MAT1B01: Curves defined by parametric equations Dr Craig 24 October 2016
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Semester Test 2 Saturday 29 October 09h30, D1 Lab 208 Scope: 4.3, 4.5, 4.7, 3.9 6.1, 6.2, 6.3, 6.5 (including proof of MVT for Integrals) 8.1, 8.2 9.3, 9.5
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Parametric curves
Parametric curves A parametric curve is a curve sketched in the plane (R 2 ) where the points of the curve are defined by a parameter, usually t. Many parametric curves do not result in a curve where y is a function of x. The variable t is used because in most instances we think of this variable as time, and then the x- and y-coordinates as the position in space of a particle or object.
When we define a parametric curve, it has more information than a normal curve. Each point is associated to a value of t (a specific time) and the curve has a particular direction. A curve defined by the parametric equations x = f(t) y = g(t) a t b has initial point (f(a), g(a)) and terminal point (f(b), g(b)).
Example: Sketch the parametric curve defined by x = t 2 2t y = t + 1 Note: we can also represent the points of this curve by a parabola of x in terms of y. However, we still need the parameter t in order to know the direction of the curve.
Example (restrictions on t) Sketch the parametric curve defined by x = t 2 2t y = t + 1 0 t 4
Example: What curve is represented by the following parametric equations? x = cos t, y = sin t, 0 t 2π
Example: Describe the curve represented by the parametric equations: x = sin 2t, y = cos 2t, 0 t 2π
Example: Find equations for the circle of radius r and centered at the point (h, k). Solution: x = h+r cos t y = k+r cos t, 0 t 2π
Example: Sketch the curve with parametric equations x = sin t, y = sin 2 t
Examples of parametric curves Go to www.fooplot.com and create your own.
Cycloids A cycloid is the curve obtained by marking a point P on a circle of radius r and then rolling this circle (think of it as a car tyre) along the x-axis. The parametric equations that define such a curve are: x = r(θ sin θ), y = r(1 cos θ), θ R
Families of parametric curves Consider the family of curves x = a + cos t y = a tan t + sin t
Example: sketch the following parametric curve (include in your sketch the direction of the curve). x = cos 2 t, y = 1 sin t, 0 t π 2 It might be useful to try and solve for y in terms of x (or x in terms of y) to get an idea of the shape of the curve.
Getting used to parametric curves Go through exercises 24 28 in Stewart (pg 642) to see how to relate the equations x = f(t) and y = g(t) to the parametric curve that they represent. Use logical reasoning to rule out certain options in the matching exercises in Q24 and Q28.
What next? Now that we have defined parametric curves and seen a few examples, we will use our techniques from calculus to find out more about their properties. Derivatives of parametric curves dy dx = dy dt dx dt if dx dt 0
Derivatives of parametric curves The derivative of a curve dy dy dx = dt dx dt is defined whenever dx dt 0 Horizontal tangent whenever dy dt = 0. We will get a vertical tangent whenever dx dy dy = 0 so long as 0. If dt dt dt = 0 we need to perform some additional checks.
Second derivatives To get the second derivative of a parametric curve, we differentiate the first derivative with respect to x: d 2 y dx = d 2 dx ( ) dy dx = d dt ( ) dy dx dx dt We can use the second derivative to explore the concavity of a parametric curve.
Example: a curve is defined by the parametric equations x = f(t) = t 2 y = g(t) = t 3 3t (a) Show that the curve has two tangents at the point (3, 0) and find their equations. (b) Find the points (x, y) on the curve where the tangent is horizontal or vertical. (c) Determine where the curve is concave upward or downward. (d) Sketch the curve.
x = f(t) = t 2 y = g(t) = t 3 3t
Tomorrow: Area under a parametric curve Arc length of parametric curves Surface area of a rotated parametric curve