Connecting a Function, Its First Derivative, and Its Second Derivative - AP Calculus AB Study Guide

The Core Idea: Connecting a Function, Its First Derivative, and Its Second Derivative

This topic is the culmination of our study of derivatives, where we synthesize information from a function's first and second derivatives to build a complete picture of the original function's behavior and sketch its graph. The first derivative, $f^{\prime }(x)$, tells us the rate of change of the function $f(x)$, revealing where it is rising (increasing) or falling (decreasing) and identifying potential peaks and valleys (relative extrema). The second derivative, $f^{\prime \prime }(x)$, tells us the rate of change of the first derivative, which in turn describes the concavity of the original function $f(x)$—whether the graph is shaped like a cup holding water (concave up) or spilling water (concave down).

By analyzing the signs of $f^{\prime }(x)$ and $f^{\prime \prime }(x)$, we can identify all the key features of $f(x)$: intervals of increase and decrease, locations of relative maxima and minima, intervals of upward and downward concavity, and points of inflection. This analytical process allows us to understand and accurately sketch the graph of a function without plotting dozens of points, relying instead on the powerful information encoded in its derivatives.

Key Definitions

The relationships between a function and its derivatives are defined by the following core principles:

• Increasing and Decreasing Behavior: The sign of the first derivative determines the direction of the function.

  • If $f^{\prime }(x)>0$ on an interval, then $f(x)$ is increasing on that interval.

  • If $f^{\prime }(x)<0$ on an interval, then $f(x)$ is decreasing on that interval.

• Concavity: The sign of the second derivative determines the curvature of the function's graph.

  • If $f^{\prime \prime }(x)>0$ on an interval, then $f(x)$ is concave up on that interval.

  • If $f^{\prime \prime }(x)<0$ on an interval, then $f(x)$ is concave down on that interval.

• Critical Points: A critical point of a function $f(x)$ is a point in the domain where $f^{\prime }(x)=0$ or $f^{\prime }(x)$ is undefined. These are the only candidates for the locations of relative (local) extrema.

• Possible Points of Inflection: A possible point of inflection for a function $f(x)$ is a point in the domain where $f^{\prime \prime }(x)=0$ or $f^{\prime \prime }(x)$ is undefined. A point of inflection occurs at such a point only if the concavity of $f(x)$ changes.

Understanding the Graphical Relationships

The most powerful application of this topic is understanding the connections between the graphs of $f(x)$, $f^{\prime }(x)$, and $f^{\prime \prime }(x)$. The features of one graph correspond directly to the features of the others.

Core Concepts & Rules

• The first derivative, $f^{\prime }(x)$, represents the rate of change (slope) of $f(x)$.

• The second derivative, $f^{\prime \prime }(x)$, represents the rate of change of $f^{\prime }(x)$, which corresponds to the concavity of $f(x)$.

• The sign of $f^{\prime }(x)$ determines if $f(x)$ is increasing ($+$) or decreasing ($-$).

• The sign of $f^{\prime \prime }(x)$ determines if $f(x)$ is concave up ($+$) or concave down ($-$).

• Critical points of $f(x)$ occur where $f^{\prime }(x)=0$ or $f^{\prime }(x)$ is undefined. These are potential locations for relative extrema.

• Possible points of inflection for $f(x)$ occur where $f^{\prime \prime }(x)=0$ or $f^{\prime \prime }(x)$ is undefined. A point of inflection is only confirmed if the sign of $f^{\prime \prime }(x)$ changes at that point.

• A complete analysis of the signs of both $f^{\prime }(x)$ and $f^{\prime \prime }(x)$ provides a robust framework for sketching the graph of $f(x)$.

Step-by-Step Example 1: Sketching a Curve from its Equation

Analyze and sketch the function $f(x)=x^3-6x^2+9x+1$.

Step 1: Find $f^{\prime }(x)$ and its critical points.

$$f^{\prime }(x)=3x^2-12x+9$$

Set $f^{\prime }(x)=0$ to find critical points:

$$3(x^2-4x+3)=0$$

$$3(x-1)(x-3)=0$$

The critical points are $x=1$ and $x=3$.

Step 2: Create a sign chart for $f^{\prime }(x)$ to determine intervals of increase/decrease.

We test values in the intervals $(-\infty ,1)$, $(1,3)$, and $(3,\infty )$.

• $f^{\prime }(0)=9$ (Positive)

• $f^{\prime }(2)=3(4)-12(2)+9=-3$ (Negative)

• $f^{\prime }(4)=3(16)-12(4)+9=9$ (Positive)

From this, we can conclude:

• Relative Maximum at $x=1$. The value is $f(1)=1-6+9+1=5$. Point: $(1,5)$.

• Relative Minimum at $x=3$. The value is $f(3)=27-54+27+1=1$. Point: $(3,1)$.

Step 3: Find $f^{\prime \prime }(x)$ and its possible points of inflection.

$$f^{\prime \prime }(x)=6x-12$$

Set $f^{\prime \prime }(x)=0$:

$$6x-12=0⟹x=2$$

The only possible point of inflection is at $x=2$.

Step 4: Create a sign chart for $f^{\prime \prime }(x)$ to determine intervals of concavity.

We test values in the intervals $(-\infty ,2)$ and $(2,\infty )$.

• $f^{\prime \prime }(0)=-12$ (Negative)

• $f^{\prime \prime }(3)=18-12=6$ (Positive)

Since the concavity changes at $x=2$, there is a point of inflection. The value is $f(2)=8-24+18+1=3$. Point: $(2,3)$.

Step 5: Sketch the graph.

Combine all the information:

• Relative Max at $(1,5)$

• Relative Min at $(3,1)$

• Point of Inflection at $(2,3)$

• Increasing on $(-\infty ,1)\cup (3,\infty )$

• Decreasing on $(1,3)$

• Concave down on $(-\infty ,2)$

• Concave up on $(2,\infty )$

The sketch would show a curve rising to $(1,5)$, falling through $(2,3)$ to $(3,1)$, and then rising again.

Step-by-Step Example 2: Analyzing a Function from the Graph of its Derivative

The graph of $f^{\prime }(x)$, the derivative of a continuous function $f(x)$, is shown below on the interval $[-4,5]$.

(Imagine a graph of $f^{\prime }(x)$ that starts at $(-4,-3)$, goes up to a peak at $(-2,3)$, crosses the x-axis at $x=0$, reaches a minimum at $(2,-3)$, and goes up to cross the x-axis at $x=4$, ending at $(5,3)$.)

Question 1: On what intervals is $f(x)$ increasing?

Solution:$f(x)$ is increasing where $f^{\prime }(x)>0$. Looking at the graph, $f^{\prime }(x)$ is above the x-axis on the intervals $(-4,0)$ and $(4,5)$.

Answer:$f(x)$ is increasing on $[-4,0]$ and $[4,5]$.

Question 2: At what x-values does $f(x)$ have a relative maximum? Justify your answer.

Solution: A relative maximum occurs where $f^{\prime }(x)$ changes from positive to negative. This happens at $x=0$.

Answer:$f(x)$ has a relative maximum at $x=0$ because $f^{\prime }(x)$ changes from positive to negative at $x=0$.

Question 3: On what intervals is $f(x)$ concave down?

Solution:$f(x)$ is concave down where $f^{\prime \prime }(x)<0$. This is equivalent to where the graph of $f^{\prime }(x)$ is decreasing. Looking at the graph, $f^{\prime }(x)$ is decreasing from its peak at $x=-2$ to its minimum at $x=2$.

Answer:$f(x)$ is concave down on the interval $(-2,2)$.

Question 4: At what x-values does $f(x)$ have a point of inflection?

Solution: A point of inflection on $f(x)$ occurs where the concavity changes, which is where $f^{\prime }(x)$ changes from increasing to decreasing or vice-versa. This corresponds to the relative extrema on the graph of $f^{\prime }(x)$.

Answer:$f(x)$ has points of inflection at $x=-2$ and $x=2$.

Using Your Calculator

While the analysis of functions, derivatives, and their connections is a primarily analytical skill, a graphing calculator is an excellent tool for verification and exploration. You can graph a function and its derivatives simultaneously to confirm their relationships.

To check the analysis of $f(x)=x^3-6x^2+9x+1$ from Example 1:

1. Press the Y= button.

2. In $Y_1$, enter the function: $X^3-6X^2+9X+1$.

3. In $Y_2$, enter the first derivative. You can use the calculator's numerical derivative function. On a TI-84, press MATH and select 8:nDeriv(. Enter it as `nDeriv(Y₁, X, X). This tells the calculator to graph the derivative of $Y₁ with respect to $X$.

4. In $Y_3$, enter the second derivative: nDeriv(Y₂, X, X).

5. Press GRAPH. You may need to adjust the WINDOW to see all key features.

How to use the graph for verification:

• Observe where the graph of $Y_2$ (the derivative) is above the x-axis ($f^{\prime }(x)>0$). You will see that the graph of $Y_1$ (the original function) is increasing in those same intervals.

• Find where $Y_2$ crosses the x-axis (e.g., using $CALC:zero$). These x-values should correspond to the relative extrema on the $Y_1$ graph.

• Observe where the graph of $Y_3$ (the second derivative) is positive. You will see that the graph of $Y_1$ is concave up in those intervals.

• Find where $Y_3$ crosses the x-axis. This x-value should correspond to the point of inflection on the Y₁` graph. ## AP Exam Quick Hit ### Common Question Types - **Given the graph of $f'(x):** This is the most common format. You will be asked to determine properties of $f(x)$, such as intervals where $f(x)$ is increasing/decreasing, the location of relative extrema, intervals of concavity, and the location of points of inflection.

• Given an equation for $f(x)$ or $f^{\prime }(x)$: You will be asked to find and, critically, justify the locations of relative extrema or points of inflection. The justification must be based on the sign changes of the appropriate derivative.

• Given a table of values for $f(x)$ and $f^{\prime }(x)$: You may be asked to determine if $f(x)$ is increasing or decreasing, or to approximate $f^{\prime \prime }(x)$ using the average rate of change of $f^{\prime }(x)$ to draw conclusions about concavity.

Common Mistakes

• Confusing the properties of $f$, $f^{\prime }$, and $f^{\prime \prime }$: A classic error is stating "$f(x)$ is increasing because the graph of $f(x)$ is positive." The correct statement is "$f(x)$ is increasing because the graph of $f^{\prime }(x)$ is positive." Be precise in your language.

• Providing insufficient justification: Stating "$f(x)$ has a relative max at $x=c$ because $f^{\prime }(c)=0$" is not enough. You must state that $f^{\prime }(x)$ changes from positive to negative at $x=c$.

• Mixing up critical points and points of inflection: Students often set $f^{\prime }(x)=0$ when looking for points of inflection. Remember: critical points come from $f^{\prime }(x)$, while points of inflection come from $f^{\prime \prime }(x)$.

• Assuming a point of inflection exists because $f^{\prime \prime }(c)=0$: A point of inflection requires a change in concavity. You must verify that the sign of $f^{\prime \prime }(x)$ changes at $x=c$. For example, $g(x)=x^4$ has $g^{\prime \prime }(0)=0$, but it is concave up on both sides of $x=0$, so there is no point of inflection.

• Misinterpreting the graph of $f^{\prime }(x)$: When given the graph of $f^{\prime }(x)$, students mistakenly look at where the $f^{\prime }(x)$ graph is increasing to determine where $f(x)$ is increasing. You must look at where the $f^{\prime }(x)$ graph is positive (above the x-axis) to determine where $f(x)$ is increasing.