AP Calculus BC Monday, 16 November 2015

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AP Calculus BC Monday, 16 November 2015 OBJECTIVE TSW (1) find the slope of a tangent line to a parametric curve, and (2) find the concavity of a parametric curve. TODAY’S ASSIGNMENT (due tomorrow) Sec. 11.2: Problems given on the next slide. TEST: Parametric Equations, Polar Equations, and Vectors is this Friday, 20 November 2015.

Sec. 11.2: Calculus with Parametric Equations Find (a) dy/dx, (b) d 2y/dx2, (c) the slope, (d) the equation of the tangent line, and (e) concavity (if possible) at the given value of the parameter. Due tomorrow, Tuesday, 17 November 2015. Find all points (if any) of horizontal and vertical tangency to the curve defined by the given parametric equations.

Sec. 11.2: Calculus with Parametric Equations 3

Sec. 11.2: Calculus with Parametric Equations How do you find the derivative of a set of parametric equations? 4

Sec. 11.2: Calculus with Parametric Equations 11.1 5

Sec. 11.2: Calculus with Parametric Equations Ex: Find dy / dx for the curve given by 6

Sec. 11.2: Calculus with Parametric Equations For higher order derivatives, use Theorem 11.1 repeatedly. Second derivative Third derivative Notice that the denominator for each higher-order derivative is always dx/dt. 7

Sec. 11.2: Calculus with Parametric Equations Ex: For the curve given by find the slope and concavity at the point (2, 3). 8

Sec. 11.2: Calculus with Parametric Equations The second derivative is 9

Sec. 11.2: Calculus with Parametric Equations We’re given the point (2, 3) & Since x = 2, that means that or t = 4. The slope at (2, 3) is: And the concavity at (2, 3) is: ∴ concave up 10

Sec. 11.2: Calculus with Parametric Equations Ex: The prolate cycloid given by crosses itself at the point (0, 2). Find the equations of both tangent lines at this point.

Sec. 11.2: Calculus with Parametric Equations Ex: The prolate cycloid given by crosses itself at the point (0, 2). Find the equations of both tangent lines at this point.

Sec. 11.2: Calculus with Parametric Equations Ex: The prolate cycloid given by crosses itself at the point (0, 2). Find the equations of both tangent lines at this point.

Sec. 11.2: Calculus with Parametric Equations A point is given; you need only determine the slope, dy/dx. Now you need to determine t. Use the original parametric equations to determine t.

Sec. 11.2: Calculus with Parametric Equations Solve one of these equations for t. The second equation would be the easiest.

Sec. 11.2: Calculus with Parametric Equations When t = /2, and the equation is

Sec. 11.2: Calculus with Parametric Equations When t = –/2, and the equation is

Sec. 11.2: Calculus with Parametric Equations Horizontal Tangents If when t = t0, then the curve represented by has a horizontal tangent at

Sec. 11.2: Calculus with Parametric Equations Vertical Tangents If when t = t0, then the curve represented by has a vertical tangent at

Sec. 11.2: Calculus with Parametric Equations Ex: Find all points (if any) of horizontal and vertical tangency to the curve defined by x = t + 1 and y = t 2 + 3t.

Sec. 11.2: Calculus with Parametric Equations Ex: Find all points (if any) of horizontal and vertical tangency to the curve defined by x = t + 1 and y = t 2 + 3t. never

Sec. 11.2: Calculus with Parametric Equations never Horizontal tangency: Vertical tangency: never NONE

Sec. 11.2: Calculus with Parametric Equations Ex: Find all points (if any) of horizontal and vertical tangency to the curve defined by x = cosθ and y = 2sin2θ.

Sec. 11.2: Calculus with Parametric Equations Find (a) dy/dx, (b) d 2y/dx2, (c) the slope, (d) the equation of the tangent line, and (e) concavity (if possible) at the given value of the parameter. Due tomorrow, Tuesday, 17 November 2015. Find all points (if any) of horizontal and vertical tangency to the curve defined by the given parametric equations.