Problem 20
Question
At a point where \(d y / d x=0\), what is special about the graph of \(y(x)\) ? Test case: \(y=x^{2}\).
Step-by-Step Solution
Verified Answer
Where \( \frac{dy}{dx} = 0 \), the graph is flat (horizontal tangent) and could indicate a local extremum (e.g., minimum or maximum). For \( y = x^2 \), it's a minimum at (0, 0).
1Step 1: Interpret the Derivative
The derivative \( \frac{dy}{dx} \) represents the rate of change of the function \( y(x) \) with respect to \( x \). When this value is zero, it implies that the function has no instantaneous change or slope at that point.
2Step 2: Identify Points of Interest
Points where \( \frac{dy}{dx} = 0 \) are often referred to as critical points. At these points, the tangent to the graph is horizontal, meaning the graph is neither increasing nor decreasing.
3Step 3: Example with \( y = x^2 \)
Calculate the derivative of \( y(x) = x^2 \), which is \( \frac{dy}{dx} = 2x \). Set the derivative equal to zero to find critical points: \( 2x = 0 \) yields \( x = 0 \) as a critical point.
4Step 4: Analyze the Point
Substitute \( x = 0 \) back into the original function \( y = x^2 \) to find \( y(0) = 0^2 = 0 \). Therefore, the critical point is at (0, 0). At this point, the graph of \( y(x) \) has a horizontal tangent and forms a minimum, as the function changes from decreasing to increasing.
Key Concepts
DerivativeHorizontal TangentRate of ChangeFunction Analysis
Derivative
A derivative is a fundamental tool in calculus used to understand how a function changes. It gives us the rate of change of a function with respect to one of its variables, typically denoted as \( \frac{dy}{dx} \) when considering a function \( y(x) \). This is essentially the slope of the tangent line to the graph at any point. When the derivative is positive, the function is increasing, and when it's negative, the function is decreasing.
For example, consider the function \( y = x^2 \). To find its derivative, we use the power rule, resulting in \( \frac{dy}{dx} = 2x \). This derivative tells us how quickly \( y \) changes as \( x \) changes.
For example, consider the function \( y = x^2 \). To find its derivative, we use the power rule, resulting in \( \frac{dy}{dx} = 2x \). This derivative tells us how quickly \( y \) changes as \( x \) changes.
- Positive derivative: Function is rising.
- Negative derivative: Function is falling.
- Zero derivative: Function is leveling out.
Horizontal Tangent
A horizontal tangent occurs when the derivative of a function equals zero at a certain point. This means that, at that point, the slope of the tangent to the graph is flat, indicating no vertical change. In mathematical terms, if \( \frac{dy}{dx} = 0 \), the tangent is horizontal.
In our example function, \( y = x^2 \), the derivative \( \frac{dy}{dx} = 2x \) becomes zero when \( x = 0 \). Thus, at \( (0, 0) \), the tangent line to the curve is horizontal. This provides a critical insight into the behavior of the graph at that point.
In our example function, \( y = x^2 \), the derivative \( \frac{dy}{dx} = 2x \) becomes zero when \( x = 0 \). Thus, at \( (0, 0) \), the tangent line to the curve is horizontal. This provides a critical insight into the behavior of the graph at that point.
- Indicates a potential local maximum or minimum.
- Signals a change in the direction of the graph.
Rate of Change
The rate of change is a concept that describes how the value of a function changes as its input changes. It's depicted by the derivative \( \frac{dy}{dx} \), which measures this rate at each point. Understanding this helps us predict and analyze the behavior of functions over specific intervals.
For instance, with \( y = x^2 \), the derivative \( 2x \) shows that as \( x \) increases or decreases, \( y \) changes at a rate proportional to \( x \).
For instance, with \( y = x^2 \), the derivative \( 2x \) shows that as \( x \) increases or decreases, \( y \) changes at a rate proportional to \( x \).
- A zero rate of change indicates a stationary point.
- A positive rate of change means the function is growing.
- A negative rate of change suggests the function is shrinking.
Function Analysis
Function analysis involves examining a function to determine its key features such as critical points, behavior, and graph structure. By understanding these elements, we can sketch, interpret, and utilize functions more effectively.
When applying function analysis to \( y = x^2 \):
When applying function analysis to \( y = x^2 \):
- Critical points are where \( \frac{dy}{dx} = 0 \). For \( x^2 \), this occurs at \( x = 0 \).
- The function is quadratic, and its graph is a parabola opening upwards, indicating that the critical point is a local minimum.
- The horizontal tangent at \((0,0)\) highlights the point where the function changes from decreasing to increasing.
Other exercises in this chapter
Problem 19
19 Suppose \(y=m x+b\) for negative \(x\) and \(y=M x+B\) for \(x \geqslant 0 .\) The graphs meet if _________ The two slopes are____________. The slope at \(x=
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