Question

Tangent lines and concavity Give an argument to support the claim that if a function is concave up at a point, then the tangent line at that point lies below the curve near that point.

Short answer

Explain your reasoning. Answer: Yes, when a function is concave up at a point, the tangent line at that point lies below the curve near that point. This is because the second derivative is positive, indicating an upward curvature, and the slope of the tangent line (first derivative) is increasing as we move from left to right on the graph. As a result, the function values will be above the tangent line near this point, which supports the claim that the tangent line lies below the curve near the point of tangency.

Step-by-step solution

Understanding Concavity

Concavity describes the curvature of a function. A function is said to be concave up if its graph bends upwards, indicating that its second derivative is positive. Conversely, a function is concave down if its graph bends downwards and its second derivative is negative. Step 2:

Understanding Tangent Lines

A tangent line is a straight line that touches a curve at a single point, without crossing it, and has the same slope as the curve at that point. The tangent line is used to approximate a function locally, around the point of tangency. Step 3:

Analyzing the Behavior of a Concave Up Function

When a function is concave up, its second derivative is positive. This means that the slope of the tangent line (first derivative) is increasing as we move from left to right on the graph. In other words, the slope of the function is getting steeper in the positive direction. Step 4:

Proving the Tangent Line Lies Below the Curve

Let's consider a function that is concave up at a point P. As we move to the right of this point, the slope of the tangent line is increasing. Hence, the tangent line (which has the same slope as the function) would approximate the function near this point more accurately in the direction of the increasing slope. Since the graph is bending upward, the function values will be above the tangent line near the point P, proving that the tangent line lies below the curve near that point. This argument supports the claim: if a function is concave up at a point, the tangent line at that point lies below the curve near that point.

Key concepts

Tangent Line

Let's first dive into the concept of a tangent line. Imagine a curve on a graph, like a mountain slope. The tangent line is like a perfectly fitted plank that just touches the mountain at one specific point. This touch is intimate, where the plank follows the mountain's slant exactly at that spot, without diving into it or jumping over it.

• It meets the curve at just one precise point.
• It shares the same steepness or slope as the curve at that exact point.

But why do we care about tangent lines? Well, they help us understand how a function behaves locally. Around the point of contact, the tangent line acts as a very close approximation of the slope of the curve, like how the slope of the mountain changes. This approximation is crucial for calculus and helps us in predicting and understanding the behavior of graphs around specific points.

Second Derivative

Now, let's talk about the second derivative. In calculus, derivatives are like a function's way of talking about motion, speed, or how fast things change. The first derivative tells us the speed or rate of change, mimicking a car's speedometer. But what if we are interested in how the speed itself is changing? This is where the second derivative steps in. If the first derivative is the speed, the second derivative is the acceleration - how fast that speed changes. It helps us understand the curvature, or bending, of a function.

• When the second derivative is positive, the function is bending upwards, like a valley.
• If it's negative, we're looking at a downward bend, like a hill.
• If the second derivative is zero, it indicates a possible inflection point where the curve might change its concavity.

Understanding the second derivative is key to determining the concavity of a graph and plays a vital role when analyzing how tangent lines relate to the overall shape of a graph.

Function Curvature

Finally, let's clarify what curvature means for a function. Imagine a flexible metal wire bent into different shapes. The way these shapes curve can relate to how functions curve on a graph. Curvature tells us about the bends and twists in the graph of a function, and it is connected deeply with the second derivative.

• When a function is concave up, it's like the crest of a wave; the graph bends upwards. The second derivative is positive, indicating the slope is increasing.
• When it is concave down, think of it like a bowl turning upside down; the graph bends downwards, and the second derivative is negative, signaling a decreasing slope.

Understanding curvature is important because it tells us not just about the direction of bending but also about the steepness and nature of the curve, aiding in visualizing how tangent lines will position relative to the entire function's graph.