The Fundamental Theorem of Calculus Preview
A concluding conceptual increment explaining that derivatives and integrals are inverse operations.
Introduction
The Fundamental Theorem of Calculus (FTC) is one of the most important results in all of mathematics. It connects the two main ideas of calculus: differentiation (finding rates of change) andintegration (finding accumulations). They are inverses of each other!
Prerequisite Connection
You understand derivatives and Riemann sums as limits.
Today's Increment
We see how differentiation and integration undo each other.
Why This Matters
The FTC makes computing integrals practical—it's the key to applied calculus.
Key Concepts
FTC Part 1: Integration then Differentiation
If you integrate then differentiate, you get back to the original function.
FTC Part 2: Differentiation then Integration
If you integrate a derivative, you get the net change in the original function.
The Practical Result
Where F is ANY antiderivative of f (F' = f).
The Big Picture
Differentiation and Integration are inverse operations, like addition/subtraction or multiplication/division.
Worked Examples
Example 1: Inverse Relationship (Basic)
If , find an antiderivative F(x).
We need F where F'(x) = x²
Since
Example 2: Using FTC to Evaluate (Intermediate)
Evaluate
Antiderivative of 2x is x²
Area = 8
Example 3: Net Change Interpretation (Advanced)
If velocity , find net displacement from t = 0 to t = 2.
Displacement = 8 units
Common Pitfalls
Forgetting the constant
Indefinite integrals need "+C". Definite integrals don't.
Wrong order of evaluation
F(b) - F(a), NOT F(a) - F(b). Upper minus lower!
Real-World Application
Engineering and Physics
The FTC connects rate of change (velocity, acceleration, power) to accumulation (position, velocity, energy). Engineers use it constantly to go between these perspectives!
Position ↔ Velocity ↔ Acceleration
Connected by differentiation and integration
Practice Quiz
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🎉 Congratulations!
You've completed the Precalculus course! You now have the conceptual foundation for Calculus: limits, derivatives, and integrals. You're ready to dive deeper into the techniques and applications of calculus.