Download presentation

Presentation is loading. Please wait.

1
**Department of Mathematics University of Leicester**

Parametric Department of Mathematics University of Leicester

2
What is it? A parametric equation is a method of defining a relation using parameters. For example, using the equation: We can use a free parameter, t, setting: and

3
What is it? We can see that this still satisfies the equation, while defining a relationship between x and y using the free parameter, t.

4
**Why do we use parametric equations**

Parameterisations can be used to integrate and differentiate equations term wise. You can describe the motion of a particle using a parameterisation: r being placement.

5
**Why do we use parametric equations**

Now we can use this to differentiate each term to find v, the velocity:

6
**Why do we use parametric equations**

Parameters can also be used to make differential equations simpler to differentiate. In the case of implicit differentials, we can change a function of x and y into an equation of just t.

7
**Why do we use parametric equations**

Some equations are far easier to describe in parametric form. Example: a circle around the origin Cartesian form: Parametric form:

8
**How to get Cartesian from parametric**

Getting the Cartesian equation of a parametric equation is done more by inspection that by a formula. There are a few useful methods that can be used, which are explored in the examples.

9
**How to get Cartesian from parametric**

Example 1: Let: So that: and

10
**How to get Cartesian from parametric**

Next set t in terms of y: Now we can substitute t in to the equation of x to eliminate t.

11
**How to get Cartesian from parametric**

Substituting in t: Which expands to:

12
**How to get Cartesian from parametric**

Example 2: Let: So that: and

13
**How to get Cartesian from parametric**

To change this we can see that: And

14
**How to get Cartesian from parametric**

And as we know that We can see that:

15
**How to get Cartesian from parametric**

Which equals: This is the Cartesian equation for an ellipse.

16
Example Example 3: let: Be the Cartesian equation of a circle at the point (a,b). Change this into parametric form.

17
Example If we set: And: Then we can solve this using the fact that:

18
Example From this we can see that: So: Therefore:

19
Example Similarly: So: Therefore:

20
**Example Compiling this, we can see that:**

Which is the parametric equation for a circle at the point (a,b).

21
Polar co-ordinates Parametric equations can be used to describe curves in polar co-ordinate form: For example:

22
Polar co-ordinates Here we can see, that if we set t as the angle, then we can describe x and y in terms of t: Using trigonometry: and

23
Polar co-ordinates These can be used to change Cartesian equations to parametric equations:

24
**Polar co-ordinates: example**

Let: Be the equation for a circle. If we set:

25
**Polar co-ordinates: example**

We can see that if we substitute these in, then the equation still holds: Therefore we can use: As a parameterisation for a circle.

26
**Finding the gradient of a parametric curve**

To find dy/dx we need to use the chain rule:

27
**How to get Cartesian from parametric: example**

Let: and Then:

28
**How to get Cartesian from parametric: example**

Then, using the chain rule:

29
**Extended parametric example**

Let: Be the Cartesian equation.

30
**Extended parametric example**

Then to change this into parametric form, we need to find values of x and y that satisfy the equation. If we set: And:

31
**Extended parametric example**

Then we have: Which expands to:

32
**Extended parametric example**

We know that: Therefore we can see that our values of x and y satisfy the equation. Therefore:

33
**Extended parametric example**

Now, as this is the placement of the particle, we can find the velocity of the particle by differentiating each term:

34
**Extended parametric example**

Next, we can find the gradient of the curve. Using the formula:

35
**Extended parametric example**

Using this: And:

36
**Extended parametric example**

Therefore the gradient is:

37
Conclusion Parametric equations are about changing equations to just 1 parameter, t. Parametric is used to define equations term wise. We can use the chain rule to find the gradient of a parametric equation.

38
**Conclusion Standard parametric manipulation of polar co- ordinates is:**

x=rcos(t) Y=rsin(t)

Similar presentations

Presentation is loading. Please wait....

OK

Arc Length and Curvature

Arc Length and Curvature

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on propagation of sound waves Free download ppt on report writing Ppt on uninterrupted power supply Ppt on marketing management by philip kotler's segment-by-segment The constitution for kids ppt on batteries Holographic 3d display ppt on tv Ppt on ram and rom function Raster and random scan display ppt on tv Ppt on 21st century skills map Ppt on orphans in india