Unit 10 · Lesson 05 · Week 29 · Fundamental Theorem & Antiderivatives
Initial Conditions and Particular Solutions
An antiderivative reverses differentiation. Indefinite integrals represent families of functions and therefore require a constant of integration; initial conditions select one member. This lesson emphasizes initial conditions and particular solutions through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in initial conditions and particular solutions.
- 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
An antiderivative reverses differentiation. Indefinite integrals represent families of functions and therefore require a constant of integration; initial conditions select one member.
\int x^n dx=\frac{x^{n+1}}{n+1}+CF'(x)=f(x)
Worked reasoning
Misconception check: The constant of integration is required for indefinite integrals because many functions share the same derivative.
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
Initial Conditions and Particular Solutions
Student Worksheet · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
Find ∫x⁴ dx.
Find ∫(3x²−2)dx.
Find F if F'=2x and F(1)=5.
Why include +C?
Find ∫e^x dx.
Check an antiderivative.
Lesson record
Mark this lesson complete only after the worksheet, correction, and reflection are finished.