Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

16. Parametric Equations & Polar Coordinates

Polar Coordinates

Calculus

16. Parametric Equations & Polar Coordinates

Polar Coordinates

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Multiple Choice

Identify whether the given equation is that of a cardioid, limaçon, rose, or lemniscate.
$r^2=-25\cos2\theta$

Cardioid

Limacon

Rose

Lemniscate

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Verified step by step guidance

Step 1: Begin by analyzing the given polar equation:

r^{2} = - 25 \cos 2 θ

. Note that the equation involves

r^{2}

, which is squared, and the cosine function with a doubled angle,

2 θ

Step 2: Recall the general forms of polar curves: Cardioids and limaçons typically involve equations of the form

r = a + b \cos θ

r = a + b \sin θ

. Roses involve equations like

r = a \cos k θ

r = a \sin k θ

. Lemniscates, however, are characterized by equations involving

r^{2}

and trigonometric functions of doubled angles, such as

r^{2} = a \cos 2 θ

r^{2} = a \sin 2 θ

Step 3: Compare the given equation

r^{2} = - 25 \cos 2 θ

with the standard forms. Notice that it matches the structure of a lemniscate equation, as it involves

r^{2}

and

\cos 2 θ

Step 4: Understand the significance of the negative sign in the equation. The negative sign does not change the classification of the curve but affects its orientation. Lemniscates can have positive or negative coefficients, which determine the direction of the lobes.

Step 5: Conclude that the given equation represents a lemniscate based on its structure and comparison with standard polar curve forms.