Problem 12 - Applied Calculus

Question

Sketch the graph of a function whose first derivative is everywhere negative and whose second derivative is positive for some \(x\) -values and negative for other \(x\) -values.

Step-by-Step Solution

Answer

A decreasing curve that changes concavity.

Step 1: Understand the First Derivative

The first derivative of a function indicates the slope of the tangent line at any given point on the graph. If the first derivative is everywhere negative, it means the function is always decreasing. Therefore, the graph must slope downwards.

Step 2: Interpret the Second Derivative

The second derivative provides information about the concavity of the graph. If the second derivative is positive for some values of \( x \), the graph is concave up (like a cup) in those intervals. Conversely, if it is negative for other \( x \) values, the graph is concave down in those intervals.

Step 3: Sketch the Graph

Begin by sketching a decreasing curve. Incorporate regions of concavity based on the second derivative: for intervals where the second derivative is positive, the graph should appear like an upward-facing "U", while in intervals where it is negative, it should appear as a downward-facing "U".

Step 4: Verify the Conditions

Ensure that the drawn graph is consistently decreasing, confirming the first derivative is negative throughout. Double-check that the graph switches between concave up and concave down as dictated by the second derivative conditions.

Key Concepts

First Derivative

In calculus, the first derivative of a function helps us understand how the function's output changes as its input changes. Imagine you're walking along a path that represents the function. Wherever the path slopes downwards, the first derivative is negative, signaling a decreasing function. This means no matter where you are on the graph, you are always moving downhill.

When the exercise specifies that the first derivative is negative everywhere, it indicates that the function never increases at any point on its graph. It's like walking down a hill that's consistently sloping downwards. This pattern is critical in shaping the graph of the function, setting a clear expectation that the graph is continuously sloping down.

Second Derivative

The second derivative offers us insights into the curvature or "shape" of the graph. It tells us how the slope of the tangent line itself is changing as you move along the curve. Imagine the second derivative as giving you the 'bends' of the path.

If the second derivative is positive, the graph bends upwards, much like a smile or a cup. This is known as concave up. In contrast, if the second derivative is negative, the graph bends downwards, forming a shape similar to a frown or a cap. This is referred to as concave down.

The second derivative helps us map out intervals where the graph changes its concavity, which is crucial in creating accurate sketches of the function.

Concavity

Concavity is all about how the graph curves. It's like assessing whether a road bends upwards or downwards as you travel along it. You'll encounter two possibilities here:

• Concave Up: When the second derivative is positive, and the graph curves upwards, like a cup ready to hold something.
• Concave Down: When the second derivative is negative, and the graph curves downwards, like a frown.

Understanding these curvatures is vital when sketching graphs, as it helps you envision the 'hills and valleys' of the function visually. The concept of concavity helps in predicting how the graph might look across different intervals, especially when combined with the principle of a decreasing function.

Decreasing Function

A decreasing function is one where, as the input increases, the output decreases. Visualize a line where, as you move to the right (increasing x-values), you find yourself consistently moving downwards.

This decrease is confirmed through having a negative first derivative everywhere on the graph. In our exercise, the whole function is decreasing, indicating there's never a point where the graph increases.

This is a significant aspect when sketching the graph, as it defines that the slope of the graph is always declining, regardless of how the graph might bend upwards or downwards due to concavity.