MAT01B1: Curves defined by parametric equations

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1 MAT01B1: Curves defined by parametric equations Dr Craig 10 October 2018

2 My details: Consulting hours: Monday 14h40 15h25 Thursday 11h20 12h55 Friday 11h20 12h55 Office C-Ring (Or, just google Andrew Craig maths.)

3 Semester Test 2 Saturday 20 October 09h00, B-LES 100/101 Scope: 4.5, 4.7, , 6.2, 6.3, 6.5 (including proof of MVT for Integrals) 8.1, , and some 10.2 (derivatives but not area)

4 Importance of the test Counts 35% of your SM Make sure your SM is higher than your goal for the module! Preparing for the test Tut questions Saturday class worksheets e-quiz/webwork, 2017 paper Past exam papers (e-exam website)

5 Tutor centre applications Forms are on Blackboard Deadline is this Friday Requirements: completed application form, certified copies of ID/passport and academic record Minimum 60% for maths modules

6 Saturday class These details are confirmed for Saturday 13 October: 09h00 to 12h00 C-LES 402 A variety of questions from sections in the test

7 Question 16 from 8.2: In the first-session tutorial on Tues (9 Oct) I mistakenly wrote the question on the board as x 2/3 + y 1/3 = 1, 0 y 1. It should have been: x 2/3 + y 2/3 = 1, 0 y 1. The question asked for the surface area when the curve was rotated about the y-axis. The correct answer is 6π 5.

8 Parametric curves

9 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.

10 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)).

11 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.

12 Example (restrictions on t) Sketch the parametric curve defined by x = t 2 2t y = t t 4

13 Example: What curve is represented by the following parametric equations? x = cos t, y = sin t, 0 t 2π

14 Example: Describe the curve represented by the parametric equations: x = sin 2t, y = cos 2t, 0 t 2π

15 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 sin t, 0 t 2π

16 Example: Sketch the curve with parametric equations x = sin t, y = sin 2 t

17 Examples of parametric curves Go to and create your own.

18 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

19 Families of parametric curves Consider the family of curves x = a + cos t y = a tan t + sin t

20

21

22 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.

23 Getting used to parametric curves Go through exercises 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.

24 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

25 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.

26 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.

27 Example: a curve is defined by the parametric equations x = f(t) = t 2 y = g(t) = t 3 3t (a) Find the values of t at the point (3, 0). (b) Show that the curve has two tangents at the point (3, 0) and find their equations. (c) Find the points (x, y) on the curve where the tangent is horizontal or vertical. (d) Determine where the curve is concave upward or downward. (e) Sketch the curve.

28 x = f(t) = t 2 y = g(t) = t 3 3t

29 This next example will be covered at the beginning of the lecture on Tuesday 16 October.

30 Example: (a) Find the tangent to the cycloid x = r(θ sin θ), y = r(1 cos θ) at the point θ = π/3. (b) Find the points when the tangent to the curve is horizontal and when it is vertical.

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