Unit 10 · Lesson 03 · Week 28 · Fundamental Theorem & Antiderivatives
The Fundamental Theorem of Calculus, Part II
The Fundamental Theorem of Calculus connects accumulation and instantaneous rate. Differentiating an accumulation function recovers the integrand, while evaluating an antiderivative computes net accumulation. This lesson emphasizes the fundamental theorem of calculus, part ii through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in the fundamental theorem of calculus, part ii.
- 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 Fundamental Theorem connect instantaneous rate with accumulated change?
Visual model
The Fundamental Theorem of Calculus connects accumulation and instantaneous rate. Differentiating an accumulation function recovers the integrand, while evaluating an antiderivative computes net accumulation.
\frac{d}{dx}\int_a^x f(t)\,dt=f(x)\int_a^b f(x)\,dx=F(b)-F(a)
Worked reasoning
Misconception check: For a variable lower bound, the derivative introduces a negative sign.
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
The Fundamental Theorem of Calculus, Part II
Teacher Answer Key · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
Differentiate F(x)=∫₀ˣ(t²+1)dt.
Answer: x²+1
Reasoning: FTC Part I recovers the integrand.
Reasoning: FTC Part I recovers the integrand.
Differentiate G(x)=∫₁^{x²} cos t dt.
Answer: 2x cos(x²)
Reasoning: Evaluate at x² and multiply by 2x.
Reasoning: Evaluate at x² and multiply by 2x.
Evaluate ∫₀² 3x² dx using FTC.
Answer: 8
Reasoning: Use the antiderivative x³.
Reasoning: Use the antiderivative x³.
Differentiate H(x)=∫ₓ³ e^t dt.
Answer: −e^x
Reasoning: A variable lower bound introduces a negative sign.
Reasoning: A variable lower bound introduces a negative sign.
What connects derivatives and integrals?
Answer: The Fundamental Theorem of Calculus
Reasoning: Accumulation and instantaneous change are inverse processes.
Reasoning: Accumulation and instantaneous change are inverse processes.
Why use a dummy variable inside an accumulation integral?
Answer: It distinguishes the integration variable from the bound.
Reasoning: The bound x remains the function input.
Reasoning: The bound x remains the function input.