Unit 10 · Lesson 02 · Week 28 · Fundamental Theorem & Antiderivatives
Accumulation Functions and Variable Bounds
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 accumulation functions and variable bounds through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in accumulation functions and variable bounds.
- 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
Accumulation Functions and Variable Bounds
Student Worksheet · 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.
Differentiate G(x)=∫₁^{x²} cos t dt.
Evaluate ∫₀² 3x² dx using FTC.
Differentiate H(x)=∫ₓ³ e^t dt.
What connects derivatives and integrals?
Why use a dummy variable inside an accumulation integral?
Lesson record
Mark this lesson complete only after the worksheet, correction, and reflection are finished.