent Line for a Parametric Curve

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Common Mistakes

  • Assuming that a parametrized curve always represents a function y = f(x) without considering that it might fail the vertical line test.
  • Neglecting to restrict the domain of the parameter, which can lead to misidentification of the portion of the curve that is being traced.
  • Incorrectly eliminating the parameter by algebraic manipulation, especially when the equations are not easily invertible.
  • Forgetting to apply the chain rule correctly when computing the derivative dy/dx from dy/dt and dx/dt.
  • Assuming that different parametrizations of the same curve will always have the same interpretation regarding the direction and timing of motion.