Problem 40 - Calculus with Concepts in Calculus

Question

In cach of the following, draw the graph of a continuous function \(f\) having the given propertics. a. \(f\) is increasing and its graph is concave upward on \((-\infty, 0)\), and \(f\) is decreasing and its graph is concave downward on \((0, \infty)\). b. \(f\) is decreasing and its graph is concave upward on \((-\infty, 1), f\) is increasing and its graph is concave upward on \((1,2)\), and \(f\) is decreasing and its graph is concave upward on \((2, \infty)\). c. \(f\) is decreasing and its graph is concave upward on \((-\infty, 1), f\) is increasing and its graph is concave upward on \((1,2)\), and \(f\) is increasing and its graph is concave downward on \((2, \infty)\). d. \(f\) is decreasing and its graph is concave downward on \((-\infty, 0), f\) is increasing and its graph is concave downward on \((0,1), f\) is increasing and its graph is concave upward on \((1,5)\), and \(f\) is decreasing and its graph is concave downward on \((5, \infty)\)

Step-by-Step Solution

Answer

Sketch graphs according to given increase/decrease and concavity intervals.

Step 1: Understand the Terms

To solve this problem, it's important to know the definitions. An 'increasing function' means that as you move from left to right along the function's graph, it rises. A 'decreasing function' means it falls. A function is 'concave upward' if it makes a 'U' shape and 'concave downward' if it makes an 'n' shape.

Step 2: Plot Graph for Part a

Draw a graph with a switch at the y-axis (x=0). For \((-\infty, 0)\), draw an increasing curve that is concave upward (looks like a left half of a U shape). For \((0, \infty)\), draw a decreasing curve that is concave downward (looks like a right half of an n shape). These two should connect smoothly at the y-axis.

Step 3: Plot Graph for Part b

Focus on three intervals: \((-\infty, 1)\), \((1, 2)\), and \((2, \infty)\). Draw a concave upward decrasing shape for \((-\infty, 1)\). For \((1,2)\), draw a concave upward increasing shape. Finally, for \((2, \infty)\), continue to make a concave upward decreasing shape. Transition points are smooth at x = 1 and x = 2.

Step 4: Plot Graph for Part c

Create a graph with three sections: \((-\infty, 1)\), \((1, 2)\), \((2, \infty)\). For \((-\infty, 1)\), display a decreasing concave upward section. On \((1, 2)\), switch to an increasing concave upward section. For \((2, \infty)\), draw an increasing concave downward section. Make connections between the segments fluid at x = 1 and x = 2.

Step 5: Plot Graph for Part d

For this graph, consider four parts: \((-\infty, 0)\), \((0, 1)\), \((1, 5)\), and \((5, \infty)\). Begin with a decreasing concave downward segment for \((-\infty, 0)\). Next, switch to an increasing concave downward curve from \((0, 1)\). Then, continue with an increasing concave upward segment from \((1, 5)\). Lastly, finish with a decreasing concave downward section for \((5, \infty)\). Ensure smooth transitions at x = 0, x = 1, and x = 5.

Key Concepts

Increasing and Decreasing Functions

When talking about increasing and decreasing functions, we refer to how the graph of a function moves as you follow it from left to right.

• An increasing function is one where, as you move horizontally from left to right (with increasing values of x), the function values (y-values) rise. Think about climbing up a hill.
• A decreasing function, on the other hand, is where, as you move right, the function values fall. This is similar to going down a slope.

Understanding these terms helps you draw graphs with specific behaviors.
To illustrate, if a function is increasing on the interval \((-\infty, 0)\), its graph travels upwards in that span. Conversely, if it is decreasing on \((0, \infty)\), the graph moves downward past the y-axis.

Concavity of Functions

Concavity gives us a deeper insight into the nature of a function's curve. It tells us about the "bending" behavior of the graph.

• A function is concave upward when the graph looks like a part of a "U" shape. It suggests that the slope is increasing, and the curve opens upwards.
• A concave downward function resembles a part of an "n" shape, where the slope is decreasing, and the curve opens downwards.

These descriptions are crucial for predicting and sketching graph behaviors in response to changes. For instance, in the problem, graph concavity changes when transitioning from one interval to another, providing valuable insight into overall function shape.

Continuous Functions

Continuous functions are a staple concept in calculus and other areas of mathematics. A function is said to be continuous if its graph is unbroken - no gaps, jumps, or holes as you trace it.

• This means you can draw the graph of the function without lifting your pen.
• Continuity also implies the function is defined and stable for all numbers in a certain range.

For instance, in the described problems, the function continues seamlessly from one interval to the next. This aspect of continuity ensures that transitions in behavior (like from increasing to decreasing) are smooth, contributing to a well-behaved graph that maintains a continuous flow.