Continuity | |
---|---|
Description | |
Exercise Name: | Continuity |
Math Missions: | Differential calculus Math Mission |
Types of Problems: | 3 |
The Continuity exercise appears under the Differential calculus Math Mission. This exercise explores the idea of continuity by the limit definition.
Types of Problems[]
There are three types of problems in this exercise:
- Find the limit from the graph: This problem takes a graph with several discontinuities. The user is asked to select which x-value satisfies the conditions of continuity described at the beginning of the problem.
- Find the value to make the function continuous: This problem has a function with a removable discontinuity. The user is asked how the function should be defined to make the function continuous at this removable discontinuity.
- Use algebra to make the piecewise function continuous: This problem describes a function with discontinuities and unknown coefficients in places. The user is asked to use algebra to find the values that can make the function continuous.
Strategies[]
Knowledge of limits, systems of equations and the concept of continuity are encouraged to ensure success on this exercise.
- Continuity informally means that one can draw the graph without lifting the paper off the pencil.
- Dicontinuities are removable when they leave a dot in the graph. One fills in the dot.
- Discontinuities are unremovable if they are at asymptote or a jump.
- On Use algebra to make the piecewise function continuous some problem require creating and solving a system of equations with multiple unknowns.
Real-life Applications[]
- Limits are used to define both the derivative and the integral.
- The concept of infinitesimals (arbitrarily close to) has applications to anything where precise answers are not always practical or possible.
- Calculus has massive applications to physics, chemistry, biology, economics and many other fields.