MTH 253 Calculus (Other Topics)Reminder: Tangent Lines to a (Cartesian) Function at a PointSlope at a Point of a Parametric CurveSlide 4Tangent Line to a Parametric Curve (Example)Slope at a Point of a Polar CurveTangent Lines to a Polar Curve (Example)MTH 253Calculus (Other Topics)Chapter 11 – Analytic Geometry in CalculusSection 11.2 – Tangent Lines & Arc Length for Parametric and Polar CurvesCopyright © 2006 by Ron Wallace, all rights reserved.Reminder: Tangent Lines to a (Cartesian) Function at a Point( )y f x=x0f(x0)Find the point (x0, f(x0))Determine y’ = f’(x)Evaluate f’(x0) = m0 0 0( ) '( )( )y f x f x x x= + -Example?Slope at a Point of a Parametric Curve( )x f t=( )y g t=Differentiate each equation, giving …'( )dxf tdt='( )dyg tdt=Divide, assuming dx/dt 0 …'( )'( )g t dy dt dyf t dx dt dx= =Therefore, the slope can be determined without eliminating the parameter.Slope at a Point of a Parametric Curve( )x f t=( )y g t=What is happening when dx/dt (i.e. f’(t)) is 0?What is happening when dy/dt (i.e. g’(t)) is 0?Vertical tangent.Horizontal tangent.What is happening when both are zero?“Singular Point” --- Analyzed Case-by-Case'( )'( )dy g tdx f t=Tangent Line to a Parametric Curve (Example)Find the equation of the line tangent to the curve when t = 2.2sinx t=25y t= -24.82cosdy tmdx t= = �-2Pt: (2sin , 5) (1.8, 1)t t - � -1 4.8( 1.8) 4.8 7.64y xx=- - -=- +When is the tangent line horizontal? vertical?'( )'( )dy g tdx f t=( )x f t=( )y g t=Slope at a Point of a Polar Curve( )r f q=From section 11.1 …cossinx ry rqq==cos sinsin cosdrrdy dy dddrdx dx drdq qqqqq qq+= =- +Essentially, parametric equations (parameter is ),so differentiating implicitly …Tangent Lines to a Polar Curve (Example)Find the points where the tangent lines are horizontal or vertical.4cos 2r q=228sin 28sin 2cos sinsin cos4cos 6cos 5 4cos 24co 4sin 6sin 5s 2dymdxq qq qqqqqq qq q= =-� �-� �=---� �� �( )r f q=horizontal tangents at = ? 90°, 24°, 156°vertical tangents at = ? 0°, 180°, 66°, 114°8sin 2drdqq=-cos sinsin cosdrrdy dy dddrdx dx drdq qqqqq qq+= =-
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