Curve Sketching with Derivatives: A Guide to Single-Variable Calculus¶

Curve sketching is an essential skill for understanding graphs of functions and solving problems involving rates of change or optimization. In single-variable calculus, we use derivatives to analyze the behavior of a function's graph at various points. The first derivative gives us information about increasing and decreasing intervals as well as local maxima/minima, while the second derivative helps us determine concavity and points of inflection.

In this guide, we will provide a step-by-step procedure to curve sketch a function using derivatives. We will explain how to find the domain, intercepts, asymptotes (if any), first derivative, critical points, second derivative, intervals of increase/decrease and concavity, and then assemble a sketch. To illustrate the process, we will provide two fully worked examples.

Step-by-step Procedure for Curve Sketching with Derivatives:¶

Find the Domain: Determine the domain of the function by examining the expression for the function and checking if there are any restrictions on the input variable(s).
Find the Intercepts: Determine the \(x\)-intercept(s) of the function by setting the function equal to zero and solving for \(x\). If the function has a constant term, also find the \(y\)-intercept by setting \(x=0\) and solving for \(y\).
Find Asymptotes: Determine if the function has any asymptotes. An asymptote is a line that the graph of the function approaches without ever touching it. Common types of asymptotes include horizontal asymptotes (limiting values as \(x \to \pm \infty\)) and vertical asymptotes (at which the function becomes unbounded).
Find the First Derivative: Compute the first derivative of the function by taking the derivative of the original expression with respect to its independent variable(s). Use the power rule, product rule, quotient rule, or chain rule as needed.
Find Critical Points: Locate any critical points where the first derivative equals zero or is undefined. These points are local maxima/minima and may affect the shape of the graph in their neighborhoods. Use the first derivative test (increasing if positive, decreasing if negative) to determine the behavior around these points.
Find the Second Derivative: Compute the second derivative of the function by taking the derivative of the first derivative with respect to its independent variable(s). This will help us analyze the concavity and inflection points of the graph.
Analyze Intervals of Increase/Decrease and Concavity: Use the sign of the first and second derivatives to determine the intervals where the function is increasing or decreasing and whether it has a positive, negative, or undefined concavity (convex up, concave down, or neither). This information will help us construct the graph more accurately.
Assemble the Sketch: Now that we have all the necessary information, we can sketch the function's graph by combining our findings from steps 1-7. Be sure to account for any asymptotes and points of inflection when assembling the final sketch.

Example 1: Finding the Domain, Intercepts, Asymptotes, Derivatives, Critical Points, Concavity, and Sketching the Graph of \(f(x) = x^3 - 6x^2 + 9x\)¶

Step-by-step Analysis:¶

Find the Domain: The function is a polynomial with no restrictions on its domain. Therefore, its domain is \((-\infty, \infty)\).
Find the Intercepts: To find the \(x\)-intercept(s), we need to set \(f(x) = 0\):

[x^3 - 6x^2 + 9x = 0]

Using synthetic division, we find that this equation has no real solutions. Thus, there are no \(x\)-intercepts. To find the \(y\)-intercept, we set \(x=0\):

[f(0) = (0)^3 - 6(0)^2 + 9(0) = 0]

Therefore, the \(y\)-intercept is \((0, 0)\).

Find Asymptotes: The function does not have any horizontal or vertical asymptotes.
Find the First Derivative: The first derivative of \(f(x)\) is:

[f'(x) = 3x^2 - 12x + 9]

Find Critical Points: To find critical points, we need to solve for where \(f'(x) = 0\):

[3x^2 - 12x + 9 = 0]

Using the quadratic formula:

[x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(3)(9)}}{2(3)} = \frac{12 \pm \sqrt{144 - 108}}{6} = \frac{12 \pm 6}{6} = 2, \frac{4}{3}]

Thus, the critical points are \(x=2\) and \(x=\frac{4}{3}\).

Find the Second Derivative: The second derivative of \(f(x)\) is:

[f''(x) = 6x - 12]

Analyze Intervals of Increase/Decrease and Concavity: To analyze the intervals, we need to find where \(f'(x) = 0\) or is undefined:

[-\frac{2x}{(x^2 + 4)^2} < 0]

The function is decreasing everywhere.

Assemble the Sketch: Combining our findings from steps 1-7, we sketch the graph of \(f(x) = \frac{1}{x^2 + 4}\):

Example 2: Finding the Domain, Intercepts, Asymptotes, Derivatives, Critical Points, Concavity, and Sketching the Graph of \(f(x) = x^3 - 6x^2 + 9x\)¶