Problem 45 - Essential Calculus

Question

Consider a function \(f\) such that \(f^{\prime}\) is increasing. Sketch graphs of \(f\) for (a) \(f^{\prime}<0\) and (b) \(f^{\prime}>0\)

Step-by-Step Solution

Answer

For \(f^{\prime}<0\), \(f\) starts high and descends with decreasing steepness. For \(f^{\prime}>0\), \(f\) starts low and ascends with increasing steepness. Both cases represent an increasing \(f'\) and its impact on \(f\).

Step 1 - Understanding the Effect of \(f^{\prime}<0\) on the Parent Function

When the derivative of a function (\(f^{\prime}\)) is negative, the function \(f\) is decreasing. The decreasing slope means that the curve descends as we move from left to right on the graph.

Step 2 - Sketching \(f\) for \(f^{\prime}<0\)

Since the derivative is negative and increasing, this means that the slope starts out very negative and gets closer to zero. Begin drawing the graph starting high and descending, but the rate of descent decreases as we move from left to right across the graph.

Step 3 - Understanding the Effect of \(f^{\prime}>0\) on the Parent Function

When the derivative of a function (\(f^{\prime}\)) is positive, the function \(f\) is increasing. The increasing slope means that the curve ascends as we move from left to right on the graph.

Step 4 - Sketching \(f\) for \(f^{\prime}>0\)

With an increasing positive derivative, the slope starts out slightly positive and gets steeper. Begin drawing the graph starting low and ascending, becoming steeper as we move from left to right across the graph.

Key Concepts

Derivative

The derivative is a central concept in calculus. It measures how a function changes as its input changes. Think of it as the function's rate of change or its "instantaneous slope." When we talk about the derivative of a function, often denoted as \( f' \), it helps us understand whether the function is increasing or decreasing at a particular point.

The sign of the derivative provides a lot of information.

• If \( f'(x) > 0 \), the function \( f \) is increasing at \( x \).
• If \( f'(x) < 0 \), the function \( f \) is decreasing at \( x \).
• An increasing derivative means the slope of \( f \) is gradually becoming steeper, either more positively or less negatively.

Every point on the graph of \( f \) can have a different derivative, influencing the curve’s shape.

Increasing Functions

Understanding increasing functions helps demonstrate the behavior of graphs over intervals.

• For functions where \( f'(x) > 0 \), the graph of the function moves upwards as you move from left to right.
• A steady increase in the derivative indicates that the slope is rising more sharply, which means the graph tilts up more steeply.
• Even when a function is mostly increasing, it can experience intervals of change in its rate of increase.

Considering our exercise, if \( f^{\prime} \) is positive, the function moves upward, and as \( f^{\prime} \) increases further, the steepness of this ascent increases, resulting in a steeper graph. The opposite happens when \( f^{\prime} \) is negative, which brings us to decreasing functions.

Graph Sketching

Sketching graphs by interpreting derivatives involves a few clear steps. It starts with understanding the derivative's influence on the graph’s slope and direction.

• If you know \( f'(x) < 0 \), start your sketch from a higher point and gradually move downwards since the function decreases.
• The descent should begin steep when \( f^{\prime} \) is very negative and gradually level off as \( f^{\prime} \) approaches zero.
• Inversely, for \( f'(x) > 0 \), the graph should start lower and move upward.
• As \( f^{\prime} \) increases, the slope steepens which requires drawing the graph line steeper over time.

To effectively sketch a graph, always consider the behavior over various intervals, noting how the derivative’s value and change affect the overall shape.