Introduction

This topic synthesizes all derivative concepts to sketch and analyze function graphs. The essential knowledge tells us that key features of functions and their derivatives can be identified and related to their graphical, numerical, and analytical representations and graphical, numerical, and analytical information from f' and f'' can be used to predict and explain the behavior of f. This culminating topic connects everything we've learned about derivatives to understand complete function behavior.

Relating f, f', and f'' Graphically

The Complete Relationship Chart

$Feature of f Increasing Decreasing Local max/min Concave up Concave down Inflection point Information from f^{'} f^{'} > 0 f^{'} < 0 f^{'} = 0 with sign change f^{'} increasing f^{'} decreasing f^{'} has local extremum Information from f^{''} - - f^{''} < 0 (max) or f^{''} > 0 (min) f^{''} > 0 f^{''} < 0 f^{''} = 0 with sign change$

Understanding the Connections

From $f$ to $f^{'}$ :

Horizontal tangent on $f$ → zero on $f^{'}$
Rising portion of $f$ → positive portion of $f^{'}$
Falling portion of $f$ → negative portion of $f^{'}$
Steep slope on $f$ → large magnitude on $f^{'}$

From $f^{'}$ to $f^{''}$ :

Horizontal tangent on $f^{'}$ → zero on $f^{''}$
Rising portion of $f^{'}$ → positive portion of $f^{''}$
Falling portion of $f^{'}$ → negative portion of $f^{''}$

Sketching f Given f'

Procedure for Sketching from f'

Identify where $f^{'} = 0$ (critical points of $f$ )
Determine where $f^{'} > 0$ ( $f$ increasing) and $f^{'} < 0$ ( $f$ decreasing)
Locate local extrema (where $f^{'}$ changes sign)
Sketch $f$ using this information and any given point

Example 1:

Given the graph of $f^{'}$ below, sketch a possible graph of $f$ given that $f (0) = 1$ .

Suppose $f^{'}$ has:

$f^{'} (x) < 0$ for $x < - 1$
$f^{'} (- 1) = 0$
$f^{'} (x) > 0$ for $- 1 < x < 2$
$f^{'} (2) = 0$
$f^{'} (x) < 0$ for $x > 2$

Analysis:

$f$ is decreasing on $(- \infty, - 1)$
$f$ has a local minimum at $x = - 1$
$f$ is increasing on $(- 1, 2)$
$f$ has a local maximum at $x = 2$
$f$ is decreasing on $(2, \infty)$

Sketch: Start at $(0, 1)$ , decrease to the left to a valley at $x = - 1$ , increase through $(0, 1)$ to a peak at $x = 2$ , then decrease to the right.

Sketching f' Given f

Procedure for Sketching the Derivative

Identify horizontal tangents on $f$ (zeros of $f^{'}$ )
Mark positive $f^{'}$ where $f$ increases, negative where $f$ decreases
Identify local extrema of $f$ (where $f^{'}$ crosses the x-axis)
Note that steeper slopes on $f$ mean larger magnitude on $f^{'}$

Example 2:

Sketch $f^{'}$ given the graph of $f (x) = x^{3} - 3 x^{2}$ .

Key features of $f$ :

Local maximum at $x = 0$
Local minimum at $x = 2$
Increasing for $x < 0$ and $x > 2$
Decreasing for $0 < x < 2$

Analysis for f':

$f^{'} (0) = 0$ (horizontal tangent at max)
$f^{'} (2) = 0$ (horizontal tangent at min)
$f^{'} > 0$ for $x < 0$ and $x > 2$
$f^{'} < 0$ for $0 < x < 2$
$f^{'}$ crosses x-axis at $x = 0$ from above (max)
$f^{'}$ crosses x-axis at $x = 2$ from below (min)

Sketch of f': Starts positive, crosses zero at $x = 0$ , stays negative until $x = 2$ , then becomes positive again.

Incorporating Concavity Information

Using f'' to Complete the Picture

Example 3:

Given $f (x) = x^{4} - 4 x^{3} + 4$ , create a complete analysis and sketch.

Find $f^{'} (x)$ : $f^{'} (x) = 4 x^{3} - 12 x^{2} = 4 x^{2} (x - 3)$ Critical points: $x = 0, 3$

Find $f^{''} (x)$ : $f^{''} (x) = 12 x^{2} - 24 x = 12 x (x - 2)$ Possible inflection points: $x = 0, 2$

Analyze increasing/decreasing:

$f^{'} > 0$ on $(3, \infty)$ → increasing
$f^{'} < 0$ on $(- \infty, 0)$ and $(0, 3)$ → decreasing
At $x = 0$ : no sign change in $f^{'}$ → neither max nor min
At $x = 3$ : $f^{'}$ changes from $-$ to $+$ → local minimum

Analyze concavity:

$f^{''} > 0$ on $(- \infty, 0)$ and $(2, \infty)$ → concave up
$f^{''} < 0$ on $(0, 2)$ → concave down
Inflection points at $x = 0$ and $x = 2$

Key points:

$f (0) = 4$

$f (2) = 16 - 32 + 4 = - 12$

$f (3) = 81 - 108 + 4 = - 23$

Sketch: Decreasing and concave up until $x = 0$ , then still decreasing but concave down until $x = 2$ , then decreasing and concave up until local min at $x = 3$ , then increasing and concave up.

Visual Exploration with Desmos

Interactive Function Analysis

To truly understand these relationships, explore these functions in Desmos:

Function 1: Enter into Desmos:

$f (x) = x^{3} - 6 x^{2} + 9 x + 1$

Then add the derivative and second derivative:

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

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

Observations to make:

Where is $f^{'} (x) = 0$ ? What happens to $f (x)$ at those points?
Where is $f^{'} (x) > 0$ ? Is $f (x)$ increasing there?
Where is $f^{''} (x) = 0$ ? What happens to the concavity of $f (x)$ ?
Where is $f^{''} (x) > 0$ ? Is $f^{'} (x)$ increasing there?

Function 2: Enter into Desmos:

$g (x) = sin (x) + 0.5 sin (2 x)$

Then add:

$g^{'} (x) = cos (x) + cos (2 x)$

$g^{''} (x) = - sin (x) - 2 sin (2 x)$

Observations to make:

How many local extrema does $g (x)$ have on $[0, 2 π]$ ?
Where does $g^{'} (x) = 0$ ? Do these match the extrema of $g (x)$ ?
Where does $g (x)$ change concavity? Does $g^{''} (x) = 0$ at those points?
Compare the "waviness" of $g (x)$ , $g^{'} (x)$ , and $g^{''} (x)$ .

Complete Curve Sketching

Comprehensive Analysis Procedure

Step 1: Find the domain

Step 2: Find intercepts

Step 3: Find critical points ( $f^{'} (x) = 0$ or undefined)

Step 4: Determine intervals of increase/decrease

Step 5: Classify critical points (max, min, or neither)

Step 6: Find possible inflection points ( $f^{''} (x) = 0$ or undefined)

Step 7: Determine concavity

Step 8: Sketch using all information

Example 4:

Sketch $f (x) = \frac{x ^{2}}{x - 2}$ .

Domain: $x \neq = 2$

Intercepts: $f (0) = 0$ ( $y$ -intercept and $x$ -intercept)

First derivative:

$f^{'} (x) = \frac{( x - 2 ) ( 2 x ) - x ^{2} ( 1 )}{( x - 2 ) ^{2}} = \frac{2 x ^{2} - 4 x - x ^{2}}{( x - 2 ) ^{2}} = \frac{x ^{2} - 4 x}{( x - 2 ) ^{2}} = \frac{x ( x - 4 )}{( x - 2 ) ^{2}}$

Critical points: $x = 0, 4$

Increasing/Decreasing:

$f^{'} > 0$ on $(- \infty, 0)$ and $(4, \infty)$
$f^{'} < 0$ on $(0, 2)$ and $(2, 4)$

Extrema:

Local maximum at $x = 0$
Local minimum at $x = 4$ where $f (4) = \frac{16}{2} = 8$

Second derivative: (complex calculation - focus on sign analysis) $f^{''} > 0$ for $x > 2$ $f^{''} < 0$ for $x < 2$

Vertical asymptote: $x = 2$

Sketch: Local max at origin, decreasing toward vertical asymptote at $x = 2$ from left, approaching from right to local min at $(4, 8)$ , then increasing.

Matching Graphs

Identifying Corresponding Derivatives

Example 5:

Match each function with its derivative.

Given three graphs of functions and three graphs of derivatives, use these strategies:

Where function has horizontal tangent, derivative crosses $x$ -axis
Where function increases, derivative is positive
Where function decreases, derivative is negative
Steeper function means larger magnitude derivative

Example 6:

Given $f^{'}$ is a parabola opening upward with zeros at $x = - 1$ and $x = 3$ , describe $f$ .