SP21 MATH 2024 - Calculus with Parametric Functions.pdf -...

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MATH 2024 Calculus with Parametric Curves Adams Slope and Tangent Lines Now that you can represent a graph in the plane by a set of parametric equations, it is natural to ask how to use calculus to study plane curves. To begin, let’s take a look at the projectile represented by the parametric equations x = 24 2 t and y = - 16 t 2 + 24 2 t as shown in the figure below: You know that these equations enable us to locate the position of the projectile at a given time. You also know that the object is initially projected at an angle of 45 . But how can you find the angle θ representing the object’s direction at some other time t ? The following theorem answers this question by giving a formula for the slope of the tangent line as a function of t . Parametric Form of the Derivative If a smooth curve C is given by the equations x = f ( t ) and y = g ( t ), then the slope of C at ( x, y ) is dy dx = dy dt dx dt , dx dt 6 = 0 Proof: Simply for illustration, consider the plane curve C generated by x = f ( t ) and y = g ( t ) for some arbitrary functions f ( t ) and g ( t ): Page 1 of 9
MATH 2024 Calculus with Parametric Curves Adams
Example 1. Find dy dx for the curve given by x = t + cos t and y = sin t .
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MATH 2024

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