Understanding the Fundamental Theorem of Calculus:

• Part 1 of the FTC states that the definite integral of a function represents the net change in its antiderivative over an interval.


  True
• The Fundamental Theorem of Calculus (FTC) eliminates the need for summation techniques to evaluate definite integrals.

  True
• What does FTC Part 1 state about the derivative of a definite integral with variable upper limit $x$?


  $f(x)$
• Part 1 of the Fundamental Theorem of Calculus (FTC) states that the definite integral $\int_{a}^{b} f(x) dx$ represents the net change in the antiderivative
• Applying FTC Part 1 to $\int_{2}^{x} t^{2} dt$ results in the derivative $x^{2}$
• If $f(x)$ is continuous on $[a, b]$, then $\frac{d}{dx}\int_{a}^{x} f(t) dt =$$f(x)$ is true according to the Fundamental Theorem of Calculus Part 1.


  True
• The Fundamental Theorem of Calculus Part 2 states that $\int_{a}^{b} f(x) dx =$$F(b) - F(a)$, where $F(x)$ is the antiderivative
• Evaluate $\int_{1}^{3} (2x) dx$ using the Fundamental Theorem of Calculus Part 2.


  $8$
• What is the derivative of \int_{a}^{x} f(t) dt</latex> if $f(x)$ is continuous according to the Fundamental Theorem of Calculus Part 1?


  $f(x)$
• The derivative of the definite integral of a continuous function $f(t)$ from $a$ to x</latex> is equal to the original function evaluated at x
• The definite integral of $f(x)$ from $a$ to $b$ equals the net change in its antiderivative
• What is the primary action taken when applying FTC Part 1 to find the derivative of a definite integral?

  Evaluate the integrand at x</latex>
• The antiderivative of $\sin(x)$ is - \cos(x)
• The Fundamental Theorem of Calculus (FTC) Part 1 states that the definite integral $\int_{a}^{b} f(x) dx$ represents the net change in the antiderivative
• What is the derivative of the definite integral \int_{a}^{x} f(t) dt</latex> according to FTC Part 1?

  $f(x)$
• Applying FTC Part 1 to \int_{2}^{x} t^{2} dt</latex> results in the derivative $x^{2}$.


  True
• The Fundamental Theorem of Calculus (FTC) allows for the evaluation of definite integrals without using summation techniques.


  True
• What does the definite integral $\int_{a}^{b} f(x) dx$ represent according to the Fundamental Theorem of Calculus?


  Net change in F(x)
• What is the derivative of $\int_{a}^{x} f(t) dt$ if $f(x)$ is continuous?


  $f(x)$
• What does the definite integral $\int_{a}^{b} f(x) dx$ equal according to the Fundamental Theorem of Calculus Part 2?


  $F(b) - F(a)$
• The Fundamental Theorem of Calculus consists of two parts that relate derivatives and integrals.
• What is the derivative of $\int_{2}^{x} t^{2} dt$ using the Fundamental Theorem of Calculus Part 1?


  $x^{2}$
• The Fundamental Theorem of Calculus Part 1 is used to find the derivative of definite integrals.


  True
• Definite integrals can be evaluated using the Fundamental Theorem of Calculus Part 2 by finding the antiderivative and computing its value at the upper and lower limits.


  True
• The Fundamental Theorem of Calculus Part 1 is used to find the area under a curve.

  False
• Evaluate $\int_{0}^{\pi} \sin(x) dx$ using FTC Part 2.


  2
• What are the two parts of the Fundamental Theorem of Calculus (FTC)?

  Derivatives and integrals
• What does the Fundamental Theorem of Calculus (FTC) Part 2 state about the derivative of a definite integral?

  It equals the original function
• The derivative of $\int_{2}^{x} t^{2} dt$ using FTC Part 1 is $x^{2}$
• What does Part 1 of the Fundamental Theorem of Calculus (FTC) state about the definite integral of a continuous function?

  Net change in antiderivative
• What is the derivative of the definite integral $\int_{a}^{x} f(t) dt$ according to FTC Part 1?


  $f(x)$
• The Fundamental Theorem of Calculus allows us to evaluate definite integrals by finding the antiderivative
• What is the derivative of $\int_{2}^{x} t^{2} dt$ using the Fundamental Theorem of Calculus Part 1?


  $x^{2}$
• Evaluating a definite integral using the Fundamental Theorem of Calculus Part 2 requires finding the antiderivative of the integrand.


  True
• The Fundamental Theorem of Calculus allows us to evaluate definite integrals without using summation techniques.


  True
• What does the Fundamental Theorem of Calculus Part 1 state regarding the derivative of a definite integral?

  $\frac{d}{dx}\int_{a}^{x} f(t) dt =$$f(x)$
• What does the Fundamental Theorem of Calculus Part 2 state regarding the value of a definite integral?

  $\int_{a}^{b} f(x) dx =$$F(b) - F(a)$
• Steps to evaluate $\int_{1}^{3} (2x) dx$ using FTC Part 2


  1️⃣ Find the antiderivative of $2x$, which is $F(x) =$$x^{2}$

  2️⃣ Compute the net change: $F(3) - F(1) =$$3^{2} - 1^{2} =$$9 - 1 =$$8$
• Evaluate $\int_{1}^{4} \frac{1}{\sqrt{x}} dx$ using FTC Part 2.


  2