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Fundamental Theorem of Calculus quiz

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  • What does the Fundamental Theorem of Calculus Part One state about the relationship between derivatives and definite integrals?
    It states that the derivative of the antiderivative of a continuous function equals the original function, or F'(x) = f(x).
  • What must be true about a function for the Fundamental Theorem of Calculus Part One to apply?
    The function must be continuous on the interval [a, b].
  • How do you find the derivative of an integral with a variable upper bound using the Fundamental Theorem of Calculus Part One?
    You replace the variable inside the integral with the upper bound variable and, if the upper bound is a function of x, multiply by its derivative (chain rule).
  • What is the result of taking the derivative of the integral from a to x of f(t) dt?
    The result is f(x).
  • If the upper bound of the integral is x^2 instead of x, what additional step must you take?
    You must multiply the result by the derivative of x^2, which is 2x.
  • What does the Fundamental Theorem of Calculus Part Two allow you to do with definite integrals?
    It allows you to evaluate a definite integral by finding the antiderivative and subtracting its value at the lower bound from its value at the upper bound.
  • How do you write the evaluation of a definite integral using the Fundamental Theorem of Calculus Part Two?
    You write it as F(b) - F(a), where F is any antiderivative of f.
  • Why is the constant of integration (+C) not needed when evaluating definite integrals?
    Because the constants cancel out when you subtract F(a) from F(b).
  • What is the geometric interpretation of a definite integral?
    It represents the area under the curve of the function between the given bounds.
  • What rule do you use to find the antiderivative of a polynomial term like 2x?
    You use the power rule: increase the exponent by one and divide by the new exponent.
  • When evaluating the definite integral of a sum or difference of functions, what can you do?
    You can take the antiderivative of each term separately and then evaluate the result at the bounds.
  • What is the solution to the definite integral of 2x from 2 to 5?
    The solution is 21.
  • If you have the definite integral from 1 to 4 of x^2 - 4x + 5 dx, what is the final answer after evaluating?
    The final answer is 6.
  • What happens if you take the derivative of the antiderivative of a function?
    You get back the original function.
  • What is the main idea to remember when using the Fundamental Theorem of Calculus Part One with a function of x as the upper bound?
    Replace the variable inside the integral with the upper bound and multiply by the derivative of the upper bound (chain rule).