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Linearization quiz #1

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  • What is the linearization of a function f(x) at a point x = a, and how is it used to approximate values of the function near a?
    The linearization of f(x) at x = a is L(x) = f(a) + f'(a)(x - a). It is used to approximate values of f(x) near a by replacing the function with its tangent line at that point, making calculations simpler and providing accurate estimates for values close to a.
  • What is the general formula for the linearization of a function f(x) at x = a?
    The formula is L(x) = f(a) + f'(a)(x - a).
  • How does linearization relate to the tangent line of a function at a point?
    Linearization is the equation of the tangent line to the function at the point x = a.
  • Why does zooming in on a smooth curve at a point make it look like a line?
    Because as you zoom in, the curve becomes nearly indistinguishable from its tangent line at that point.
  • How can linearization be used to approximate values of a function near x = a?
    By substituting values close to a into the linearization formula, you get estimates for f(x) that are close to the actual values.
  • What is the linearization of f(x) = x^2 at a = 1?
    The linearization is L(x) = 2x - 1.
  • If you use linearization to approximate f(1.05) for f(x) = x^2 at a = 1, what value do you get?
    You get L(1.05) = 1.1 as the approximate value.
  • How accurate is the linear approximation when x is close to a compared to when x is farther from a?
    The approximation is more accurate when x is close to a and becomes less accurate as x moves farther from a.
  • Why is linearization useful in real-world applications?
    It simplifies complex functions to linear ones, making calculations and estimations easier, especially when exact values are hard to compute.
  • What happens to the accuracy of linearization if you try to approximate values far from the point of tangency?
    The accuracy decreases as you move farther from the point of tangency because the function and its tangent line diverge.