Question

Sketching a Graph Sketch the graph of a function \(f\) such that \(f^{\prime}>0\) for all \(x\) and the rate of change of the function is decreasing.

Short answer

The graph should be like an upward curve which is concave down. This curve indicates that the function is always increasing, but at a slowing rate as \(x\) increases.

Step-by-step solution

Draw the axes

Start by sketching the \(x\) and \(y\) axes. Normally these will be horizontal and vertical lines intersecting at the point known as the origin, which is also the point (0, 0).

Draw a curve representing function \(f\)

Given the conditions of the problem, function \(f\) should be drawn as a curve starting at the lower left, rising as it moves to the right. The curve should indicate that the function is always increasing, but the steepness should be decreasing, suggesting the rate of change of the function is decreasing.

Noting Specifics

It's also worth noting that there isn't one 'correct' graph: any graph that shows a function increasing everywhere but at a decreasing rate will be correct. There are no further specifics given, so there's no need to show the function passing through any particular points or having any specific slopes at certain points

Key concepts

Increasing Function

An increasing function is one that climbs upward as you move from left to right along the x-axis. This upward movement means that for any two points on the function, if the second point is to the right of the first, its y-value will be greater. In mathematical terms, if we have two points, say, \texttt{a} and \texttt{b}, such that \texttt{a
In the context of graphing, once you draw the x and y axes, you'll sketch a curve that consistently moves upward. However, there's an important nuance to note. An increasing function where the rate of change is decreasing will look like a hill that's becoming less steep as you climb it. This detail is crucial for both understanding the nature of functions and accurately representing them on a graph.

Rate of Change

The rate of change is a measurement of how much a quantity, such as a function value, changes with respect to a change in another quantity, such as time or distance. In calculus, when we speak of the rate of change for a function, we often refer to the derivative of the function. The derivative provides us with the function's instant rates of change, which geometrically equals the slopes of tangent lines along the function's curve. For the function mentioned in our original exercise, the rate of change is decreasing. This means that as you progress along the curve from left to right, the steepness of the curve diminishes.

While sketching a graph, representing a decreasing rate of change entails making the slope of the curve less pronounced at each point as you move to the right. It's like drawing a hill that flattens out gradually. This concept helps students determine the function's concavity and predict its future behavior.

Sketching Graphs

Sketching graphs correctly is a vital skill in calculus that helps visualize concepts and functions. When graphing functions like the one in our example, start by drawing the axes with a clear indication of the origin. Next, understanding the behavior of the function is key. The exercise calls for a graph where the function is always increasing, and yet the steepness is decreasing. So, the sketch will start low on the y-axis, then rise to the right, smoothly reducing its incline as it goes.

Remember, there's a lot of room for creativity as long as you maintain the overall characteristics of the function's behavior. No graph will look exactly the same, and that's alright. The aim is to get a visual representation that accurately shows all qualitative aspects of the function: increasing nature and decreasing steepness, or rate of change, with no need for specific points unless stated.

First Derivative Test

The first derivative test is a method used in calculus to determine where a function is increasing or decreasing. By taking the function's derivative, denoted as \( f'(x) \), you can identify the critical points where \( f'(x) = 0 \) or where the derivative does not exist. For our function, the first derivative is positive, \( f'(x) > 0 \), for all values of \( x \). This indicates that the function is increasing everywhere. However, the twist here is that the rate of change of the function is decreasing, suggested by a decreasing slope.

Where the derivative is decreasing, even though it's still positive, it tells us the function's slope is flattening out. Sketching this requires that at each subsequent point to the right, the slope should appear less steep than at the previous point. The first derivative test doesn't just tell us if a function is moving up or down—it also provides a way to predict the shape and direction of the graph's curve.