Problem 50 - Calculus of a Single Variable

Question

True or False? In Exercises \(47-50\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(y=f(x), f\) is increasing and differentiable, and \(\Delta x>0\) , then \(\Delta y \geq d y\)

Step-by-Step Solution

Answer

The statement is false. The change in \(y\) (\(\Delta y\)) isn't necessarily always greater than or equal to the exact differential \(dy\) even if \(f\) is differentiable and increasing, and \(\Delta x > 0\).

Step 1: Understanding the concepts

Firstly, an increasing function is a function that increases as the input (in this case, \(x\)) increases. On the other hand, a differentiable function is a function that has a derivative everywhere in its domain. The change in \(y\) (\(\Delta y\)), also known as the difference quotient, represents the change in the output of the function for a certain change in the input \(\Delta x\). Lastly, \(dy\) is the exact differential or the change in the function evaluated with the derivative.

Step 2: Comparing \(\Delta y\) and \(dy\)

In an increasing, differentiable function, if \(\Delta x > 0\), then it's also true that \(\Delta y > 0\) and \(dy > 0\). This is because, as \(x\) increases, \(y\) also increases (since \(f\) is increasing), and \(dy\) represents the rate of change of the function. However, \(\Delta y\) won't necessarily be greater than or equal to \(dy\). This is because \(\Delta y\) approximates the change in \(y\), while \(dy\) gives the exact change, calculated using the derivative.

Step 3: Concluding the statement's veracity

Because \(\Delta y\) can sometimes approximate the change in \(y\) closer than \(dy\) and other times not, the statement 'If \(y = f(x)\), \(f\) is increasing and differentiable, and \(\Delta x > 0\), then \(\Delta y \geq dy\)' is false.

Key Concepts

Increasing Function

An increasing function is one where the output, or the value of the function, grows as the input, typically represented by \(x\), increases. This means that for any two points \(x_1\) and \(x_2\) where \(x_1 < x_2\), the function value at \(x_1\) will be less than or equal to the function value at \(x_2\) (i.e., \(f(x_1) \leq f(x_2)\)).
An increasing function doesn't mean a steep rise, just a non-decreasing trend over the interval.

• It is common in various applications like economics and natural sciences.
• An increasing function will always have a positive slope, represented by its derivative.

Knowing if a function is increasing helps predict behavior as input values change, ensuring outputs move in a predictable direction.

Differentiable Function

A differentiable function is a function that has a derivative at every point in its domain. This derivative represents the function's rate of change with respect to the input variable. In simpler terms, you can draw a tangent line to the curve at any point, and this tangent line slope is the derivative.
To determine if a function is differentiable, consider:

• The function must be continuous over its domain. Sudden jumps or breaks disqualify it.
• No sharp corners or cusps should appear on its graph, as these would cause undefined derivatives.

Having a differentiable function means calculations involving rates of change and slopes are possible, allowing for detailed analysis of how changes in \(x\) impact \(y\).

Change in y (Δy)

The change in \(y\), denoted as \(\Delta y\), reflects how much the output of a function varies when the input changes by \(\Delta x\). It is essentially the difference between two function values, \(f(x + \Delta x) - f(x)\).
This change can give us an idea about how a function behaves over an interval:

• This difference may not perfectly match the actual rate of change, especially if the interval \(\Delta x\) is large.
• It is more of an approximation for larger \(\Delta x\), whereas \(dy\) aims to capture the exact change.

Understanding \(\Delta y\) is vital for studying how much a function's output changes over a certain input range.

Exact Differential (dy)

The exact differential, expressed as \(dy\), represents the infinitesimal change in a function based on its derivative. It is computed as \(dy = f'(x) \cdot dx\) where \(f'(x)\) is the derivative and \(dx\) is an infinitesimally small change in \(x\).
Unlike \(\Delta y\), \(dy\) provides a precise calculation of change:

• It uses the derivative to tell us exactly how \(y\) changes with a tiny \(dx\).
• This is essential for accurate calculations in physics and engineering where small changes matter.

Understanding \(dy\) helps in precision modeling and serves as a foundation for integral calculus, providing insights into continuous growth and changes.