Time Differentiation Slides TIME BASED DIFFERENTIATION
Foundations of Related Rates
Draft 1.0
A Dynamic Shift
Until now, we've looked at how y changes as x moves.
"If \( y = x^2 \), and \( x \) is a function of time, how fast is \( y \) changing right now?"
Static vs. Dynamic Rates
The Static View
Change in position relative to a coordinate.
\(\frac{dy}{dx}\)
The Dynamic View
Change in position relative to Time (t).
\(\frac{dx}{dt}\) and \(\frac{dy}{dt}\)
The Hidden Chain Rule
When variables are functions of time, we treat them as "inner functions."
Differentiate \( y^2 \) with respect to \( t \):
\(\frac{d}{dt}[y^2]\)
\(2y \cdot \frac{dy}{dt}\)
The "Rate Tail": Always multiply by the derivative of the variable with respect to time.
Blueprint Practice
Find the derivative of each equation with respect to \( t \).
1. \( x^2 + y^2 = 25 \) Pythagorean Blueprint
2. \( A = \pi r^2 \) Expanding Circle
3. \( V = s^3 \) Growing Cube
Rate Tails Worksheet Rate Tails Worksheet
Topic: Differentiation with Respect to Time (\(t\))
Name:
Date:
The "Rate Tail" Rule
When differentiating with respect to time (\(t\)), every variable that is not \(t\) must be treated as a function of time. After taking the normal derivative, multiply by the "rate tail" (e.g., \(\frac{dx}{dt}\), \(\frac{dy}{dt}\), \(\frac{dr}{dt}\)).
1 Single Expressions
Find the derivative of each expression with respect to \(t\).
A. \( x^3 \)
B. \( 4y^2 \)
C. \( \pi r^2 \)
D. \( \frac{1}{3} \pi h^3 \)
2 Full Equations
Differentiate both sides of the equation with respect to \(t\).
E. \( x^2 + y^2 = 100 \) (Pythagorean Identity)
F. \( A = s^2 \) (Area of a Square)
G. \( V = \frac{4}{3} \pi r^3 \) (Volume of a Sphere)
Reflection
If an object is constant (like a ladder length that doesn't change), what is its derivative with respect to time? Why?
Modeling Dynamic Change Slides MODELING DYNAMIC CHANGE
The Language of Related Rates
Lesson 02
Case Study: The Coast Guard Report
"The oil spill is spreading in a circular pattern. The radius is growing at 5 meters per hour. We need to know how fast the total area is covered when the radius hits 50 meters."
Translating...
Radius growing: \(\frac{dr}{dt} = 5\)
How fast is area covered: Find \(\frac{dA}{dt}\)
When radius is 50: \(r = 50\)
The Golden Rule: Don't Freeze Too Early
Constants
Values that never change during the problem. They can be plugged in immediately.
Example: The length of a ladder leaning against a wall.
Snapshots
Values that are constantly changing. They can only be used after differentiation.
Example: The distance from the ladder base to the wall.
The G/F/W Framework
G
GIVEN
What rates or values do we know? (e.g., \(\frac{dx}{dt} = 2\))
F
FIND
What is the target rate we are looking for? (e.g., \(\frac{dA}{dt} = ?\))
W
WHEN
At what specific "snapshot" in time are we calculating? (e.g., \(r = 10\))
Practice: Spot the Variable
Identify which values are Constants and which are Snapshots.
A 10ft ladder leaning against a wall slides down. Ladder = Constant
A balloon is inflated with air at 3 cubic cm/sec. Volume = Snapshot
Water drains from a conical tank at 5 gal/min. Height = Snapshot
GFW Translation Worksheet GFW Translation Guide
Converting Words into Calculus Notation
Name:
Period:
G: GIVEN
The rates or values we already know from the text.
F: FIND
The specific rate of change we are solving for.
W: WHEN
The "snapshot" moment or specific condition.
1 A circle's radius is increasing at a rate of 3 cm/sec. How fast is the area increasing when the radius is 10 cm?
Given
Find
When
2 Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm³/s. How fast is the radius of the balloon increasing when the diameter is 50 cm?
Given
Find
When
3 A 13-ft ladder is leaning against a wall. If the bottom of the ladder is pulled away from the wall at 2 ft/s, how fast is the top of the ladder sliding down the wall when the bottom is 5 ft from the wall?
Given
Find
When
Reverse Translation
Given the notation, write a one-sentence "story" for what is happening.
Given: \(\frac{ds}{dt} = 4\)
Given: \(\frac{dA}{dt} = 20\)
Geometry in Motion Slides GEOMETRY IN MOTION
Expanding Circles and Squares
Lesson 03
The Ripple Effect
A pebble is dropped into a pond. The radius of the ripple increases at a constant rate.
"Wait—if the radius grows at a steady 1 foot per second, does the Area grow at a steady rate too?"
Differentiating the Formula
We start with the relationship between variables at all times.
Geometric Equation
\( A = \pi r^2 \)
Differentiated with \( t \)
\( \frac{dA}{dt} = 2\pi r \frac{dr}{dt} \)
Note: The rate of change of Area (\(\frac{dA}{dt}\)) depends on BOTH how fast it's growing (\(\frac{dr}{dt}\)) and how big it already is (\(r\)).
The Deadly Mistake
Plugging in numbers BEFORE you differentiate.
WRONG
Premature Substitution
Area = \(\pi (10)^2\)
Area = \(100\pi\)
\(\frac{dA}{dt} = 0\) (Error: A constant's derivative is zero)
RIGHT
The Calculus Path
\( A = \pi r^2 \)
Differentiate first:
\(\frac{dA}{dt} = 2\pi r \frac{dr}{dt}\)
Substitute now:
\(\frac{dA}{dt} = 2\pi (10)(1)\)
Challenge: Metal Plate
"A square metal plate is being heated. Its side length is increasing at a rate of 0.2 cm/min. How fast is the area increasing when the side length is 20 cm?"
Step 1: Relate Equation
\( A = s^2 \)
Step 2: Differentiate
\( \frac{dA}{dt} = 2s \frac{ds}{dt} \)
Geometry Lab Worksheet Geometry Lab Activity
Topic: Expanding Circles and Squares
Name:
Date:
The Golden Rule of Timing
Differentiate the relationship first. Substitute the snapshot values (\(r\), \(h\), \(x\)) last. Constant rates (\(\frac{dr}{dt}\), etc.) are the only numbers that can stay throughout the process.
1
The Expanding Ripple
A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius of the outer ripple is increasing at a constant rate of 2 ft/sec.
Relate Equation (Area of a Circle)
Differentiate with respect to \(t\)
GFW Setup
G:
F:
W:
Calculate how fast the area of the ripple is increasing when the radius is 5 feet:
2
The Heated Square Plate
A square metal plate is being heated. As it expands, its area increases at a rate of 12 cm²/min. How fast is the side length increasing when the area is 100 cm²?
Equation & Differentiated Form
GFW Setup
G:
F:
W:
Show your work and solve for the rate of change of the side length:
Wait! Unit Check
In Problem 2, your final answer should be in length per time. Does your calculated rate match the units used in the problem (cm, cm², min)?
Write your final answer with units here:
The Protocol Slides THE SOLVING PROTOCOL
Engineering a Flawless Setup
Lesson 04
Setup > Solution
In Calculus, 90% of mistakes happen before the first derivative is even taken.
"If you can translate the story into the right math, the calculus solves itself."
The 4-Step Framework
01
SKETCH
Visualize the scene and label all variables.
02
LIST (GFW)
Identify Given, Find, and When snapshot.
03
RELATE
Find the formula that ties variables together.
04
SOLVE
Differentiate implicitly and plug in values.
Step 1: Sketch & Label
A sketch isn't art; it's an inventory.
Draw the fixed shapes.
Never label numbers for changing variables.
Use letters (\(x\), \(y\), \(r\)) for dynamic parts.
Drawing Board
x
y
13 ft
Step 3: Finding the Relation
Ask: "What geometric law governs this sketch?"
Right Triangles
\( a^2 + b^2 = c^2 \)
Expanding Circles
\( A = \pi r^2 \)
Growing Solids
\( V = s^3 \)
Protocol Mastery Worksheet The 4-Step Protocol
Procedural Mastery in Related Rates
Name:
Date:
1 SKETCH
2 LIST (GFW)
3 RELATE
4 SOLVE
MISSION 01: The radius of a circular oil spill is growing at 4 ft/min. How fast is the area growing when the radius is 10 ft?
01. Sketch & Label
02. GFW List
G:
F:
W:
03. Relate Equation
04. Differentiate & Solve
MISSION 02: Each side of a square is increasing at a rate of 6 cm/s. At what rate is the area of the square increasing when the area of the square is 16 cm²?
01. Sketch & Label
02. GFW List
G:
F:
W:
03. Relate Equation
04. Differentiate & Solve
Calculus Detective Slides CASE FILE #05
CALCULUS DETECTIVE WORK
Finding the Fatal Flaw
Investigation Unit
The Suspicious Balloon
EXHIBIT A
"The balloon is deflating. The volume is decreasing at 10 cubic inches per second. The student calculates the rate of change of the radius as..."
\(\frac{dr}{dt} = 0.5 \text{ in/s}\)
Something is wrong.
If the balloon is shrinking, why is the derivative positive?
The first rule of detection: The math must match the physics.
The "Usual Suspects" of Errors
Snapshot Freeze
Substituting changing values (like radius) into the equation before differentiating.
Unit Mismatch
Mixing inches and feet or ignoring squared/cubed units in rates.
Sign Sabotage
Treating a decreasing rate as a positive number (ignoring the negative sign).
The Reality Check
Before you submit, ask these three questions:
1. SIGN CHECK
Is it shrinking? (Result should be negative). Is it growing? (Result should be positive).
2. DIMENSION CHECK
Are my final units consistent? Area rates should be sq-units/time; length rates should be units/time.
3. MAGNITUDE CHECK
Does the number make sense? If a ripple grows by 1 inch, the area shouldn't grow by 1,000 miles.
Your Mission
Open your Detective Log. Analyze the "crime scene" solutions and identify exactly where the math went rogue.
Time to Debug the Calculus.
Detective Log Worksheet Calculus Detective Log
CASE FILE: UNIT-05-RELATED-RATES
OPERATIVE:
DATE:
Mission Briefing
Below are three "Crime Scenes"—solutions to related rates problems that contain critical calculation errors. Your task: Identify the Error Type (Snapshot Freeze, Unit Mismatch, or Sign Sabotage), Find the Mistake, and Provide the Corrective Action.
SCENE 01
The Premature Evaluation
Evidence
The Problem:
Find \(dA/dt\) of a circle when \(r = 10\) and \(dr/dt = 2\).
1. \(A = \pi r^2\)
Error Identification
Corrective Action (Show Correct Work)
SCENE 02
The Vanishing Negative
Evidence
The Problem:
A square area is shrinking at 20 cm²/s. Find \(ds/dt\) when \(s = 5\).
1. \(A = s^2\)
Error Identification
Corrective Action (Show Correct Work)
SCENE 03
The Unit Collision
Evidence
The Problem:
A cube's edge increases at 1 ft/min. Find \(dV/dt\) (in in³/min) when the edge is 2 inches.
1. \(V = s^3\)
Error Identification
Corrective Action (Show Correct Work)
CASE CLOSED
