Book cover for Calculus of a Single Variable

Calculus of a Single Variable

Ron Larson, Bruce Edwards

ISBN #9781285060286

10th Edition

6,214 Questions

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101,749 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section integrates geometric and algebraic methods to study conic sections – parabolas, ellipses, and hyperbolas – and their properties such as reflective behavior and eccentricity. The standard equations derived through completing the square lead to a deeper understanding of each conic’s geometry and applications in real-world problems including satellite orbits and radar. Additionally, the introduction of parametric equations demonstrates a robust method for describing plane curves, effectively linking geometric visualization with time-dependent motion analysis.

Learning Objectives

1

Explain the geometric definitions and derivations of conic sections (parabola, ellipse, hyperbola) and how they are generated from the intersection of a plane and a double-napped cone.

2

Analyze and derive the standard equations of parabolas, ellipses, and hyperbolas using geometric properties and techniques such as completing the square.

3

Illustrate the reflective properties and eccentricity of conics, and how these properties are applied in real-world contexts such as satellite orbits, radar detection, and optical systems.

4

Interpret and construct parametric equations to represent plane curves, and convert between parametric and rectangular forms.

5

Apply calculus concepts (arc length, integration) to analyze conic sections and related curves.

Key Concepts

CONCEPT

DEFINITION

Conic Section

A curve obtained as the intersection of a plane and a double-napped cone, including circles, parabolas, ellipses, and hyperbolas.

Parabola

The locus of all points equidistant from a fixed point (focus) and a fixed line (directrix). Its standard equation involves a vertex, a focus, and an axis of symmetry.

Ellipse

The set of all points for which the sum of the distances from two fixed points (foci) is constant. It has a major axis, minor axis, and a center.

Hyperbola

The set of all points where the absolute difference of the distances from two fixed points (foci) is constant. Its graph consists of two separate branches with asymptotes which help in sketching the curve.

Eccentricity

A measure of how much a conic section deviates from being circular. For ellipses and hyperbolas, it is defined as e = c/a, where c is the distance from the center to the focus and a is the length of the semi-major axis.

Parametric Equations

A pair (or set) of equations that express the coordinates of points on a curve as functions of a parameter, commonly denoted by t. They describe both the shape of the curve and the time sequence of motion along it.

Example Problems

Example 1

Matching In Exercises $1-6,$ match the equation with its graph. [The graphs are labeled (a), (b), (c), (e), and (f). $$ y^{2}=4 x $$

Example 2

Matching In Exercises $1-6,$ match the equation with its graph. [The graphs are labeled (a), (b), (c), (e), and (f). $$ (x+4)^{2}=-2(y-2) $$

Example 3

Matching In Exercises $1-6,$ match the equation with its graph. [The graphs are labeled (a), (b), (c), (e), and (f). $$ \frac{y^{2}}{16}-\frac{x^{2}}{1}=1 $$

Example 4

Matching In Exercises $1-6,$ match the equation with its graph. [The graphs are labeled (a), (b), (c), (e), and (f). $$ \frac{(x-2)^{2}}{16}+\frac{(y+1)^{2}}{4}=1 $$

Example 5

Matching In Exercises $1-6,$ match the equation with its graph. [The graphs are labeled (a), (b), (c), (e), and (f). $$ \frac{x^{2}}{4}+\frac{y^{2}}{9}=1 $$
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Step-by-Step Explanations

QUESTION

Given a parabola defined by a quadratic expression, how do you determine the focus by converting it to standard form?

STEP-BY-STEP ANSWER:

Step 1: Write the given quadratic equation and group the x-terms (or y-terms) appropriately.
Step 2: Complete the square to rewrite the equation in its standard form: (x - h)² = 4p(y - k) (or similarly for a horizontal parabola).
Step 3: Identify the vertex (h, k) and the parameter p, where the absolute value of p denotes the distance from the vertex to the focus.
Step 4: Determine the position of the focus relative to the vertex based on the sign of p.
Final Answer: The focus is located at (h, k + p) for a vertical parabola or (h + p, k) for a horizontal parabola.

Finding the Focus of a Parabola

QUESTION

How do you convert the general second-degree equation into the standard form of an ellipse?

STEP-BY-STEP ANSWER:

Step 1: Start with the general equation Ax² + Cy² + Dx + Ey + F = 0.
Step 2: Group the x and y terms and complete the square for each group.
Step 3: Divide the entire equation by the constant on the right-hand side to achieve the form (x - h)²/a² + (y - k)²/b² = 1.
Step 4: Identify the center (h, k), the lengths of the semi-major axis (a), and the semi-minor axis (b), ensuring that a > b if the major axis is horizontal (or vice versa).
Final Answer: The standard form of the ellipse is (x - h)²/a² + (y - k)²/b² = 1 with foci located at distances c, where c² = a² - b².

Deriving the Standard Equation of an Ellipse

QUESTION

How do you find the asymptotes of a hyperbola given its standard equation?

STEP-BY-STEP ANSWER:

Step 1: Write the hyperbola’s equation in standard form. For a horizontal transverse axis, it is (x - h)²/a² - (y - k)²/b² = 1.
Step 2: Recognize that the asymptotes are the lines that the hyperbola approaches as x or y becomes large.
Step 3: Derive the equations of the asymptotes from the standard form: y - k = ± (b/a)(x - h) for a horizontal hyperbola (or y - k = ± (a/b)(x - h) for a vertical hyperbola).
Final Answer: The asymptotes of the hyperbola (x - h)²/a² - (y - k)²/b² = 1 are given by y - k = ± (b/a)(x - h).

Determining the Asymptotes of a Hyperbola

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Common Mistakes

  • Failing to properly complete the square when converting a general quadratic equation to standard form.
  • Mixing up the definitions of eccentricity for ellipses and hyperbolas.
  • Overlooking the significance of the sign of the parameter p in parabolic equations, which determines the direction the parabola opens.
  • Confusing a curve’s graph with its underlying parametric equations, leading to incorrect interpretations of motion.
  • Misapplying the formulas for asymptotes by not correctly identifying the correct standard form of the hyperbola.