Understanding the Fundamental Theorem of Calculus:

Cards (39)

  • 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 xx?

    f(x)f(x)
  • Part 1 of the Fundamental Theorem of Calculus (FTC) states that the definite integral ∫abf(x)dx\int_{a}^{b} f(x) dx represents the net change in the antiderivative
  • Applying FTC Part 1 to ∫2xt2dt\int_{2}^{x} t^{2} dt results in the derivative x2x^{2}
  • If f(x)f(x) is continuous on [a,b][a, b], then ddx∫axf(t)dt=\frac{d}{dx}\int_{a}^{x} f(t) dt =f(x) f(x) is true according to the Fundamental Theorem of Calculus Part 1.

    True
  • The Fundamental Theorem of Calculus Part 2 states that ∫abf(x)dx=\int_{a}^{b} f(x) dx =F(b)−F(a) F(b) - F(a), where F(x)F(x) is the antiderivative
  • Evaluate ∫13(2x)dx\int_{1}^{3} (2x) dx using the Fundamental Theorem of Calculus Part 2.

    88
  • What is the derivative of \int_{a}^{x} f(t) dt</latex> if f(x)f(x) is continuous according to the Fundamental Theorem of Calculus Part 1?

    f(x)f(x)
  • The derivative of the definite integral of a continuous function f(t)f(t) from aa to x</latex> is equal to the original function evaluated at x
  • The definite integral of f(x)f(x) from aa to bb 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)\sin(x) is - \cos(x)
  • The Fundamental Theorem of Calculus (FTC) Part 1 states that the definite integral ∫abf(x)dx\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)f(x)
  • Applying FTC Part 1 to \int_{2}^{x} t^{2} dt</latex> results in the derivative x2x^{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 ∫abf(x)dx\int_{a}^{b} f(x) dx represent according to the Fundamental Theorem of Calculus?

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

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

    F(b)−F(a)F(b) - F(a)
  • The Fundamental Theorem of Calculus consists of two parts that relate derivatives and integrals.
  • What is the derivative of ∫2xt2dt\int_{2}^{x} t^{2} dt using the Fundamental Theorem of Calculus Part 1?

    x2x^{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 ∫0πsin⁡(x)dx\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 ∫2xt2dt\int_{2}^{x} t^{2} dt using FTC Part 1 is x2x^{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 ∫axf(t)dt\int_{a}^{x} f(t) dt according to FTC Part 1?

    f(x)f(x)
  • The Fundamental Theorem of Calculus allows us to evaluate definite integrals by finding the antiderivative
  • What is the derivative of ∫2xt2dt\int_{2}^{x} t^{2} dt using the Fundamental Theorem of Calculus Part 1?

    x2x^{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?
    ddx∫axf(t)dt=\frac{d}{dx}\int_{a}^{x} f(t) dt =f(x) f(x)
  • What does the Fundamental Theorem of Calculus Part 2 state regarding the value of a definite integral?
    ∫abf(x)dx=\int_{a}^{b} f(x) dx =F(b)−F(a) F(b) - F(a)
  • Steps to evaluate ∫13(2x)dx\int_{1}^{3} (2x) dx using FTC Part 2

    1️⃣ Find the antiderivative of 2x2x, which is F(x)=F(x) =x2 x^{2}
    2️⃣ Compute the net change: F(3)−F(1)=F(3) - F(1) =32−12= 3^{2} - 1^{2} =9−1= 9 - 1 =8 8
  • Evaluate ∫141xdx\int_{1}^{4} \frac{1}{\sqrt{x}} dx using FTC Part 2.

    2