Domanda del libro di testo
40–41. {Use of Tech} Slopes of tangent lines
b. Find the slope of the lines tangent to the curve at the origin (when relevant).
r =3 − 6 cos θ
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Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 12, Problema 12.R.1e
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
e. The hyperbola y²/2 - x²/4 = 1 has no x-intercept.
Guida verificata passo dopo passo1
Recall that the x-intercepts of a curve occur where the graph crosses the x-axis, which means the y-coordinate is zero. So, to find the x-intercepts, set \(y = 0\) in the equation of the hyperbola.
Substitute \(y = 0\) into the equation \(\frac{y^2}{2} - \frac{x^2}{4} = 1\), which simplifies to \(\frac{0^2}{2} - \frac{x^2}{4} = 1\), or \(-\frac{x^2}{4} = 1\).
Multiply both sides of the equation by \(-4\) to isolate \(x^2\): \(x^2 = -4\).
Since \(x^2 = -4\) has no real solutions (because the square of a real number cannot be negative), there are no real values of \(x\) that satisfy the equation when \(y=0\).
Therefore, the hyperbola \(\frac{y^2}{2} - \frac{x^2}{4} = 1\) has no x-intercepts, confirming the statement is true.
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Definition of x-intercept
An x-intercept is a point where a graph crosses the x-axis, meaning the y-coordinate is zero. To find x-intercepts, set y = 0 in the equation and solve for x. If no real solutions exist, the graph has no x-intercepts.
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Percorso guidato
05:43
Definition of the Definite Integral
Equation of a hyperbola
A hyperbola is a conic section defined by an equation of the form (y²/a²) - (x²/b²) = 1 or (x²/a²) - (y²/b²) = 1. It represents two separate curves and its shape depends on the signs and values of a² and b².
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Percorso guidato
5:50Asymptotes of Hyperbolas
Solving for intercepts in conic sections
To determine intercepts in conic sections, substitute the corresponding coordinate (x=0 for y-intercept, y=0 for x-intercept) into the equation. Analyze the resulting equation for real solutions to confirm the existence of intercepts.
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Percorso guidato
5:33Parabolas as Conic Sections
Pratica correlata
Domanda del libro di testo
A polar conic section Consider the equation r² = sec2θ
a. Convert the equation to Cartesian coordinates and identify the curve.
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Domanda del libro di testo
19–20. Area bounded by parametric curves Find the area of the following regions. (Hint: See Exercises 103–105 in Section 12.1.) The region bounded by the y-axis and the parametric curve
The region bounded by the x-axis and the parametric curve x=cost, y=sin2t, for 0≤t≤π/2
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Domanda del libro di testo
58–59. Tangent lines Find an equation of the line tangent to the following curves at the given point. Check your work with a graphing utility.
x²/16 - y²/9 = 1; (20/3, -4)
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Domanda del libro di testo
53–57. Conic sections
d. Make an accurate graph of the curve.
x = 16y²
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Domanda del libro di testo
24–26. Sets in polar coordinates Sketch the following sets of points.
4 ≤ r² ≤ 9
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