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Antiderivatives quiz #1 Flashcards

Antiderivatives quiz #1
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  • What is an antiderivative?
    An antiderivative is a function whose derivative is the given function; it is the reverse process of differentiation.
  • What is the general form of the antiderivative of 2x?
    The general antiderivative of 2x is x^2 + c, where c is an arbitrary constant.
  • Why do we add a constant 'c' when finding an antiderivative?
    We add a constant 'c' because the derivative of any constant is zero, so the original function could have any constant term.
  • How do you find a particular antiderivative given a point, such as f(2) = 3 for f'(x) = 2x?
    First, find the general antiderivative (x^2 + c), then substitute x = 2 and f(2) = 3 to solve for c.
  • What is the particular antiderivative of f'(x) = 2x if f(2) = 3?
    The particular antiderivative is x^2 - 1.
  • What is the antiderivative of the constant function f(x) = 3?
    The antiderivative is 3x + c.
  • What is the antiderivative of f(x) = 0?
    The antiderivative is a constant, c.
  • How can you check if your antiderivative is correct?
    Take the derivative of your antiderivative; if it matches the original function, your answer is correct.
  • What is the general antiderivative of f(x) = 3x^2?
    The general antiderivative is x^3 + c.
  • Given f'(x) = 3x^2 and f(1) = 5, what is the particular antiderivative?
    The particular antiderivative is x^3 + 4.
  • What does the constant 'c' represent in the general antiderivative?
    The constant 'c' represents any possible vertical shift of the function, accounting for all functions with the same derivative.
  • If the derivative of F(x) is f(x), what is the relationship between F(x) and f(x)?
    F(x) is an antiderivative of f(x), meaning F'(x) = f(x).
  • What is the antiderivative of f(x) = x^n, where n ≠ -1?
    The antiderivative is (1/(n+1)) x^(n+1) + c.
  • Why is there an infinite family of antiderivatives for a given function?
    Because adding any constant to an antiderivative does not change its derivative, so there are infinitely many possible antiderivatives differing by a constant.
  • What is the process for finding a particular antiderivative when given an initial condition?
    First, find the general antiderivative, then substitute the given point to solve for the constant c, and write the final function with this value.