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Curve Sketching quiz

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  • What is the domain of the function f(x) = x^3 - 3x^2 + 4?
    The domain is all real numbers, from negative infinity to positive infinity.
  • How do you find the x-intercepts of a polynomial function?
    Set the function equal to zero and solve for x, often by factoring the polynomial.
  • What is the y-intercept of f(x) = x^3 - 3x^2 + 4?
    The y-intercept is 4, found by evaluating f(0).
  • Does the function f(x) = x^3 - 3x^2 + 4 have any asymptotes?
    No, this polynomial function does not have any asymptotes.
  • How do you test a function for symmetry?
    Plug in -x for x and compare the result to the original function to check for even or odd symmetry.
  • What does the sign of the first derivative tell you about a function?
    It tells you whether the function is increasing (positive) or decreasing (negative) on an interval.
  • How do you find critical points of a function?
    Set the first derivative equal to zero or find where it does not exist, then solve for x.
  • What are the critical points of f(x) = x^3 - 3x^2 + 4?
    The critical points are x = 0 and x = 2.
  • On which intervals is f(x) = x^3 - 3x^2 + 4 increasing?
    It is increasing on (-∞, 0) and (2, ∞).
  • How do you determine where a function is concave up or concave down?
    Use the sign of the second derivative: positive means concave up, negative means concave down.
  • What is the inflection point of f(x) = x^3 - 3x^2 + 4?
    The inflection point is at x = 1, where the concavity changes.
  • How do you use the first derivative test to find local extrema?
    Check where the sign of the first derivative changes at critical points to identify local maxima or minima.
  • Where does f(x) = x^3 - 3x^2 + 4 have a local maximum?
    There is a local maximum at x = 0.
  • Where does f(x) = x^3 - 3x^2 + 4 have a local minimum?
    There is a local minimum at x = 2.
  • What is the general process for sketching the graph of a function using calculus?
    Analyze the domain, intercepts, asymptotes, and symmetry, then use the first and second derivatives to find intervals of increase/decrease, concavity, and local extrema.