Problem 1 - Calculus Early Transcendentals

Question

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 b. \(n(t)=\) number of quarters needed to park legally in a metered parking space for \(t\) minutes c. \(T(t)=\) temperature \(t\) minutes after midnight in Chicago on January 1 d. \(p(t)=\) number of points scored by a basketball player after \(t\) minutes of a basketball game

Step-by-Step Solution

Answer

Answer: The functions a(t) and T(t) are continuous for all values in their domain.

Step 1: Analyze the continuity of the altitude of a skydiver a(t)

The function a(t) represents the altitude of a skydiver t seconds after jumping from a plane. The skydiver's altitude changes smoothly as they fall, and there are no sudden jumps or gaps in this movement. Therefore, a(t) is continuous for all values in its domain.

Step 2: Analyze the continuity of the number of quarters needed for parking n(t)

The function n(t) represents the number of quarters needed to park legally in a metered parking space for t minutes. The number of quarters changes only at specific time intervals (e.g., every 15 minutes) and remains constant in between these intervals. This means that as t increases, n(t) will have sudden jumps, indicating that it is not continuous for all values in its domain.

Step 3: Analyze the continuity of the temperature after midnight T(t)

The function T(t) represents the temperature t minutes after midnight in Chicago on January 1. Temperature changes continuously over time, and it is unlikely that there are any sudden jumps or gaps in the temperature. Hence, T(t) is continuous for all values in its domain.

Step 4: Analyze the continuity of the number of points scored in a basketball game p(t)

The function p(t) represents the number of points scored by a basketball player after t minutes of a game. The points scored in a basketball game change discretely - the number of points does not increase smoothly but instead increases by a certain value (e.g., 1, 2, or 3) whenever a player scores. Therefore, there are sudden jumps in p(t), and it is not continuous for all values in its domain. In summary: 1. a(t) is continuous for all values in its domain 2. n(t) is not continuous for all values in its domain 3. T(t) is continuous for all values in its domain 4. p(t) is not continuous for all values in its domain