Problem 75 - Calculus Single Variable

Question

Suppose that \(f\) is defined on an open interval centered at \(c .\) Suppose also that $$ \ell_{R}=\lim _{h \rightarrow 0^{+}} \frac{f(c+h)-f(c)}{h} $$ and $$ \ell_{L}=\lim _{h \rightarrow 0^{-}} \frac{f(c+h)-f(c)}{h} $$ exist. Let $$ \begin{array}{ll} T_{R}(x)=f(c)+\ell_{R}(x-c) & \text { for } x \geq c \\\ T_{L}(x)=f(c)+\ell_{L}(x-c) & \text { for } x \leq c \end{array} $$ Define \(\alpha_{f}(c)\) to be the radian measure of the angle through which \(T_{R}\) must be rotated counterclockwise about \((c, f(c))\) to coincide with \(T_{L} .\) We may think of \(\alpha_{f}(c)\) as the angle of the corner at \(P=(c, f(c))\). a. For what values of \(\alpha_{f}(c)\) is there actually a corner at \(P ?\) Explain. b. For what value of \(\alpha_{f}(c)\) is there a tangent line at \(P\). Explain. c. If the graph of \(f\) has a vertical tangent at \(P,\) is \(\alpha_{f}(c)\) defined? Explain.

Step-by-Step Solution

Answer

a. A corner exists when \( \ell_{L} \neq \ell_{R} \). b. A tangent line exists when \( \alpha_{f}(c) = 0 \). c. \( \alpha_{f}(c) \) is not defined with a vertical tangent since derivative is infinite.

Step 1: Understand Corner at Point P

A corner at point \( P = (c, f(c)) \) occurs when the left derivative \( \ell_{L} \) and the right derivative \( \ell_{R} \) exist, are both finite, but are not equal. This results in an abrupt change in the direction of the curve, forming a corner. Thus, for a corner to exist, \( \ell_{L} eq \ell_{R} \). The angle \( \alpha_{f}(c) \) is defined as the measure of the rotation needed to align \( T_{R} \) with \( T_{L} \). Therefore, \( \alpha_{f}(c) \) will be non-zero (not equal to \( 0 \)) when there is a corner.

Step 2: Understand Tangent Line at Point P

A tangent line at \( P \) implies no corner, meaning \( \ell_{L} = \ell_{R} \). When \( \ell_{L} = \ell_{R} \), the graph is smooth at \( P \) and the right and left tangents align to form a single line. Thus, the angle \( \alpha_{f}(c) \) would be \( 0 \), indicating no rotation is needed to align the two tangents.

Step 3: Vertical Tangent and Angle Definition

A vertical tangent at \( P \) means that either \( \ell_{L} = \infty \) or \( \ell_{R} = \infty \). This situation leads to an undefined slope, which creates a vertical line. The angle \( \alpha_{f}(c) \), involving the rotation of tangents, is not defined because the concept of rotation does not apply when considering a vertical line.

Key Concepts

Left Derivative

The left derivative of a function at a point is focused on analyzing the behavior of the function as you approach the point from the left-hand side. In mathematical terms, it is denoted by \(\ell_{L}\) and is expressed as

• \(\ell_{L} = \lim_{h \rightarrow 0^{-}} \frac{f(c+h)-f(c)}{h}\)

This equation represents the limit of the average rate of change of the function as the interval \(h\) approaches zero from the negative direction—meaning, as it approaches from values less than zero.
The left derivative evaluates how steep the slope of the function is approaching from the left towards a specific point \(c\). If this derivative exists and is finite, it tells us that the function behaves regularly in regards to its slope on this side. In simpler terms, the function does not have any sudden spikes or vertical lines as you near the point from the left.

Right Derivative

The right derivative is fundamentally similar to the left derivative but evaluates the behavior of the function as it approaches the point from the right-hand side. It's represented by \(\ell_{R}\) and defined as

• \(\ell_{R} = \lim_{h \rightarrow 0^{+}} \frac{f(c+h)-f(c)}{h}\)

Here, the limit is taken as \(h\) approaches zero from positive values, which means you're approaching the point \(c\) from the right side.
The essence of the right derivative is to determine how the function behaves in terms of its slope or incline from the right direction, near point \(c\).
A finite right derivative indicates a well-behaved function on its right side. Thus, this helps determine potential changes or continuity issues that might arise as you move towards \(c\) from the right. Understanding both derivatives aids in discussing the overall behavior of the function around the point.

Corner

A corner in the graph of a function at a given point \((c, f(c))\) is characterized by a sharp turn, which occurs when the left and right derivatives exist, are finite, but are not equal. This situation is quite intuitive:

• If \(\ell_{L} eq \ell_{R}\), there is an abrupt change in direction, resulting in a corner.

When the slope from the left is different from the slope from the right, the graph does not transition smoothly through the point but rather takes an unexpected jagged turn.
Exploring this concept involves finding the angle \(\alpha_{f}(c)\), which illustrates how much one tangent needs to rotate around the point \(P\) to meet the other. If \(\alpha_{f}(c) eq 0\), it signals a difference between the two derivatives and hence a corner formation.
This concept has broad applications in real-world scenarios where measuring sudden changes or discontinuities is essential.

Tangent Line

A tangent line refers to a straight line that "just touches" a curve at a given point, where the curve has the same slope as the tangent. If at point \((c, f(c))\), the left and right derivatives are equal \(\ell_{L} = \ell_{R}\), the graph of the function is smooth and has a tangent line there.

• \(\alpha_{f}(c) = 0\) indicates the tangent line's presence as no rotation is needed for alignment.

This means both \(T_{L}(x)\) and \(T_{R}(x)\) perfectly coincide at the point, smoothly transitioning without forming any corners. In simple terms, a tangent line represents continuity and smooth transitions since there is no angle to correct between the tangents.
Understanding this concept is beneficial in calculus and geometry, helping students predict and analyze how functions might behave near a specific point. Recognizing where tangent lines exist aids in mapping out complex curves with greater accuracy.