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Chapter 2: Problem 13

Using your own words, explain geometrically why the derivative is undefined where a curve has a corner point.

Short Answer

Expert verified
Corners lack a consistent slope, causing the derivative to be undefined.

Step by step solution

01

Understanding the Derivative

The derivative of a function at a point represents the slope of the tangent line to the curve at that exact point. In simple terms, it's the rate at which the function's value is changing at that point.
02

Identifying a Corner Point

A corner point on a curve is a point where the direction of the curve changes abruptly. The curve does not have a single, smooth tangent line at a corner, because it looks like two different lines meeting sharply.
03

Visualizing Tangent Lines

For the derivative to be defined, there must be a unique tangent line at the point on the curve, which implies a consistent slope from both sides of the point. However, at a corner point, approaching from the left yields one slope, while approaching from the right yields a completely different slope.
04

Concluding Undefined Derivative

Since the slopes from both directions do not match at a corner point, a unique tangent line cannot be drawn. Thus, the derivative is undefined at corners, as it requires a single, consistent slope to define a tangent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Corner point
In geometry, a corner point on a curve signifies a sudden change in direction at a specific point. Imagine a road with a sharp turn 鈥� it represents a shift that is neither smooth nor gradual. Similar to how a sharp turn on a path occurs instantly, a corner point does the same on a graph.

At a corner point, the curve transitions abruptly between two different paths. As a result, instead of having a smooth, rounded curve, the graph appears angular, like an arrow tip.
  • There are two distinct sides meeting at the corner.
  • Each side forms a different segment of the graph.
  • This sudden change interrupts the continuity of the curve.
Thus, corner points are areas of geometric interest where standard lines and slope calculations become more challenging.
Tangent line
A tangent line is a straight line that touches a curve at a single point, reflecting the immediate direction of the curve at that exact location. It is like the surface of the ocean gently kissing the horizon, sharing only a single point of contact.

To determine the tangent line of a curve at a given point, consider:
  • The smoothness and continuity of the curve around the point.
  • The idea that the closer you zoom closer to the curve at that point, the more it resembles the tangent line.
The tangent effectively captures the essence of the curve's direction at that point. But, when you encounter a corner point, this becomes improbable because the curve's direction changes too abruptly to support a single, definitive tangent line.
Slope
The slope of a curve at any point is an indicator of how steep the curve is at that point, which is quantified by the derivative. Slope determines the angle or steepness of a line relative to the horizontal axis, often described as "rise over run."

In calculus, slope is key to understanding how a function behaves at various points:
  • A positive slope means the curve is increasing as you move along it.
  • A negative slope indicates the curve is decreasing.
  • A zero slope suggests the curve is flat at that point, like the top of a hill.
Evaluating the slope involves assessing how a curve behaves as it is approached from either side of a point. At a corner point, you'll find the mystery 鈥� it has different slopes from the left and right, creating confusion and preventing the formation of a singular slope representative of that corner.
Undefined derivative
The derivative is a core concept in calculus representing how a function changes at a specific point. Think of the derivative as calculating the exact steepness of a hill at any moment. When a derivative is undefined, like at a corner point, it's similar to the idea of our inability to describe the slope of the hill right at a sharp turn.

Here's why the derivative is often undefined at corner points:
  • Approaching from the left side results in one slope value.
  • Approaching from the right side leads to a different slope value.
  • The mismatch means no single, consistent slope can adequately describe the curve at that point.
This absence of a unique slope leads to an undefined derivative 鈥� the cornerstone of calculus at corner points, revealing the limitations of mathematical description at abrupt direction changes.

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Most popular questions from this chapter

Use the Generalized Power Rule to find the derivative of each function. $$g(x)=\left(2 x^{2}-7 x+3\right)^{4}$$

The windchill index (revised in 2001 ) for a temperature of 32 degrees Fahrenheit and wind speed \(x\) miles per hour is \(W(x)=55.628-22.07 x^{0.16}\). a. Graph the windchill index on a graphing calculator using the window \([0,50]\) by \([0,40]\). Then find the windchill index for wind speeds of \(x=15\) and \(x=30\) mph. b. Notice from your graph that the windchill index has first derivative negative and second derivative positive. What does this mean about how successive 1-mph increases in wind speed affect the windchill index? c. Verify your answer to part (b) by defining \(1 / 2\) to be the derivative of \(y_{1}\) (using NDERIV), evaluating it at \(x=15\) and \(x=30\), and interpreting your answers.

Use the Generalized Power Rule to find the derivative of each function. $$y=x^{4}+(1-x)^{4}$$

a. Show that the definition of the derivative applied to the function \(f(x)=\sqrt{x}\) at \(x=0\) gives \(f^{\prime}(0)=\lim _{h \rightarrow 0} \frac{\sqrt{h}}{h}\). b. Use a calculator to evaluate the difference quotient \(\frac{\sqrt{h}}{h}\) for the following values of \(h\) : \(0.1,0.001\), and \(0.00001\). [Hint: Enter the calculation into your calculator with \(h\) replaced by \(0.1\), and then change the value of \(h\) by inserting zeros.] c. From your answers to part (b), does the limit exist? Does the derivative of \(f(x)=\sqrt{x}\) at \(x=0\) exist? d. Graph \(f(x)=\sqrt{x}\) on the window \([0,1]\) by \([0,1]\). Do you see why the slope at \(x=0\) does not exist?

Use the Generalized Power Rule to find the derivative of each function. $$f(x)=\left(x^{2}+1\right)^{3}$$
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