A tutorial on how to use the first and second derivatives, in calculus, to study the properties of the graphs of functions.
In what follows, \( C \) represents constant of integration where applicable.
Theorems
To graph functions in calculus we first review several theorems. Three theorems are used to find maxima and minima using first and second derivatives, and they are combined here with two additional theorems to study complete function graphs.
Theorem 4: If \( f \) is defined on the interval \( [I_1, I_2] \) and differentiable on the interval \( (I_1, I_2) \), then:
- 4.a: If \( f'(x) > 0 \) on \( (I_1, I_2) \), then \( f \) is increasing on \( [I_1, I_2] \).
- 4.b: If \( f'(x) < 0 \) on \( (I_1, I_2) \), then \( f \) is decreasing on \( [I_1, I_2] \).
- 4.c: If \( f'(x) = 0 \) on \( (I_1, I_2) \), then \( f \) is constant on \( [I_1, I_2] \).
Theorem 5: If \( f \) is twice differentiable on the interval \( (I_1, I_2) \), then:
- 5.a: If \( f''(x) > 0 \) on \( (I_1, I_2) \), then \( f \) is concave up on \( [I_1, I_2] \).
- 5.b: If \( f''(x) < 0 \) on \( (I_1, I_2) \), then \( f \) is concave down on \( [I_1, I_2] \).
Examples with Detailed Solutions
We present examples of graphing functions using the theorems from "using first and second derivatives" and Theorems 4 and 5 above.
Click on each solution box to view the step-by-step curve sketching guide.
Example 1
Use first and second derivative theorems to graph the function \( f \) defined by:
\[ f(x) = x^2 \]Show Solution to Example 1
Step 1: Find the first derivative, stationary points, and the sign of \( f'(x) \) to determine intervals of increase/decrease.
\[ f'(x) = 2x \]Stationary points are solutions to \( f'(x) = 2x = 0 \), giving \( x = 0 \).
\( f'(x) \) is negative on \( (-\infty, 0) \), so \( f \) decreases there (Theorem 4.b). \( f'(x) \) is positive on \( (0, +\infty) \), so \( f \) increases there (Theorem 4.a). By Theorem 2, \( f \) has a minimum at \( x = 0 \).
Step 2: Find the second derivative, its sign, and concavity information.
\[ f''(x) = 2 \quad (\text{always positive}) \]This confirms a minimum at \( x = 0 \) since \( f''(0) = 2 > 0 \). The graph is concave up on \( (-\infty, +\infty) \) (Theorem 5.a).
Step 3: Find intercepts and extrema.
\( y \)-intercept: \( f(0) = 0 \). \( x \)-intercepts: \( x^2 = 0 \implies x = 0 \).
Minimum point is at \( (0, 0) \).
Step 4: Sketch the graph.
As \( x \to \pm\infty \), \( f(x) = x^2 \to +\infty \).
Example 2
Use first and second derivative theorems to graph the function \( f \) defined by:
\[ f(x) = x^3 - 4x^2 + 4x \]Show Solution to Example 2
Step 1: First derivative and stationary points.
\[ f'(x) = 3x^2 - 8x + 4 \]Solving \( 3x^2 - 8x + 4 = 0 \) yields critical points \( x = 2 \) and \( x = \dfrac{2}{3} \).
Step 2: Second derivative and inflection points.
\[ f''(x) = 6x - 8 \]Solving \( 6x - 8 = 0 \) gives \( x = \dfrac{4}{3} \), where concavity changes (inflection point).
Step 3: Intercepts and end behavior.
\( y \)-intercept: \( f(0) = 0 \).
\( x \)-intercepts: factor \( x(x^2 - 4x + 4) = x(x - 2)^2 = 0 \), giving \( x = 0 \) and \( x = 2 \) (multiplicity 2).
As \( x \to +\infty \), \( f(x) \to +\infty \). As \( x \to -\infty \), \( f(x) \to -\infty \).
Step 4: Table of signs and graph.
