Problem 6 - Active Calculus

Question

For each of the following prompts, give an example of a function that satisfies the stated criteria. A formula or a graph, with reasoning, is sufficient for each. If no such example is possible, explain why. a. A function \(f\) that is continuous at \(a=2\) but not differentiable at \(a=2\). b. A function \(g\) that is differentiable at \(a=3\) but does not have a limit at \(a=3\). c. A function \(h\) that has a limit at \(a=-2,\) is defined at \(a=-2,\) but is not continuous at \(a=-2\) d. A function \(p\) that satisfies all of the following: \- \(p(-1)=3\) and \(\lim _{x \rightarrow-1} p(x)=2\) $$\text { - } p(0)=1 \text { and } p^{\prime}(0)=0$$ \- \(\lim _{x \rightarrow 1} p(x)=p(1)\) and \(p^{\prime}(1)\) does not exist

Step-by-Step Solution

Answer

a. \( f(x) = |x - 2| \) b. Not possible. c. \( h(x) = \begin{cases} x^2 & \text{if } x eq -2 \ 3 & \text{if } x = -2 \end{cases} \) d. Piecewise function satisfying all the conditions.

Step 1: Part (a): A function that is continuous at \(a=2\) but not differentiable at \(a=2\)

The absolute value function \( f(x) = |x - 2| \) is a perfect example. This function is continuous at \(x = 2\), but it is not differentiable at \(x = 2\) because the left-hand and right-hand derivatives are not equal.

Step 2: Part (b): A function that is differentiable at \(a=3\) but does not have a limit at \(a=3\)

Such a function does not exist. If a function is differentiable at a point \(x\), it must have a limit at that point. Differentiability implies continuity, and continuity implies the existence of a limit.

Step 3: Part (c): A function that has a limit at \(a=-2\), is defined at \(a=-2\), but is not continuous at \(a=-2\)

Consider the function \( h(x) = \begin{cases} x^2 & \text{if } x eq -2 \ 3 & \text{if } x = -2 \end{cases} \). The limit as \(x\) approaches \(-2\) is \(4\), the function is defined at \(x = -2\) with \( h(-2) = 3 \), but since \( h(-2) eq \lim_{x \to -2} h(x) \), the function is not continuous at \(-2\).

Step 4: Part (d): A function that satisfies multiple criteria

Construct the function piecewise as follows:

Key Concepts

continuous functions

Continuous functions are those which have no interruptions, jumps, or breaks in their graphs. This means that for a function to be continuous at a point \(a\), the following conditions must be met:

• The function must be defined at \(a\).
• The limit of the function as \(x\) approaches \(a\) must exist.
• The value of the function at \(a\) must equal the limit of the function as \(x\) approaches \(a\).

For example, the absolute value function \(f(x) = |x - 2|\) is continuous at \(x = 2\) because it meets all the above conditions. However, it is not differentiable at \(x = 2\). This makes it a good example of a function that is continuous but not differentiable.

differentiable functions

Differentiable functions not only need to be continuous but also must have a defined derivative at every point in their domain. Differentiability ensures the existence of a unique tangent at every point of the function. Important points to note:

• If a function is differentiable at a point \(a\), it means the function is also continuous at \(a\).
• Differentiability implies a unique slope or rate of change at that point.
• If the function has sharp corners or cusps at any point, it is not differentiable there.

For instance, the function \(g(x)\) being differentiable at \(a=3\) but not having a limit at \(a=3\) is impossible. Differentiability inherently requires the function to have a limit at that point because differentiability implies continuity.

limits and continuity

Understanding limits and the role they play in continuity is essential. A limit helps us investigate the behavior of a function as it gets arbitrarily close to a point, but not necessarily reaching it. For continuity at a point \(a\), the following must be true:

• \(\text{lim}_{x \to a} f(x) = L\): The function approaches a specific value \(L\) as \(x\) gets close to \(a\).
• \(f(a)\) is defined.
• \(\text{lim}_{x \to a} f(x) = f(a)\).

A classic example of a function that has a limit but is not continuous at a point is the piecewise function \(h(x)\) given by:
\[ h(x) = \begin{cases} x^2 & \text{if } x e -2 \, 3 & \text{if } x = -2 \end{cases} \]. This function is defined at \(x = -2\) and the limit as \(x\) approaches \(-2\) exists and equals \(4\), but \(h(-2) = 3\). Since \(h(-2)\) is not equal to the limit, the function is not continuous at \(-2\).

piecewise functions

Piecewise functions are defined by different expressions depending on the input value. They allow us to create functions with specific behaviors at certain points. A piecewise function can meet various criteria by combining multiple distinct functions. For example, the function \(p(x)\) satisfying:

• \(p(-1)=3\) and \(\text{lim}_{x \rightarrow -1} p(x)=2\)
• \(p(0)=1\) and \(p^{\text{prime}}(0)=0\)
• \(\text{lim}_{x \rightarrow 1} p(x)=p(1)\) and \(p^{\text{prime}}(1)\) does not exist.

To achieve the specified properties, we can define each segment matching the criteria. Piecewise functions are excellent for modelling real-world scenarios where different rules apply in different contexts.