Chapter 13 — Graphing with First and Second Derivatives

13.1 Learning Objectives

By the end of this chapter, students should be able to:

• Interpret first derivatives graphically.
• Interpret second derivatives graphically and physically.

Determine where functions:

• increase
• decrease

flatten

curve upward

curve downward

Identify critical points.

Identify local maxima and minima.

Understand concavity intuitively.

Locate possible inflection points.

Connect derivatives to graph shape.

Analyze functions structurally rather than memorizing isolated facts.

13.2 Big Picture — Derivatives Reveal Shape

Earlier chapters taught students:

how to compute derivatives

Now we ask a deeper question:

What do derivatives TELL us about graphs?

This chapter transforms derivatives from:

symbolic procedures

into:

geometric insight.

Derivatives reveal:

steepness

direction

curvature

turning behavior

acceleration

local shape

Calculus now begins reading graphs dynamically.

13.3 The First Derivative — Measuring Slope

Recall:

f

′

(x)

measures:

tangent slope

instantaneous rate of change

But slope tells us much more than steepness.

Slope also reveals:

whether a graph rises

or

falls

This becomes foundational for graph analysis.

13.4 Positive First Derivative

Suppose:

f

′

(x)>0

This means:

tangent slopes positive

Graph behavior:

rising left-to-right

Students should visualize:

uphill motion

The function is:

increasing.

13.5 Negative First Derivative

Suppose:

f

′

(x)<0

Then:

tangent slopes negative

Graph behavior:

falling left-to-right

Students should visualize:

downhill motion

The function is:

decreasing.

13.6 Zero First Derivative

Suppose:

f

′

(x)=0

Then:

tangent line horizontal

Possible behaviors:

local maximum

local minimum

flat inflection

plateau

Students must understand:

zero derivative alone does NOT guarantee a maximum or minimum.

13.7 Critical Points

Critical points occur where:

f

′

(x)=0

or:

derivative undefined

These points are important because:

graph behavior may change there.

Possible outcomes:

peaks

valleys

cusps

turning points

13.8 Local Maximum

A local maximum occurs when:

graph rises

then:

graph falls

Visualize:

top of hill

Derivative behavior:

Before point:

f

′

(x)>0

After point:

f

′

(x)<0

The derivative changes:

positive → negative

13.9 Local Minimum

A local minimum occurs when:

graph falls

then:

graph rises

Visualize:

bottom of valley

Derivative behavior:

Before point:

f

′

(x)<0

After point:

f

′

(x)>0

Derivative changes:

negative → positive

13.10 Worked Example — First Derivative Analysis

Suppose:

f(x)= x

2

Derivative:

f

′

(x)=2x

Analyze sign.

When:

x<0

Derivative negative.

Graph decreases.

When:

x>0

Derivative positive.

Graph increases.

At:

x =0

Derivative zero.

Graph changes:

decreasing → increasing

Therefore:

local minimum at:

x =0

13.11 Visualization of x

2

Students should visualize:

parabola descending toward origin

flattening

then rising upward

Derivative behavior perfectly matches graph shape.

This is one of the central insights of calculus.

13.12 The Second Derivative — Curvature

Now calculus studies something deeper.

The first derivative measures:

slope

The second derivative measures:

how slope itself changes.

This reveals:

curvature

bending behavior

acceleration of growth

13.13 Understanding Concavity Intuitively

Suppose:

graph bends upward like bowl

This is:

concave up

Suppose:

graph bends downward like upside-down bowl

This is:

concave down

Students should visualize:

curvature direction.

13.14 Positive Second Derivative

Suppose:

f

′′

(x)>0

Then:

slopes increasing

Graph bends upward.

Concave up.

Visualize:

smile shape

bowl shape

13.15 Negative Second Derivative

Suppose:

f

′′

(x)<0

Then:

slopes decreasing

Graph bends downward.

Concave down.

Visualize:

frown shape

upside-down bowl

13.16 Why Concavity Matters

Concavity reveals:

acceleration of change

Examples:

vehicle speeding up

population accelerating

economic growth increasing rapidly

Second derivatives measure:

how rates themselves evolve.

13.17 Worked Example — Second Derivative

Suppose:

f(x)= x

3

First derivative:

f

′

(x)=3x

2

Second derivative:

f

′′

(x)=6x

Analyze signs.

When:

x<0

Second derivative negative.

Concave down.

When:

x>0

Second derivative positive.

Concave up.

13.18 Inflection Points

An inflection point occurs where:

concavity changes

Possible transition:

concave down → concave up

or:

concave up → concave down

Students should visualize:

graph changing bending direction.

13.19 Inflection Example

For:

f(x)= x

3

Second derivative:

6x

At:

x =0

Second derivative changes sign.

Therefore:

inflection point at origin.

13.20 First Derivative vs Second Derivative

Students must distinguish carefully.

First Derivative

Describes:

rising/falling behavior

Second Derivative

Describes:

bending/curvature behavior

These are fundamentally different ideas.

13.21 Physical Interpretation

Suppose:

s(t)

represents position.

Then:

s

′

(t)

represents velocity.

And:

s

′′

(t)

represents acceleration.

This is one of the most important physical interpretations in calculus.

13.22 Why Acceleration Is a Second Derivative

Velocity measures:

change in position

Acceleration measures:

change in velocity

So:

acceleration becomes derivative of derivative.

This layered structure appears constantly in physics.

13.23 Graphical Interpretation of Acceleration

Suppose:

velocity increasing

Then:

acceleration positive

Suppose:

velocity decreasing

Then:

acceleration negative

Second derivatives describe:

how motion itself evolves.

13.24 A Systematic Graphing Strategy

When analyzing a graph:

Step 1

Find first derivative.

Step 2

Find critical points.

Step 3

Analyze where derivative:

positive

negative

Step 4

Determine increasing/decreasing intervals.

Step 5

Find second derivative.

Step 6

Analyze concavity.

Step 7

Locate inflection points.

This structure organizes graph analysis clearly.

13.25 Worked Example — Full Analysis

Suppose:

f(x)= x

3

−3x

First derivative:

f

′

(x)=3x

2

−3

Set equal to zero:

3x

2

−3=0

x

2

=1

Critical points:

x =±1

Second derivative:

f

′′

(x)=6x

Analyze:

concavity

turning behavior

Students should sketch mentally:

cubic shape

turning points

changing curvature

13.26 Why Derivative Graphing Matters

Before graphing calculators:

derivatives allowed mathematicians to analyze curves deeply.

Even today:

calculus provides structural understanding

beyond simple plotting.

Derivatives reveal:

WHY graphs behave the way they do.

13.27 Common Student Mistakes

Mistake 1 — Confusing Increasing with Concavity

Increasing:

first derivative

Concavity:

second derivative

Mistake 2 — Assuming Zero Derivative Means Maximum

Could also be:

minimum

flat inflection

Mistake 3 — Forgetting Sign Analysis

Students must analyze intervals carefully.

Mistake 4 — Losing Geometric Interpretation

Calculus should remain visual and conceptual.

13.28 Visualization Strategy

Students should continually ask:

Is graph rising?

Is graph falling?

Is slope increasing?

Is slope decreasing?

Is curve bending upward?

Is curve bending downward?

This creates genuine graphical intuition.

13.29 Why This Chapter Is Important

This chapter marks a major transition.

Students now begin:

reading graphs dynamically

predicting behavior structurally

interpreting derivatives visually

This is one of the major goals of differential calculus.

13.30 Practice Problems

A. First Derivative Interpretation

Suppose:

f

′

(x)>0

Describe graph behavior.

Suppose:

f

′

(x)<0

Describe graph behavior.

Suppose:

f

′

(x)=0

Describe possible graph behavior.

Explain why positive derivative means increasing.

Explain why negative derivative means decreasing.

B. Critical Points

Find critical points of:

f(x)= x

2

Find critical points of:

f(x)= x

3

−3x

Explain why critical points matter.

Explain difference between:

maximum

minimum

Explain why zero derivative alone proves little.

C. Second Derivative and Concavity

Find second derivative of:

f(x)= x

3

Determine concavity of:

f(x)= x

2

Determine concavity of:

f(x)=−x

2

Explain meaning of concave up.

Explain meaning of concave down.

D. Inflection Points

Find inflection point of:

f(x)= x

3

Explain why inflection points matter.

Explain why curvature changes.

Explain graphical meaning of second derivative sign change.

Explain why inflection points do not necessarily occur at maxima/minima.

E. Physical Interpretation

Suppose:

s(t)= t

2

Find:

velocity

acceleration

Suppose:

s(t)= t

3

Find:

velocity

acceleration

Explain why acceleration is second derivative.

Explain why changing velocity matters physically.

Explain relationship between:

position

velocity

acceleration

F. Conceptual Problems

Explain why derivatives reveal graph shape.

Explain why calculus analyzes local behavior.

Explain why curvature matters physically.

Explain why first and second derivatives describe different behaviors.

Explain why graph analysis becomes powerful in science and engineering.

13.31 Selected Solutions

Problem 6

Suppose:

f(x)= x

2

Derivative:

f

′

(x)=2x

Set equal to zero:

2x =0

Critical point:

x =0

Problem 11

Suppose:

f(x)= x

3

First derivative:

3x

2

Second derivative:

6x

Problem 16

For:

f(x)= x

3

Second derivative:

6x

Changes sign at:

x =0

Therefore:

inflection point at:

0

Problem 21

Suppose:

s(t)= t

2

Velocity:

s

′

(t)=2t

Acceleration:

s

′′

(t)=2

Constant positive acceleration.

13.32 Chapter Summary

In this chapter we introduced:

increasing/decreasing behavior

critical points

local maxima/minima

concavity

second derivatives

inflection points

acceleration

graphical interpretation of derivatives

Most importantly:

students learned that derivatives reveal:

the dynamic structure and shape of graphs

allowing calculus to analyze:

motion

curvature

growth

physical systems

with remarkable precision.