6.4 The Fundamental Theorem of Calculus and Accumulation Functions

• What does the Fundamental Theorem of Calculus establish a relationship between?

  Integration and differentiation
• Part 2 of the Fundamental Theorem states that differentiation "undoes" integration
• Part 2 of the Fundamental Theorem of Calculus states that if $F(x)$ is an antiderivative of $f(x)$, then $f(x) =$$\frac{d}{dx}F(x)$.


  True
• Match the element with its description according to Part 1 of the Fundamental Theorem of Calculus:

  \int_{a}^{b} f(x) dx</latex> ↔️ Definite integral of $f(x)$ from $a$ to $b$

  $F(x)$ ↔️ Antiderivative of $f(x)$

  $F(b) - F(a)$ ↔️ Difference in the antiderivative evaluated at $b$ and $a$
• What does the definite integral $\int_{a}^{b} f(x) dx$ equal according to the Fundamental Theorem of Calculus?


  $F(b) - F(a)$
• What is the formula for an accumulation function F(x)</latex> if $f(x)$ is the original function?


  $F(x) =$$\int_{a}^{x} f(t) dt$
• Match the part of the Fundamental Theorem with its statement:

  Part 1 ↔️ $\int_{a}^{b} f(x) dx =$$F(b) - F(a)$

  Part 2 ↔️ $f(x) =$$\frac{d}{dx}F(x)$
• If $F'(x) =$$f(x)$, then $f(x)$ is the derivative
• Differentiation and integration are inverse operations.


  True
• The variable $a$ in the accumulation function $F(x) =$$\int_{a}^{x} f(t) dt$ represents a fixed starting point.
• What is the first step in evaluating a definite integral using the Fundamental Theorem of Calculus?

  Find the antiderivative
• The Fundamental Theorem of Calculus consists of two parts.


  True
• What does Part 1 of the Fundamental Theorem of Calculus provide a method to calculate?

  Definite integrals
• The Fundamental Theorem of Calculus highlights the fundamental connection between integration and differentiation
• What is the key idea behind Part 1 of the Fundamental Theorem of Calculus for calculating definite integrals?

  Find an antiderivative
• Part 1 of the Fundamental Theorem of Calculus states that if $f(x)$ is continuous on $[a, b]$, then $\int_{a}^{b} f(x) dx =$$< blank_{s}tart > F(b) - F(a) < / blank_{e}nd > < distractors > F(a) +$$F(b) < / distractors > < cloze_{e}nd > < truefalse_{s}tart > < line > 31 < / line > < statement_{s}tart > If < latex > F(x)$ is an antiderivative of $f(x)$, then $f(x) =$$\frac{d}{dx}F(x)$.<statement_end><answer_start>True<answer_end><truefalse_end>

  
  

  <cloze_start>An accumulation function represents the total change of another function over an interval
• Differentiating an accumulation function $F(x)$ gives back the original function $f(x)$.


  True
• Differentiation and integration are inverse operations according to the Fundamental Theorem of Calculus.


  True
• Differentiation and integration are inverse operations.
• What is the formal definition of an accumulation function $F(x)$?


  F(x) = \int_{a}^{x} f(t) dt</latex>
• What is the relationship between the derivative of $F(x)$ and $f(x)$?


  $\frac{d}{dx}F(x) =$$f(x)$
• Match the real-world scenario with its corresponding accumulation function:

  Population Growth ↔️ $F(t) =$$\int_{t_{0}}^{t} r(x) dx$

  Inventory Management ↔️ $F(t) =$$\int_{0}^{t} (p(x) - s(x)) dx$

  Area Calculation ↔️ $F(x) =$$\int_{a}^{x} h(t) dt$