Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 29-34, sketch the graph of a continuous function fon \([0,6]\) that satisfies all the stated conditions. \(f(0)=1 ; f(6)=3\); increasing and concave down on \((0,6)\)

Step-by-Step Solution

Answer

Draw an increasing, concave down curve from (0, 1) to (6, 3).

Step 1: Understand the Problem

We need to sketch a graph of a function \( f \) that is defined on the interval \([0, 6]\). The function starts at 1, \(f(0) = 1\), and ends at 3, \(f(6) = 3\). It must be increasing and concave down on \((0, 6)\).

Step 2: Define the Properties of the Graph

The function needs to be increasing, meaning as \(x\) moves from 0 to 6, the value of \(f(x)\) should not decrease. It also needs to be concave down, indicating that its slope decreases over the interval \((0, 6)\) even though \(f(x)\) itself is increasing.

Step 3: Sketch the Graph

Start by plotting the point \((0, 1)\). Since the function is increasing and ends at \((6, 3)\), draw a curve that starts at \((0, 1)\) and ends at \((6, 3)\). Ensure the curve is consistently increasing. To achieve the concave down condition, make the curve arched like an upside-down "U" shape, so it is steep at the beginning and flattens as it approaches \(x = 6\).

Key Concepts

Continuous Function

A continuous function is a type of function without any interruptions or gaps over its domain. It means that you can draw the graph of this function without lifting your pencil off the paper. This feature is significant as it ensures there are no abrupt changes in the value of the function. For example, suppose you are evaluating the function between two points in a continuous interval, say from 0 to 6. Due to the continuity, every intermediate point in this interval is also covered by the function smoothly. This makes continuous functions predictable and easy to work with in calculus. With continuity, the outputs smoothly follow the inputs, which is critical when dealing with real-world phenomena like sound waves or temperature changes, where nature doesn’t make jumps. In the given exercise, the function is continuous over ":[0, 6] , meaning that no value within this interval is skipped, thus making it easier to sketch the function.

Concave Down

When a function is concave down, it resembles an upside-down bowl or an arch. This property tells us about the curvature of the graph. Specifically, a concave down function has a decreasing slope as you move along the curve. In practical terms, if you picture the tangent line to the curve at any point, it will always lie above the curve. This happens because an increasing function like in this case, although increasing in value, has a slowing rate of increase. Hence, a function that is concave down starts steep, but its growth slows down as it progresses. In the context of our exercise, being concave down means that the curve starts to flatten out as it approaches the endpoint ":[(6, 3)] . Drawing the function with this characteristic ensures that the visual representation accurately captures the diminishing rate of growth - it should look like a smile turned upside down!

Increasing Function

An increasing function is one where the output values get larger as you move along the x-axis. This means that as x increases, so do the values of the function. This is straightforward on a graph: if you pick any two points, ":[a] and ":[b] with ":[a < b] , it will always hold that ":[f(a) < f(b)] . This ensures that the function consistently rises from left to right. In our specific exercise, from ":[0, 6] , the function needs to rise from a height of 1 to a height of 3. This is essential because even though the function is concave down, the values continue to increase through the entire interval. The flattest part near the end point represents the completion of the increasing trend since the slope continues to decrease. An increasing function can be thought of like climbing a hill; you are always going upwards, even if the path becomes less steep as you move forward.