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

Conic Sections

Calculus

16. Parametric Equations & Polar Coordinates

Conic Sections

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Multiple Choice

How can you slice a vertically oriented 3D cone to get a 2D parabola?

Slice the cone with a horizontal plane.

Slice the cone with a slightly tilted plane.

Slice the cone with a heavily tilted plane.

Slice the cone with a vertical plane.

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Verified step by step guidance

Step 1: Understand the geometry of a cone. A cone is a three-dimensional shape with a circular base that tapers smoothly to a point called the apex. When slicing a cone, the intersection of the plane and the cone determines the resulting 2D shape.

Step 2: Recall the conic sections. Conic sections are the curves obtained by intersecting a cone with a plane. These include circles, ellipses, parabolas, and hyperbolas, depending on the angle and orientation of the slicing plane.

Step 3: To obtain a parabola, the slicing plane must be tilted at an angle such that it is parallel to the slant height of the cone. This ensures the intersection forms a parabola, which is defined as a curve where every point is equidistant from a fixed point (focus) and a fixed line (directrix).

Step 4: A horizontal plane will produce a circle, while a slightly tilted plane will produce an ellipse. A heavily tilted plane will produce a hyperbola. To achieve a parabola, the plane must be tilted at just the right angle relative to the cone's axis.

Step 5: Visualize the slicing process. Imagine the cone and the plane intersecting. The parabola is formed when the plane cuts through one nappe (half) of the cone and is parallel to the cone's slant height. This geometric relationship is key to understanding why the resulting shape is a parabola.