Problem 25 - Calculus Concepts

Question

Most piecewise-defined continuous functions have discontinuities at their break points. Consider, however, piecewise-defined continuous functions that are continuous at their break points. Is it possible to draw a tangent line at a break point for such a function? Discuss how and why this might or might not happen. Use these two functions as examples: $$ \begin{array}{l} f(x)=\left\\{\begin{array}{ll} -x^{2}+8 & \text { when } x \leq 2 \\ x^{3}-9 x+14 & \text { when } x>2 \end{array}\right. \\ g(x)=\left\\{\begin{array}{ll} x^{3}+9 & \text { when } x \leq 3 \\ 5 x^{2}-3 x & \text { when } x>3 \end{array}\right. \end{array} $$

Step-by-Step Solution

Answer

g(x) allows a tangent at the break point; f(x) does not.

Step 1: Checking Continuity at Break Points

To determine if a tangent line can be drawn at the break point of a piecewise function, we first check if the function is continuous at that point. For the function \( f(x) \), we need to check continuity at \( x = 2 \). Evaluate the left-hand limit, right-hand limit, and the function value at \( x = 2 \): - Left-hand limit as \( x \) approaches 2: \( \lim_{x \to 2^-} (-x^2 + 8) = -2^2 + 8 = 4 \).- Right-hand limit as \( x \) approaches 2: \( \lim_{x \to 2^+} (x^3 - 9x + 14) = 2^3 - 9(2) + 14 = 4 \).- Function value at \( x = 2 \): \( f(2) = -2^2 + 8 = 4 \). Therefore, \( f(x) \) is continuous at \( x = 2 \) because all these equal 4.

Step 2: Check Differentiability at Break Point

A tangent line can only exist at a point where the function is differentiable. For differentiability at \( x = 2 \), the left-hand derivative and right-hand derivative must be equal. - Left-hand derivative: \( \frac{d}{dx}(-x^2 + 8) = -2x \). Evaluating at \( x = 2 \), left-hand derivative = \( -2(2) = -4 \).- Right-hand derivative: \( \frac{d}{dx}(x^3 - 9x + 14) = 3x^2 - 9 \). Evaluating at \( x = 2 \), right-hand derivative = \( 3(2)^2 - 9 = 12 - 9 = 3 \). As the derivatives are not equal, \( f(x) \) is not differentiable at \( x = 2 \), hence no tangent line can be drawn.

Step 3: Evaluate Continuity of g(x) at Break Point

Next, examine continuity for the function \( g(x) \) at its break point \( x = 3 \):- Left-hand limit as \( x \) approaches 3: \( \lim_{x \to 3^-} (x^3 + 9) = 3^3 + 9 = 36 \).- Right-hand limit as \( x \) approaches 3: \( \lim_{x \to 3^+} (5x^2 - 3x) = 5(3)^2 - 3(3) = 45 - 9 = 36 \).- Function value at \( x = 3 \): \( g(3) = 3^3 + 9 = 36 \). Thus, \( g(x) \) is continuous at \( x = 3 \).

Step 4: Check Differentiability of g(x) at Break Point

For \( g(x) \), determine differentiability at \( x = 3 \):- Left-hand derivative: \( \frac{d}{dx}(x^3 + 9) = 3x^2 \). Evaluating at \( x = 3 \), left-hand derivative = \( 3(3)^2 = 27 \).- Right-hand derivative: \( \frac{d}{dx}(5x^2 - 3x) = 10x - 3 \). Evaluating at \( x = 3 \), right-hand derivative = \( 10(3)-3 = 30-3 = 27 \). As both derivatives are equal at \( x = 3 \), \( g(x) \) is differentiable at this point, thus a tangent line can be drawn.

Key Concepts

Continuity

In mathematics, a function is said to be continuous at a point if it does not have any abrupt changes or breaks at that point. For a piecewise-defined function, checking the continuity at its break point is crucial. To determine the continuity at a break point, you need to evaluate three things:

• The left-hand limit as you approach the break point.
• The right-hand limit as you approach from the other side.
• The actual value of the function at the break point.

If these three values are equal, the function is continuous at that point. In the exercise, for the function \( f(x) \), the check at \( x = 2 \) revealed that all those values are 4, indicating continuity. Similarly, for \( g(x) \) at \( x = 3 \), all values were 36. Therefore, despite being a piecewise function, both functions are continuous at their respective break points.

Differentiability

Differentiability extends the idea of continuity by examining whether we can compute a derivative at a point, which represents the function's rate of change there. For a function to be differentiable at a point, two things must be true:

• The function must be continuous at that point.
• The left-hand derivative and the right-hand derivative at that point must be equal.

For \( f(x) \) at \( x = 2 \), though it was continuous, the left-hand derivative \(-4\) and right-hand derivative \(3\) were not equal. This indicates non-differentiability, hence no tangent line.

Conversely, for \( g(x) \) at \( x = 3 \), both derivatives were \(27\), confirming differentiability. Thus, a tangent line can be drawn here. This reveals a nuanced understanding that continuity does not always imply differentiability, especially at break points.

Tangent line

A tangent line to a curve at a given point is a straight line that just "touches" the curve at that point and gives the slope of the curve there. Establishing a tangent line involves:

• Ensuring the function is differentiable at the point.
• Using the derivative to find the slope of the tangent line.

Once differentiability is confirmed, like for \( g(x) \) at \( x = 3 \), the derivative provides the slope, here 27. You can then use the point-slope form of the line equation to write the equation of the tangent line.

For \( f(x) \) at \( x = 2 \), the lack of differentiability means there is no single straight line that fully represents the rate of change or "slope" at that point, thus no tangent line can be formed there.

Limit evaluation

Limit evaluation is a fundamental concept in determining a function's behavior as it approaches a specific point. It's essential for assessing both continuity and differentiability. Here's how it's typically done:

• Calculate the left-hand limit as you approach the break point from one side.
• Calculate the right-hand limit as you approach from the other side.

These limits help verify if a function is continuous by showing if both sides match the function's actual value at the break point. In the exercise, evaluating limits at the break points for \( f(x) \) and \( g(x) \) helped confirm their continuity.

Furthermore, examining these limits at the derivative level helps determine if the function is differentiable, as equal limit slopes on both sides of the point indicate the presence of a tangent line, as seen with \( g(x) \). This underscores how thoroughly understanding limits supports deeper insight into the nature of piecewise-defined functions.