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Calculus with Parametric Curves quiz

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  • How do you find the derivative dy/dx for parametric equations x(t) and y(t)?
    You divide dy/dt by dx/dt, so dy/dx = (dy/dt) / (dx/dt).
  • What must be true about dx/dt when finding dy/dx for parametric equations?
    dx/dt must not be zero, since division by zero is undefined.
  • What is the formula for the arc length s of a parametric curve from t = a to t = b?
    s = ∫ from a to b of sqrt[(dx/dt)^2 + (dy/dt)^2] dt.
  • How do you find the second derivative d²y/dx² for parametric equations?
    Differentiate the first derivative dy/dx with respect to t, then divide by dx/dt.
  • What is the geometric meaning of the derivative at a point on a parametric curve?
    It represents the slope of the tangent line at that point.
  • How do you find the equation of the tangent line to a parametric curve at t = t₀?
    Find x(t₀), y(t₀), and the slope dy/dx at t₀, then use point-slope form: y - y₀ = m(x - x₀).
  • What rule is used to differentiate a rational function like dy/dx = f(t)/g(t)?
    The quotient rule: [g(t)f'(t) - f(t)g'(t)] / [g(t)]².
  • How do you find higher-order derivatives (third, fourth, etc.) for parametric curves?
    Differentiate the previous derivative with respect to t and divide by dx/dt each time.
  • What is the relationship between the arc length formula and the distance formula?
    The arc length formula sums infinitely many small distances along the curve, similar to the distance formula for two points.
  • What identity simplifies the arc length integral for x(t) = 2cos(t), y(t) = 2sin(t)?
    cos²(t) + sin²(t) = 1, the Pythagorean identity.
  • What is the arc length of the curve x(t) = 2cos(t), y(t) = 2sin(t) from t = 0 to t = π?
    The arc length is 2π.
  • What is the derivative of x(t) = t² + 3t with respect to t?
    dx/dt = 2t + 3.
  • What is the derivative of y(t) = 2t³ - 4 with respect to t?
    dy/dt = 6t².
  • How do you compute dy/dx at t = 1 for x(t) = t² + 3t and y(t) = 2t³ - 4?
    Plug t = 1 into dy/dt and dx/dt, then divide: dy/dx = 6/5.
  • What is the general process for finding the tangent line to a parametric curve?
    Evaluate x and y at the given t, find the slope dy/dx at that t, and use point-slope form to write the equation.