Problem 56 - Calculus of a Single Variable

Question

Can you find a function \(f\) such that \(f(-2)=-2, f(2)=6,\) and \(f^{\prime}(x)<1\) for all \(x ?\) Why or why not?

Step-by-Step Solution

Answer

No, there can be no function which satisfies all three given conditions because the slope calculated between the two given points of the function exceeds the derivative limit set by the third condition.

Step 1: Determine slope through two points

Firstly, to find a function that satisfies the first two conditions, consider that \(f(-2)=-2\) and \(f(2)=6\) can be thought of as points (-2,-2) and (2,6) on a Cartesian plane. We can therefore find the slope of the line that passes through these points using the formula \(m=(y_2-y_1)/(x_2-x_1)\), where (x1,y1) and (x2,y2) are the points in question. Substituting (-2, -2) as (x1, y1) and (2,6) as (x2, y2) into the formula gives \(m=(6-(-2))/(2-(-2))=2\). Therefore, the slope of the line connecting the two points given by the first two conditions of the function is 2.

Step 2: Check slope against derivative condition

In the third condition, we are told that \(f'(x)<1\) for all \(x\), which means the slope of the tangent line to the function must be less than 1 at all points. However, we have just found that the slope of the line between the two given points of the function is 2, which is greater than 1. Thus, these conditions are incompatible.

Step 3: Formulate conclusion

Based on the contradiction found in steps 1 and 2, it can be concluded that there can be no function which satisfies all three conditions. The slope obtained from the first two conditions exceeds the limit set by the third condition.

Key Concepts

Derivative of a Function

The derivative of a function is a mathematical concept that measures how a function changes as its input changes. Essentially, it's akin to the 'speedometer' of a function, showcasing the rate at which the function's value is changing at any given point. In formal terms, if you have a function denoted as \( f(x) \), its derivative at a point \( x \) is represented as \( f'(x) \) or \( \frac{df}{dx} \).

For a linear function like \( f(x) = mx + b \), where \( m \) and \( b \) are constants, the derivative is simply the slope \( m \). However, for non-linear functions, the derivative varies depending on the input value \( x \). Imagine you're driving a car on a winding road—the derivative tells you how sharply the road is turning at any moment. A positive derivative means the function is increasing, while a negative one implies it's decreasing. If the derivative is zero, the function has reached a flat point, neither increasing nor decreasing.

Finding the derivative is a crucial part of calculus, widely used to optimize and predict behavior in physics, economics, and nearly all branches of engineering.

Slope of a Function

The slope of a function at a given point represents the steepness or incline of the graph at that point. In simpler terms, it tells us how fast the \( y \)-value of a function is changing relative to changes in the \( x \)-value. If you're picturing a hill, the slope describes how steep the hill is as you walk along it.

For a straight line, the slope is consistent throughout and can be calculated using the difference between any two points on the line: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Yet, in the world of more complex functions, the slope can change at each point. This is where the concept of the derivative and the slope intersect—because the derivative of a function at a particular point is numerically equal to the slope of the tangent line at that point on the graph of the function.

A slope greater than zero indicates an upward trend as one moves along the function from left to right, whereas a slope less than zero indicates a downward trend. A slope equal to zero would mean the function is flat at that point. Slopes are integral to understanding the behavior of functions, as they give us insight into their dynamics and tendencies.

Cartesian Plane

The Cartesian plane, also known as the coordinate plane, is a foundational concept in algebra and calculus. It's essentially a grid used to plot points, lines, and curves, facilitating a visual representation of mathematical relationships. The plane is named after René Descartes, who formulated the idea of using two intersecting axes as a way to link algebra and geometry.

Composed of a horizontal axis (the x-axis) and a vertical axis (the y-axis), the Cartesian plane creates a two-dimensional space where each point is defined by an ordered pair (x, y). The point where the axes intersect is the origin, designated as (0, 0). The position of a point is determined by how far along the x-axis (horizontally) and the y-axis (vertically) the point lies.

Understanding the Cartesian plane is crucial for graphing functions and interpreting their behavior. For example, the straight line connecting two points lets us visualize the slope or rate of change of a function. As a powerful visual tool, the Cartesian plane allows students and mathematicians to explore and communicate complex mathematical concepts effectively.