📚 Topic Summary
The Fundamental Theorem of Calculus (FTC) links differentiation and integration. Essentially, it states that differentiation and integration are inverse processes. The theorem has two parts. The first part says that if you integrate a function $f(x)$ from a constant $a$ to a variable $x$, and then differentiate the result, you get back the original function $f(x)$. The second part provides a way to evaluate definite integrals: if $F(x)$ is an antiderivative of $f(x)$, then the definite integral of $f(x)$ from $a$ to $b$ is simply $F(b) - F(a)$. Understanding this theorem is key to solving many calculus problems.
🧠 Part A: Vocabulary
Match the terms with their definitions:
| Term |
Definition |
| 1. Definite Integral |
A. A function whose derivative is the given function. |
| 2. Antiderivative |
B. The process of finding the area under a curve. |
| 3. Differentiation |
C. The value of the integral between defined limits. |
| 4. Integration |
D. The process of finding the derivative of a function. |
| 5. Fundamental Theorem of Calculus |
E. Establishes the relationship between integration and differentiation. |
✏️ Part B: Fill in the Blanks
The Fundamental Theorem of Calculus connects ________ and ________. The first part states that differentiating the integral of a function returns the ________ function. The second part allows us to evaluate definite integrals by finding the ________ and evaluating it at the limits of ________.
🤔 Part C: Critical Thinking
Explain, in your own words, why the Fundamental Theorem of Calculus is considered "fundamental" to the study of calculus.