Problem 19 - Brief Applied Calculus

Question

Using only straight lines, sketch a function that (a) is continuous everywhere and (b) is differentiable everywhere except at \(x=1\) and \(x=3\).

Step-by-Step Solution

Answer

Draw a continuous line with corner points at \(x=1\) and \(x=3\).

Step 1: Draw the Base of the Function

To start, create a base that is continuous and differentiable everywhere, such as a straight horizontal line, say at \(y = 1\). This will serve as the foundation to ensure continuity.

Step 2: Introduce Discontinuity Points in Derivative

At \(x = 1\) and \(x = 3\), modify the function to disrupt differentiability, such as by adding corners or cusps. From \(x = 0\) to \(x = 1\), draw a slanting line upwards to meet the base line at \(x = 1\). From \(x = 1\) to \(x = 3\), continue with a straight horizontal line. Then from \(x = 3\), introduce another slant, downwards in this case.

Step 3: Check the Continuity

Ensure the entire function is connected without any breaks, implying continuity. There should be no sudden jumps between \(x = 0\) and \(x = 4\) as the lines meet at the same y-values between \(x=1\) and \(x=3\).

Step 4: Verify Differentiability Conditions

Verify that the function is differentiable everywhere except at \(x = 1\) and \(x = 3\), where sharp turns were introduced. This means that the derivative does not exist at these points due to abrupt changes in direction.

Key Concepts

Differentiability

In calculus, differentiability is about determining whether a function has a defined derivative at a specific point. A function is differentiable at a point if its derivative exists at that point.
Differentiability essentially means you can draw a tangent line at that point, which will touch the curve smoothly. If a function is differentiable everywhere, it is quite smooth with no sharp turns or cusps.

• **Smoothness:** To be differentiable at a point, the function should not have any abrupt changes like sharp corners.
• **Existence of a Derivative:** A derivative of a function at a point gives the slope of the tangent line at that point. If this derivative does not exist, the function is not differentiable there.

For example, in the given exercise, the function is not differentiable at points where there are sharp turns, namely at \(x = 1\) and \(x = 3\). Here, the changes in direction create points where the derivative does not exist, showcasing a lack of smoothness.

Continuity

Continuity in calculus refers to a function that has no breaks, jumps, or holes in its graph. In simple terms, a continuous function can be drawn without lifting your pen.
To check for continuity at a point, a function must meet three main conditions:

• The function is defined at that point.
• The function approaches the same value from both left and right as the point is approached (the limits from left and right match).
• The value that the function approaches from both sides equals the function's value at that point.

In the exercise, ensuring that the lines connect smoothly without gaps ensures continuity. Even though the function has sharp turns at \(x = 1\) and \(x = 3\), it remains continuous because the y-values do not jump. The function does not break at these points; it simply changes direction, which is a different concern addressed by differentiability.

Piecewise Functions

Piecewise functions are defined by different expressions over different parts of their domain. Think of them as a combination of distinct "pieces" that can exhibit different behaviors.
With piecewise functions, each piece works like a section of the function that operates according to its own rules within a specific interval.

• **Defined Intervals:** Each piece of a piecewise function operates over a specific interval.
• **Different Expressions:** The function can have drastically different behaviors over separate intervals.

In the exercise, the function features straight lines, each functioning over a defined interval to maintain continuity, except for the points causing differentiability issues. From \(x = 0\) to \(x = 4\), the function pieces together, forming one unified graph by using different line segments. This piecewise nature helps to easily meet the conditions of the problem—static in some parts and sloped in others, without breaking the overall continuity.