Problem 34 - Calculus and its applications

Question

Draw a graph that is continuous for all \(x\), with no corners, but not differentiable at \(x=-1\) and \(x=2\).

Step-by-Step Solution

Answer

Use the function \( f(x) = (x^2-1)|x+1| + |x-2| \) to create a continuous graph with non-differentiable points at \(x=-1\) and \(x=2\).

Step 1: Understand the Requirements

We need to draw a function that is continuous across all values of \(x\), meaning there will be no breaks or jumps in the graph, but it should not be differentiable at \(x=-1\) and \(x=2\). A function is typically not differentiable where there is a sharp turn, cusp, or vertical tangent line.

Step 2: Ensure Continuity

To ensure the function is continuous everywhere, the graph should flow smoothly without lifting the pencil from the paper at any point. Possible functions for continuity are polynomials or piecewise functions where each piece connects seamlessly.

Step 3: Create Non-Differentiable Points

Introduce features at \(x=-1\) and \(x=2\) that make the function non-differentiable. A sharp corner or cusp can be created at these points. One way to achieve this is by using absolute value functions, such as \(|x+1|\) and \(|x-2|\), which create V-shaped graphs at \(x = -1\) and \(x = 2\), respectively.

Step 4: Combine for Final Function

Design a function like \( f(x) = (x^2-1)|x+1| + |x-2| \). In this function, \((x^2-1)\) ensures the entire graph is smoothly connected since it's a polynomial, while\(|x+1|\) and \(|x-2|\) create non-differentiable points at \(x=-1\) and \(x=2\).

Step 5: Sketch the Graph

Plot \( f(x) = (x^2-1)|x+1| + |x-2| \). You'll see a smooth curve with a continuous domain. At \(x = -1\) and \(x = 2\), the graph forms sharp corners making it non-differentiable there but still continuous.

Key Concepts

Continuous functions

A continuous function is one of the core concepts in calculus. It is essential for understanding how a graph behaves over a range of values. In simple terms, a function is continuous if you can draw its graph without lifting your pencil from the paper. There are no breaks, holes, or jumps in the graph. Continuous functions have a smooth progression from one point to another.

A few more key details about continuous functions include:

• No Gaps: The function must have no interruptions in its domain, meaning it can be traced consistently across all values of \(x\).
• Limit Exists: At every point in the domain, the limit of the function as \(x\) approaches a value is equal to the function's value at that point.
• Real-World Examples: Many natural phenomena can be modeled with continuous functions, such as temperature changes over a day or the speed of a car over time.

Understanding continuous functions allows us to map real-world situations effectively, making it possible to predict behaviors and analyze trends.

Differentiability

Differentiability expands on the concept of continuity. It explores where a function's graph not only connects smoothly but also has a defined tangent at each point. A function is differentiable at a point if it has a derivative at that point. This tells us about the graph's slope and indicates how steep the curve is at any given point.

It's essential to distinguish the difference between continuity and differentiability:

• Continuous but Not Differentiable: A function can be continuous everywhere but not differentiable at certain points, such as corners or cusps where the slope is undefined.
• Smooth Transition: A differentiable function must transition smoothly without any abrupt changes in direction.
• Mathematical Notation: Mathematically, a function \(f(x)\) is differentiable at a point \(a\) if the derivative \(\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\) exists.

Being differentiable gives us more insight into a function, allowing for advanced analysis like calculating motion or analyzing growth rates.

Absolute value function

The absolute value function offers a fascinating look into creating functions that can be continuous yet not differentiable at certain points. Represented by \(|x|\), this function outputs the positive version of any input, giving it a distinct V-shaped graph.

Some important characteristics of the absolute value function include:

• Graph Shape: The hallmark of an absolute value function is its V-shape, creating points (or 'corners') at the vertex.
• Not Differentiable at Vertex: The corner point means the function is not differentiable at that point, although it remains continuous.
• Applications in Piecewise Functions: Absolute value functions are often used in piecewise definitions, bringing versatility in describing more complex graphs with different behaviors at specific points.
• Equation Form: Functions like \( |x+1| \) or \( |x-2| \) are often used to introduce variability at specific values of \(x\), creating an impactful feature on the graph.

The absolute value function is crucial in calculus for modeling non-differentiable points while maintaining continuity, bridging the gap between smooth and sharp elements in graphical representations.