Problem 26 - Differential Equations and Linear Algebra

Question

Sketch the given function and determine whether it is piecewise continuous on \([0, \infty)\). $$f(t)=\left\\{\begin{array}{cc} 1, & 0 \leq t \leq 1 \\ 1 /(t-1), & t > 1 \end{array}\right.$$

Step-by-Step Solution

Answer

The given function has two parts: \(f(t) = 1\) on \([0,1]\) and \(f(t) = \frac{1}{t-1}\) on \((1, \infty)\). The first part is a horizontal line segment at \(f(t) = 1\), and the second part is a rational function that approaches a vertical asymptote at \(t = 1\) and approaches \(0\) as \(t \to \infty\). The function is not piecewise continuous on \([0, \infty)\) since there's a jump at \(t = 1\).

Step 1: Sketch the constant function part

For the first function part, we have \(f(t) = 1\) where \(0 \leq t \leq 1\). This is a constant function, so it's simply a straight horizontal line at \(f(t) = 1\) from \(t = 0\) to \(t = 1\). The line segment should include the endpoints (\(0,1\)) and (\(1,1\)) as the interval is closed.

Step 2: Sketch the rational function part

For the second function part, we have \(f(t) = \frac{1}{t-1}\) where \(t > 1\). This is a rational function - to graph it, we can observe that it has a vertical asymptote at \(t = 1\), where the denominator is zero. As \(t\) approaches \(1\) from the right, the function approaches infinity. We can also see that as \(t\) goes to infinity, the function approaches zero from the positive side. Start sketching the graph for this part starting from a bit larger than 1 (because it is not defined at 1) and going to the right. The graph will approach the vertical line \(t = 1\), and get closer and closer to the \(t\) axis as we go to the right.

Step 3: Combine the two parts

Now we can join the two parts together. We have a horizontal line segment at \(f(t) = 1\) for \(0 \leq t \leq 1\) and the rational function part starting from just above \(t=1\) and going to infinity. This will create a single graph of the function \(f(t)\). The change from the constant part to the rational part is not smooth, and there's a jump at \(t = 1\).

Step 4: Determine piecewise continuity

We can now determine if the function is piecewise continuous on the interval \([0, \infty)\). Both parts of the function are continuous individually on their respective domains, but there's a jump at \(t = 1\), showing a discontinuity point. Therefore, the function is not piecewise continuous on the given interval.

Key Concepts

Continuity in Functions

A function is continuous if, at every point in its domain, you can draw it without lifting your pen from the paper. It's a smooth and unbroken graph around every point. Continuity is significant because it ensures there are no sudden jumps or holes, which can be essential for modeling real-world phenomena and ensuring predictions based on these models are reliable. When determining continuity:

• Check if the function exists at every point in its domain.
• Ensure there are no breaks, jumps, or holes.
• Verify that the left-hand and right-hand limits at a point are equal to the function's value at that point.

For the given piecewise function:- The first portion, with a constant value of 1, is continuous between 0 and 1 as it's a horizontal line with no breaks.- The second portion, beginning just after 1 and extending to infinity, has a discontinuity at \(t = 1\) due to the vertical asymptote, creating a jump discontinuity.Therefore, the function isn't piecewise continuous over its entire domain because it has a significant discontinuous point at \(t = 1\).

Function Graphing

Graphing functions helps visualize how they behave across different domains. In piecewise functions, carefully sketching each part keeps clear the separate behaviors as the function changes over intervals. Consider the sketching process for a piecewise function like the one given:- **Constant parts**, such as \(f(t) = 1\) for \(0 \leq t \leq 1\), are straightforward. These are flat, horizontal lines on the graph as they do not vary with \(t\).- **Rational parts**, like \(f(t) = \frac{1}{t-1}\) for \(t > 1\), need special attention as they can have unique features like asymptotes.Identify points and characteristics:

• Where does the graph meet the t-axis or another axis?
• Determine any asymptotes—lines that the graph approaches but never actually touches.
• Consider behavior as \(t\) becomes infinitely large.

Joining these parts forms a complete graph, showing how the function behaves across its full domain and highlights any discontinuities, such as the jump at \(t = 1\).

Rational Functions

Rational functions are quotients of polynomials. They have unique properties that make them interesting to graph, such as asymptotes and undefined points.Definitions and key features of rational functions:

• These functions have the form \(f(t) = \frac{P(t)}{Q(t)}\), where \(P(t)\) and \(Q(t)\) are polynomials.
• Vertical asymptotes occur where the denominator equals zero and the function becomes undefined. These are where rational functions can "jump" to infinity.
• Horizontal or slant asymptotes may occur as \(t\) approaches infinity or negative infinity, reflecting how the function behaves as it "levels out."

For our specific function:- At \(t = 1\), the denominator equals zero in \(f(t) = \frac{1}{t-1}\). This creates a vertical asymptote at \(t = 1\), showcasing the function's undefined point.- As \(t\) increases beyond 1, \(f(t)\) gradually approaches zero, which is typical for rational functions with a horizontal asymptote at the t-axis. These characteristics underscore the essential nature of rational functions in modeling dynamics where certain inputs lead to dramatic changes in outputs.