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Chapter 2: Problem 48

Sketch a graph of a function whose derivative is always positive.

Short Answer

Expert verified
The graph of the function \( y = x \) can serve as an example of a function whose derivative is always positive as it constantly increases with increasing \( x \).

Step by step solution

01

Understand the Problem

A derivative is a measure of how a function changes as its input changes. In simpler terms, it could be said that the derivative of a function at any given point is the slope of the tangent line at that point on the function's graph. When a function's derivative is positive, the function is increasing. From this understanding, an increasing function should be drawn.
02

Sketch the Function

To depict a function whose derivative is always positive, consider a simple linear function. Let's say \( y = x \). This is a simple straight line graph which passes through the origin and has a uniform slope. The derivative of \( y = x \) is always 1, which signifies that the function always increases as \( x \) increases. Hence, sketching \( y = x \) would depict a function whose derivative is always positive. However, other functions could also have this property. It's just easier to depict it with a function like \( y = x \), which is a simple straight line with a positive slope.

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Key Concepts

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

Increasing Function
An increasing function is a type of mathematical function where as the input value grows, the output value also rises. For a function to be identified as increasing, the derivative must be positive for every point in its domain. The derivative represents the rate at which the output changes with respect to a change in the input. Therefore, when every value of the derivative is positive, it ensures that the direction of the function is always upwards. This is seen with functions like the linear equation \( y = x \), where each increase in \( x \) results in a consistent increase in \( y \).
  • A positive derivative indicates an increasing function.
  • As the function progresses, its graph demonstrates a continual upward movement.
  • Common increasing functions include linear and exponential equations.
It's pivotal to differentiate between "increasing" and "strictly increasing." While increasing functions do not decrease, strictly increasing functions never stay constant over any interval.
Tangent Line
A tangent line to a curve is a straight line that touches the curve at a single point and represents the immediate rate of change. It is an essential concept when discussing derivatives, as the derivative value at a particular point on the function can be visualized as the slope of this tangent line. This means if you were to draw a tangent at any point along the line \( y = x \), you'd notice that the slope (or the derivative) consistently remains at 1.
  • A tangent line intersects the curve at one single point.
  • It helps in visualizing and understanding the behavior of the function at that specific point.
  • The slope of the tangent line gives a direct insight into the direction and speed of change of the function.
Understanding the relationship between the tangent line and the curve itself is crucial in grasping how the function behaves at any given point.
Positive Slope
A positive slope is an indicator that a line or curve on a graph is inclining upwards as you move from left to right. When discussing a function, having a positive slope means that the function is increasing. The derivative of a function tells us the slope of the tangent line at any point. Therefore, if the derivative is consistently positive, this means each tangent line will have a positive slope.
  • Positive slope means an upward direction across the graph.
  • It reflects the function's increase over time or input value.
  • Linear functions with constant positive derivatives, such as \( y = x \), have a steady positive slope.
Recognizing positive slopes in functions helps in predicting the general behavior and future value trends of the function.

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Most popular questions from this chapter

Use the position function \(s(t)=-4.9 t^{2}+v_{0} t+s_{0}\) for free-falling objects. A projectile is shot upward from the surface of Earth with an initial velocity of 120 meters per second. What is its velocity after 5 seconds? After 10 seconds?

Consider the third-degree polynomial \(f(x)=a x^{3}+b x^{2}+c x+d, \quad a \neq 0\) Determine conditions for \(a, b, c\), and \(d\) if the graph of \(f\) has (a) no horizontal tangents, (b) exactly one horizontal tangent, and (c) exactly two horizontal tangents. Give an example for each case.

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Verify that the two families of curves are orthogonal where \(C\) and \(K\) are real numbers. Use a graphing utility to graph the two families for two values of \(C\) and two values of \(K\). $$ x^{2}+y^{2}=C^{2}, \quad y=K x $$
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