Problem 48 - Calculus Early Transcendentals

Question

Functions from graphs Use the graphs of \(f^{\prime}\) and \(f^{\prime \prime}\) to find the critical points and inflection points of \(f\), the intervals on which \(f\) is increasing and decreasing, and the intervals of concavity. Then graph \(f\) assuming \(f(0)=0\)

Step-by-Step Solution

Answer

Question: Given the graphs of \(f'(x)\) and \(f''(x)\), find the critical points and inflection points of the function \(f(x)\), and determine the intervals of increase, decrease, concavity, and graph the function with the condition \(f(0) = 0\).

Step 1: Find the critical points of \(f\)

Analyze the graph of \(f'(x)\) and locate the points where it is equal to 0 or undefined. These are the critical points of \(f(x)\).

Step 2: Determine intervals of increase and decrease

Using the graph of \(f'(x)\), identify the intervals where it is positive and negative. The function \(f(x)\) will be increasing on intervals where \(f'(x) > 0\) and decreasing on intervals where \(f'(x) < 0\).

Step 3: Find the inflection points of \(f\)

Now, analyze the graph of \(f''(x)\) and locate the points where it is equal to 0 or undefined. These are the inflection points of \(f(x)\).

Step 4: Determine intervals of concavity

Utilizing the graph of \(f''(x)\), identify the intervals where it is positive and negative. The function \(f(x)\) will be concave up on intervals where \(f''(x) > 0\) and concave down on intervals where \(f''(x) < 0\).

Step 5: Graph the function \(f(x)\) with the given information

Using \(f(0) = 0\), the critical points, inflection points, intervals of increase and decrease, and intervals of concavity, graph the function \(f(x)\). In conclusion, after analyzing the given graphs of \(f'(x)\) and \(f''(x)\), we have found the critical and inflection points of the function \(f(x)\). Additionally, we have determined under what intervals the function increases, decreases, is concave up, and is concave down. Lastly, we have graphed the function \(f(x)\) under the condition \(f(0) = 0\).

Key Concepts

Increasing and Decreasing Intervals

Understanding when a function is climbing uphill or sliding downward is a foundational concept in calculus. This is described as identifying the increasing and decreasing intervals of a function. We gain this insight by looking at the first derivative of the function, denoted as f'(x).

When f'(x) > 0, the function f(x) is increasing because the slope of the tangent to the curve is positive. Imagine being on a hike, and the path slopes upwards; that's an increasing interval. Conversely, when f'(x) < 0, the function f(x) is decreasing, akin to descending down a hill. If the graph of the function isn't available, plotting f'(x) and checking where it crosses the x-axis (changing from positive to negative or vice versa) can help us determine these intervals.

Concavity and Inflection Points

The curvature of a graph, or its concavity, tells us how the function bends and twists. To determine this, we look at the second derivative of the function, signified by f''(x).

If f''(x) > 0, the function exhibits concave upward behaviour, resembling the open basin of a spoon. On the flip side, if f''(x) < 0, the function is concave downward like an arch. Inflection points occur where the concavity switches, essentially where f''(x) = 0 or is undefined. These points are significant as they signify a change in the curve's direction – from a frown to a smile, per se. Understanding concavity is essential, not just for plotting graphs, but for optimizing functions in economics, physics, and engineering.

Graphing Functions

Graphing a function is a powerful tool that provides a visual representation of its behaviour over a range of values. To sketch the function accurately, we combine the information from its derivatives.

Start by marking critical points (where f'(x) = 0 or is undefined) on a graph to signify where the function's increase-decrease behaviour changes. Then, plot points of inflection where the concavity of the function changes (where f''(x) = 0 or is undefined). Finally, using the intervals where the function is increasing or decreasing and where it is concave up or down, piece together the shape of the graph. This visual representation provides a clearer understanding of the function's properties and is invaluable not only for solving calculus problems but also for modelling and analyzing real-world scenarios.

First and Second Derivatives

The first and second derivatives of a function are like the heartbeat of calculus, providing rich insights into the function's behaviour. The first derivative, f'(x), describes the rate of change or the slope of the function at any point. It identifies where the function has peaks, valleys, or flattens out – these are the critical points.

The second derivative, f''(x), speaks about the acceleration or the bend of the graph of the function. It helps determine the shape of the graph by identifying concavity and points of inflection. Understanding these derivatives is crucial in various applications, from predicting optimum profit in business to understanding motion in physics. The intertwined dance of the first and second derivatives gives a comprehensive picture of a function's dynamics.