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Calculus with Parametric equations Example 2 Area under a curve Arc Length: Length of a curveCalculus with Parametric equationsLet C be a parametric curve described by the parametric equationsx f (t), y g (t). If the function f and g are differentiable and y is also a, dyand dxare related bydifferentiable function of x, the three derivatives dydxdtdtthe Chain rule:dy dxdy dtdx dtusing this we can obtain the formula to compute dyfrom dxand dy:dxdtdtdy dxdydtdxdtAnnette Pilkingtonifdx6 0dtLecture 35: Calculus with Parametric equations
Calculus with Parametric equations Example 2 Area under a curve Arc Length: Length of a curveCalculus with Parametric equationsLet C be a parametric curve described by the parametric equationsx f (t), y g (t). If the function f and g are differentiable and y is also a, dyand dxare related bydifferentiable function of x, the three derivatives dydxdtdtthe Chain rule:dy dxdy dtdx dtusing this we can obtain the formula to compute dyfrom dxand dy:dxdtdtdy dxIdydtdxdtifdx6 0dtThe value of dygives gives the slope of a tangent to the curve at anydxgiven point. This sometimes helps us to draw the graph of the curve.Annette PilkingtonLecture 35: Calculus with Parametric equations
Calculus with Parametric equations Example 2 Area under a curve Arc Length: Length of a curveCalculus with Parametric equationsLet C be a parametric curve described by the parametric equationsx f (t), y g (t). If the function f and g are differentiable and y is also a, dyand dxare related bydifferentiable function of x, the three derivatives dydxdtdtthe Chain rule:dy dxdy dtdx dtusing this we can obtain the formula to compute dyfrom dxand dy:dxdtdtdy dxIIdydtdxdtifdx6 0dtThe value of dygives gives the slope of a tangent to the curve at anydxgiven point. This sometimes helps us to draw the graph of the curve.The curve has a horizontal tangent when dy 0, and has a verticaldx .tangent when dydxAnnette PilkingtonLecture 35: Calculus with Parametric equations
Calculus with Parametric equations Example 2 Area under a curve Arc Length: Length of a curveCalculus with Parametric equationsLet C be a parametric curve described by the parametric equationsx f (t), y g (t). If the function f and g are differentiable and y is also a, dyand dxare related bydifferentiable function of x, the three derivatives dydxdtdtthe Chain rule:dy dxdy dtdx dtusing this we can obtain the formula to compute dyfrom dxand dy:dxdtdtdy dxIIIdydtdxdtdx6 0dtifThe value of dygives gives the slope of a tangent to the curve at anydxgiven point. This sometimes helps us to draw the graph of the curve.The curve has a horizontal tangent when dy 0, and has a verticaldx .tangent when dydxThe second derivatived2ydx 2can also be obtained fromd 2yd dy ( ) dx 2dx dxAnnette Pilkingtond dy(dt dxdxdt)ifdydxanddx.dtdx6 0dtLecture 35: Calculus with Parametric equationsIndeed,
Calculus with Parametric equations Example 2 Area under a curve Arc Length: Length of a curveExample 1Example 1 (a) Find an equation of the tangent to the curvex t 2 2ty t 3 3twhent 2Annette PilkingtonLecture 35: Calculus with Parametric equations
Calculus with Parametric equations Example 2 Area under a curve Arc Length: Length of a curveExample 1Example 1 (a) Find an equation of the tangent to the curvex t 2 2ty t 3 3twhent 2I When t 2, the corresponding point on the curve isP (4 4, 8 6) (8, 2).Annette PilkingtonLecture 35: Calculus with Parametric equations
Calculus with Parametric equations Example 2 Area under a curve Arc Length: Length of a curveExample 1Example 1 (a) Find an equation of the tangent to the curvex t 2 2ty t 3 3twhent 2I When t 2, the corresponding point on the curve isP (4 4, 8 6) (8, 2).I W
Calculus with Parametric equationsExample 2Area under a curveArc Length: Length of a curve Example 1 Example 1 (b) Find the point on the parametric curve where the tangent is
