Calculus AB/BC – 1.10 Exploring Types of Discontinuities

Summary

In this lesson, Mr. Bean explains the concept of continuity and discontinuities in calculus, using various examples to illustrate different types. He focuses on removable discontinuities (holes), non-removable discontinuities (vertical asymptotes), and jump discontinuities. Through examples involving both algebraic and trigonometric functions, he demonstrates how to identify when a function has discontinuities by analyzing factors like zero denominators. The lesson includes hands-on factoring exercises, and concludes with an overview of continuous functions with no discontinuities, reinforcing key concepts for students to master.

Takeaways

• 😀 Continuity in functions means you can graph it without lifting your pencil. Discontinuities occur when you need to lift the pencil.
• 😀 A function is continuous if you can draw it without interruption; discontinuities appear when you need to lift your pencil during graphing.
• 😀 There are three main types of discontinuities: holes (removable), vertical asymptotes (non-removable), and jump discontinuities (non-removable).
• 😀 A hole in the graph is removable if you can fill it in to make the graph continuous. This happens when a factor cancels out.
• 😀 Vertical asymptotes are non-removable discontinuities because there’s no point to 'fill in' to make the function continuous.
• 😀 Jump discontinuities typically appear in piecewise functions and cannot be removed by filling in a hole.
• 😀 When analyzing fractions for discontinuities, check the denominator for zero. A denominator equal to zero indicates a potential discontinuity.
• 😀 If a factor cancels out between the numerator and denominator, it represents a removable discontinuity (a hole).
• 😀 A non-cancelled factor in the denominator creates a vertical asymptote, a non-removable discontinuity.
• 😀 In trigonometric functions like tangent, discontinuities occur when the denominator equals zero, often requiring you to solve for specific angles on the unit circle.

Q & A

What is the definition of a continuous function according to Mr. Bean?

A continuous function is one that can be drawn without lifting your pen or pencil. If you can draw the function without interruption, it is considered continuous.

What happens when you have to lift your pencil while drawing a graph?

When you lift your pencil, it indicates a discontinuity in the graph. This marks the point where the function is no longer continuous.

What is a 'hole' in a function and how is it classified?

A 'hole' is a discontinuity in a graph that is considered removable. It occurs when a function has a gap, but if you fill in that gap, the graph becomes continuous again.

What is the difference between a removable and non-removable discontinuity?

A removable discontinuity occurs when a hole can be filled in to make the graph continuous. A non-removable discontinuity occurs when you cannot simply fill in the gap, such as with a vertical asymptote or jump discontinuity.

How do you identify a vertical asymptote in a function?

A vertical asymptote is identified when the denominator of a function equals zero, and the function does not have a corresponding value for that point.

What is the process of finding discontinuities in a rational function?

To find discontinuities, factor both the numerator and denominator of the rational function. Set the denominator equal to zero and solve for the x-values where the denominator is zero. If a factor cancels out, it’s a hole; if it doesn’t, it’s a vertical asymptote.

What is the significance of the numerator and denominator when finding discontinuities?

The numerator gives the values where the function equals zero, while the denominator gives the potential discontinuities. If a factor from the numerator cancels with the denominator, it indicates a hole. If not, it results in a vertical asymptote.

How does the concept of 'jump discontinuities' differ from other discontinuities?

A jump discontinuity occurs when the function has a break or jump between two pieces, usually in piecewise functions. Unlike holes or vertical asymptotes, jump discontinuities cannot be removed by filling in a point.

When dealing with trigonometric functions, how do you determine discontinuities?

For trigonometric functions like tangent, you check when the denominator equals zero. For example, for tangent (sin(x)/cos(x)), discontinuities occur when cos(x) equals zero, which happens at odd multiples of π/2.

How do you solve for discontinuities in trigonometric functions like tangent?

You solve for discontinuities by setting the denominator equal to zero and solving for x. For example, for tan(2x), you solve when cos(2x) equals zero, which leads to multiple possible x-values.