Which of the following functions are continuous for all values in their domain? Justify your answers.


a. a(t)=altitude of a skydiver t seconds after jumping from a plane

Verified step by step guidance

1. Step 1: Understand the concept of continuity. A function is continuous at a point if the limit of the function as it approaches the point from both sides is equal to the function's value at that point. A function is continuous over an interval if it is continuous at every point in that interval.
2. Step 2: Consider the function a(t) = altitude of a skydiver t seconds after jumping from a plane. This function represents the altitude of a skydiver as a function of time.
3. Step 3: Analyze the behavior of the function a(t). Initially, the skydiver is at a certain altitude, and as time progresses, the altitude decreases as the skydiver falls.
4. Step 4: Determine if there are any points of discontinuity. In the context of a skydiver's altitude, there are no sudden jumps or breaks in the altitude as time progresses, assuming no external forces like parachute deployment are considered.
5. Step 5: Conclude that the function a(t) is continuous for all values in its domain, as the altitude changes smoothly over time without any abrupt changes.

Key Concepts

Continuity of Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For a function to be continuous over its entire domain, it must be continuous at every point in that domain. This means there are no breaks, jumps, or asymptotes in the function's graph.

Domain of a Function

The domain of a function is the set of all possible input values (or 'x' values) for which the function is defined. Understanding the domain is crucial for determining continuity, as a function may be continuous within its domain but not defined outside of it. For example, a function that involves division cannot have inputs that make the denominator zero.

Real-World Applications of Functions

In real-world scenarios, such as the altitude of a skydiver over time, functions often model physical phenomena. These functions can be continuous or discontinuous based on the context. For instance, a skydiver's altitude is typically a continuous function until they reach the ground, where the function may become discontinuous due to the sudden change in altitude.