Unit 04 · Lesson 05 · Week 11 · The Derivative from First Principles
Differentiability and Corners, Cusps & Vertical Tangents
The derivative is the limit of average rates of change over shrinking intervals. Geometrically it is tangent slope; contextually it is instantaneous rate with output units divided by input units. This lesson emphasizes differentiability and corners, cusps & vertical tangents through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in differentiability and corners, cusps & vertical tangents.
- Move among symbolic, graphical, numerical, and contextual representations.
- Show complete reasoning with correct notation, domain conditions, units, and intervals.
- Verify a conclusion using a second method or representation.
Essential question
How does the limiting slope of secant lines define instantaneous change?
Visual model
The derivative is the limit of average rates of change over shrinking intervals. Geometrically it is tangent slope; contextually it is instantaneous rate with output units divided by input units.
f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}y-f(a)=f'(a)(x-a)
Worked reasoning
Misconception check: The derivative is not the ratio f(x)/x. It is a limit of change in output divided by change in input.
Three learning pathways
Foundation
Reconstruct definitions, annotate a worked example, and solve representative problems with guided prompts.
Core
Complete mixed representations, justify every major step, and verify exact and approximate results.
Extended
Generalize a result, analyze an error or edge case, and connect the lesson to modelling or proof-informed reasoning.
Communication and modelling
State what is known, identify the theorem or structure used, show symbolic work, interpret the result in context, and explain how the conclusion was checked. Calculator or CAD evidence must include settings, units, and a written interpretation.
Khaemenes Academy · Open-Age Calculus I
Differentiability and Corners, Cusps & Vertical Tangents
Teacher Answer Key · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
Using first principles, find f'(2) for f(x)=x².
Answer: 4
Reasoning: The derivative of x² from the difference quotient is 2x; at 2 it is 4.
Reasoning: The derivative of x² from the difference quotient is 2x; at 2 it is 4.
Write the tangent-line equation to y=x² at x=1.
Answer: y−1=2(x−1)
Reasoning: The point is (1,1) and the derivative is 2x.
Reasoning: The point is (1,1) and the derivative is 2x.
What does f'(a)>0 indicate locally?
Answer: f is increasing near a.
Reasoning: A positive derivative means positive tangent slope.
Reasoning: A positive derivative means positive tangent slope.
Name a point where |x| is not differentiable.
Answer: x=0
Reasoning: The graph has a corner at the origin.
Reasoning: The graph has a corner at the origin.
If position is measured in meters and time in seconds, give velocity units.
Answer: meters per second
Reasoning: Derivative units are output units divided by input units.
Reasoning: Derivative units are output units divided by input units.
What feature of f often corresponds to a zero of f'?
Answer: A horizontal tangent or stationary point
Reasoning: Zero derivative means tangent slope zero.
Reasoning: Zero derivative means tangent slope zero.