Day -1: Average Rate of Change (AROC) & Secant Lines & The 'Shrinking Interval' (Secant to Tangent) & Instantaneous Rate of Change (IROC) & Tangent Lines & Geometric Models & Similarity & Absolute Values and Graph Transformations

Average Rate of Change (AROC) & Secant Lines

AROC measures the 'average' steepness of a function over a fixed interval. Geometrically, it represents the slope of the Secant Line—a straight line that cuts through two specific points on a curve.

Core Theorem

The AROC Formula: For a function

f(x)

on the interval

[a, b]

, the average rate of change is

m_{sec} = \frac{f(b) - f(a)}{b - a}

.

Worked Examples

Read these first — the full solution is shown, with the reasoning behind each move.

Worked Example 01Easy

Find the AROC of

f(x) = \sqrt{x}

on the interval

[4, 9]

.

How to think about it

Key idea: Calculate the y-values for $x=4$ and $x=9$ first.

Full solution

1. Step 1: Evaluate Points

f(4) = 2

and

f(9) = 3

. Points are

(4, 2)

and

(9, 3)

.

2. Step 2: Slope Formula

m = \frac{3 - 2}{9 - 4} = \frac{1}{5} = 0.2

.

Step-by-Step SOP

1
Identify the Interval
Note the start ( $a$ ) and end ( $b$ ) x-values.
2
Map to Y-values
Find the height of the graph at those two points.
3
Calculate Slope
Use $\frac{\text{Change in Y}}{\text{Change in X}}$ .

Practice — Your Turn

Try each one before opening the hint or the solution.

Practice 01Medium

A cup of coffee cools from

180^{\circ}F

to

100^{\circ}F

over 20 minutes. What is the AROC of the temperature?

Need a hint?

Think about the units. It should be 'degrees per minute'.

Show solution

1. Step 1: Change in Temp

\Delta T = 100 - 180 = -80^{\circ}F

.

2. Step 2: AROC

\frac{-80}{20} = -4^{\circ}F/min

. The negative sign means it's cooling down.

Common Pitfalls

⚠
Mixing up the OrderIf you use $f(b)-f(a)$ on top, you MUST use $b-a$ on the bottom. Swapping one but not the other flips the sign of your answer.

The 'Shrinking Interval' (Secant to Tangent)

This is the 'Big Bang' of Calculus. By moving the two points of a secant line closer and closer together, the gap between them eventually vanishes, leaving us with a line that touches the curve at exactly one point.

Core Theorem

The Tangent Intuition: As the interval width

\Delta x \to 0

, the Secant Line 'morphs' into the Tangent Line.

Worked Examples

Read these first — the full solution is shown, with the reasoning behind each move.

Worked Example 01Medium

Given the table of slopes for

f(x)

starting at

x=3

with different interval widths

h

:
-

h=0.1, m=6.1

-

h=0.01, m=6.01

-

h=0.001, m=6.001

What is the estimated slope of the tangent line at

x=3

?

How to think about it

Key idea: Look at the number these slopes are 'chasing' as $h$ disappears.

Full solution

1. Analysis

As

h

gets smaller, the slope values are clearly approaching the whole number 6.

2. Prediction

The Instantaneous Rate of Change at

x=3

is 6.

Step-by-Step SOP

1
Pick a Pivot
Choose the point where you want to find the 'instant' speed.
2
Create a Sequence
Calculate slopes over smaller and smaller intervals (0.1, 0.01, 0.001).

Common Pitfalls

⚠
The 0/0 ProblemIn algebra, dividing by zero is forbidden. In Calculus, we use 'Limits' to talk about getting infinitely close to zero without actually being zero.

Instantaneous Rate of Change (IROC) & Tangent Lines

IROC is the exact rate of change at a single moment in time. Geometrically, it is the slope of the Tangent Line.

Core Theorem

IROC =

\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}

. (The formal derivative definition we'll see in Day 1).

Worked Examples

Read these first — the full solution is shown, with the reasoning behind each move.

Worked Example 01Hard

Rank the IROC (slopes) at points A, B, and C on a curve where A is a steep climb, B is the peak, and C is a gentle descent.

How to think about it

Key idea: Steep up = Large positive. Flat = Zero. Down = Negative.

Full solution

1. Visual Rank

Point A (Positive) > Point B (Zero) > Point C (Negative).

Step-by-Step SOP

1
The Surfer Test
Imagine a surfer on the wave. The board represents the tangent line; its tilt is the IROC.
2
Local Linearity
Zoom in. If the curve looks like it's going up-right, the IROC is positive.

Practice — Your Turn

Try each one before opening the hint or the solution.

Practice 01Medium

If a position graph is a straight line

s(t) = 5t + 2

, what is the IROC at

t=10

?

Need a hint?

Does the slope of a straight line ever change?

Show solution

1. Analysis

A straight line has a constant slope. Here,

m=5

everywhere.

2. Conclusion

The IROC is 5, regardless of the value of

t

.

Common Pitfalls

⚠
Tangent vs. NormalRemember, the tangent line 'skims' the surface in the same direction as the curve. It doesn't cut through it perpendicularly.

Geometric Models & Similarity

Calculus often measures how physical quantities change. To do this, you must first have the 'container' formulas ready.

Core Theorem

Key Formulas:
1. Circle: Area

A = \pi r^2

, Circumference

C = 2\pi r

.
2. Sphere: Volume

V = \frac{4}{3}\pi r^3

, Surface Area

S = 4\pi r^2

.
3. Cone: Volume

V = \frac{1}{3}\pi r^2 h

.
4. Similarity: If two triangles are similar, the ratios of their corresponding sides are equal:

\frac{r}{h} = \frac{R}{H}

.

Worked Examples

Read these first — the full solution is shown, with the reasoning behind each move.

Worked Example 01Medium

A 1.8m tall man walks away from a 5m tall lamppost. If his distance from the post is

x

and the length of his shadow is

s

, express

s

in terms of

x

.

How to think about it

Key idea: Draw two right triangles: the small one (man + shadow) and the large one (lamppost + total distance).

Full solution

1. Step 1: Set up the Proportion

Using similar triangles:

\frac{\text{man's height}}{\text{shadow length}} = \frac{\text{lamp height}}{\text{total distance}}

.

2. Step 2: Plug in Variables

\frac{1.8}{s} = \frac{5}{x + s}

.

3. Step 3: Solve for s

1.8(x + s) = 5s \implies 1.8x + 1.8s = 5s \implies 1.8x = 3.2s \implies s = \frac{1.8}{3.2}x = \frac{9}{16}x

.

Step-by-Step SOP

1
Dimension Reduction
In multi-variable shapes (like cones), always use similarity or given constraints to reduce the formula to a single variable before differentiating.

Practice — Your Turn

Try each one before opening the hint or the solution.

Practice 01Easy

The Melting Iceball: A spherical snowball is melting. If its surface area

S

is currently

36\pi

cm², find its volume

V

.

Need a hint?

Use the surface area to find the radius

r

first, then plug into the volume formula.

Show solution

1. Step 1: Find Radius

S = 4\pi r^2 = 36\pi \implies r^2 = 9 \implies r = 3

cm.

2. Step 2: Calculate Volume

V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (3)^3 = \frac{4}{3}\pi (27) = 36\pi

cm³.

3. Note

Interestingly, for a sphere with

r=3

, the numerical values of

S

and

V

are the same (

36\pi

).

Practice 02Hard

The Trough Problem: A water trough is 10m long and its ends are isosceles triangles with width 4m and height 3m. If the water level is

h

, find the volume

V

of water.

Need a hint?

Volume = (Area of triangular cross-section)

\times

length. Use similarity to find the water's width

w

.

Show solution

1. Step 1: Find Water Width

By similar triangles on the end face:

\frac{w}{h} = \frac{4}{3} \implies w = \frac{4}{3}h

.

2. Step 2: Area of Cross-section

Area

= \frac{1}{2} \cdot w \cdot h = \frac{1}{2} \cdot (\frac{4}{3}h) \cdot h = \frac{2}{3}h^2

.

3. Step 3: Total Volume

V = \text{Area} \cdot \text{length} = \frac{2}{3}h^2 \cdot 10 = \frac{20}{3}h^2

.

Absolute Values and Graph Transformations

To understand limits ( $epsilon-delta$ ) and continuity, you must view functions not just as lines, but as distances and movements.

Core Theorem

1. Distance Interpretation:

|x - a| < L

means the distance between

x

and

a

is less than

L

. (i.e.,

a-L < x < a+L

).
2. Transformations: For

y = af(b(x-h)) + k

:
-

h, k

: Horizontal/Vertical shift.
-

a, b

: Vertical/Horizontal stretch or compression.
3. Piecewise Splitting:

|f(x)| = f(x)

if

f(x) \geq 0

, and

-f(x)

if

f(x) < 0

.

Worked Examples

Read these first — the full solution is shown, with the reasoning behind each move.

Worked Example 01Medium

Write

f(x) = |x - 3| + 2

as a piecewise function and describe its graph.

How to think about it

Key idea: Find the 'critical point' where the expression inside the absolute value is zero.

Full solution

1. Step 1: Case x < 3

x-3

is negative, so

|x-3| = -(x-3)

.

f(x) = -x + 3 + 2 = -x + 5

.

2. Step 2: Case x ≥ 3

x-3

is positive, so

|x-3| = x-3

.

f(x) = x - 3 + 2 = x - 1

.

3. Step 3: Transformation View

This is the graph of

y = |x|

shifted Right 3 units and Up 2 units. The 'corner' (non-differentiable point) is at

(3, 2)

.

Step-by-Step SOP

1
Breaking Absolute Values
When dealing with $|g(x)|$ in calculus, always split the domain based on the roots of $g(x)$ to remove the bars.

Practice — Your Turn

Try each one before opening the hint or the solution.

Practice 01Medium

Define

f(x) = \frac{|x-2|}{x-2}

as a piecewise function. What happens at

x=2

?

Need a hint?

Check the sign of

(x-2)

for values greater than and less than 2.

Show solution

1. Case x > 2

|x-2| = x-2

, so

f(x) = \frac{x-2}{x-2} = 1

.

2. Case x < 2

|x-2| = -(x-2)

, so

f(x) = \frac{-(x-2)}{x-2} = -1

.

3. Conclusion

The function is

1

for

x>2

and

-1

for

x<2

. It is undefined at

x=2

(a jump discontinuity).

Practice 02Easy

Solve the inequality

|2x - 5| < 3

and interpret it as a distance.

Need a hint?

The distance between

2x

and

5

is less than 3.

Show solution

1. Step 1: Split

-3 < 2x - 5 < 3

.

2. Step 2: Add 5

2 < 2x < 8

.

3. Step 3: Divide by 2

1 < x < 4

.

Common Pitfalls

⚠
The Horizontal Stretch TrapIn $f(bx)$ , if $b=2$ , the graph is compressed by half, NOT doubled. Inside moves are always counter-intuitive!

Worked Examples

1. Step 1: Evaluate Points

2. Step 2: Slope Formula

Identify the Interval

Map to Y-values

Calculate Slope

Practice — Your Turn

1. Step 1: Change in Temp

2. Step 2: AROC

Worked Examples

1. Analysis

2. Prediction

Pick a Pivot

Create a Sequence

Worked Examples

1. Visual Rank

The Surfer Test

Local Linearity

Practice — Your Turn

1. Analysis

2. Conclusion

Worked Examples

1. Step 1: Set up the Proportion

2. Step 2: Plug in Variables

3. Step 3: Solve for s

Dimension Reduction

Practice — Your Turn

1. Step 1: Find Radius

2. Step 2: Calculate Volume

3. Note

1. Step 1: Find Water Width

2. Step 2: Area of Cross-section

3. Step 3: Total Volume

Worked Examples

1. Step 1: Case x < 3

2. Step 2: Case x ≥ 3

3. Step 3: Transformation View

Breaking Absolute Values

Practice — Your Turn

1. Case x > 2

2. Case x < 2

3. Conclusion

1. Step 1: Split

2. Step 2: Add 5

3. Step 3: Divide by 2

Gary Chang