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AP Calculus BC – Chapter 10 Parametric, Vector, and Polar Functions 10.1: Parametric Functions

AP Calculus BC – Chapter 10 Parametric, Vector, and Polar Functions 10.1: Parametric Functions. Goals : Find derivatives and second derivatives of parametrically-defined functions. Calculate lengths of parametrically-defined curves and calculate surface areas. Derivatives:.

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AP Calculus BC – Chapter 10 Parametric, Vector, and Polar Functions 10.1: Parametric Functions

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  1. AP Calculus BC – Chapter 10Parametric, Vector, and Polar Functions 10.1: Parametric Functions Goals: Find derivatives and second derivatives of parametrically-defined functions. Calculate lengths of parametrically-defined curves and calculate surface areas.

  2. Derivatives: Derivative at a Point: A parametrized curve x=f(t), y=g(t), a ≤ t ≤ b, has a derivative at t=t0 if f and g have derivatives at t=t0. The curve is differentiable if it is differentiable at every parameter value. The curve is smooth if f’ and g’ are continuous and not simultaneously zero.

  3. Derivative Formula: A parametrized curve can yield one or more ways to define y as a function of x. Recall that at a point where both the curve and y(x) are differentiable and dx/dt ≠ 0, the derivatives dx/dt, dy/dt, and dy/dx are related by the formula:

  4. Second Derivative: Parametric Formula for d2y/dx2: If the equations x=f(t), y=g(t) define y as a twice-differentiable function of x, then at any point where dx/dt ≠ 0,

  5. Length of a Smooth Curve: Arc Length of a Smooth Parametrized Curve: If a smooth curve x=f(t), y=g(t), a ≤ t ≤ b, is traversed exactly once as t increases from a to b, the curve’s length is

  6. Surface Area: Surface Area (from a Smooth Parametrized Curve): If a smooth curve x=f(t), y=g(t), a ≤ t ≤ b, is traversed exactly once as t increases from a to b, then the areas of the surfaces generated by revolving the curve about the coordinate axes are as follows. 1. Revolution about the x-axis (y ≥ 0): 2. Revolution about the y-axis (x ≥ 0):

  7. Assignments: • HW 10.1: • Quick Review & Exploration. • Concepts Worksheet 1.4. • HW 10.1: #3, 9, 16, 21, 27, 30, 31, 32.

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