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Tangent Lines & Derivatives quiz

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  • What is the main difference between a secant line and a tangent line on a curve?
    A secant line intersects the curve at two points, while a tangent line touches the curve at only one point.
  • What formula is used to find the slope of a tangent line at a specific point c?
    The formula is limit as x approaches c of (f(x) - f(c)) / (x - c).
  • Why do we need to simplify the expression when calculating the slope of a tangent line using limits?
    We need to simplify to avoid division by zero, which occurs when x approaches c.
  • How can the difference of squares help in simplifying the limit expression for tangent lines?
    Factoring the difference of squares allows us to cancel terms and eliminate division by zero.
  • What is the slope of the tangent line to f(x) = x^2 at x = 1?
    The slope is 2.
  • What is the slope of the secant line between two points on f(x) = x^2, and how does it compare to the tangent line's slope?
    The secant line's slope is steeper (e.g., 4) compared to the tangent line's slope (e.g., 2) at x = 1.
  • What do we call the slope of a tangent line at a single point?
    It is called the instantaneous rate of change or the derivative.
  • What is the first step in finding the equation of a tangent line to a function at a given x value?
    Plug the x value of interest (c) into the function to find the corresponding y value, f(c).
  • After finding the slope of a tangent line, what form do we use to write its equation?
    We use the point-slope form: y - y₁ = m(x - x₁).
  • What is the slope of the tangent line to f(x) = 3x^2 - 4 at x = -2?
    The slope is -12.
  • What is the equation of the tangent line to f(x) = 3x^2 - 4 at x = -2?
    The equation is y = -12x - 16.
  • What is the general formula for the derivative of a function f(x)?
    The derivative is f'(x) = limit as h approaches 0 of (f(x+h) - f(x)) / h.
  • What does the variable h represent in the limit definition of the derivative?
    h represents the difference between two x-values, which approaches zero to find the instantaneous rate of change.
  • What is the derivative of f(x) = x^2 for any x value?
    The derivative is f'(x) = 2x.
  • How can you use the general derivative to find the slope of the tangent line at any specific x value?
    Plug the desired x value into the derivative formula to get the slope at that point.