Presentation on theme: "Department of Mathematics University of Leicester"— Presentation transcript:
1Department of Mathematics University of Leicester ParametricDepartment of MathematicsUniversity of Leicester
2What 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
3What is it?We can see that this still satisfies the equation, while defining a relationship between x and y using the free parameter, t.
4Why 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.
5Why do we use parametric equations Now we can use this to differentiate each term to find v, the velocity:
6Why 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.
7Why do we use parametric equations Some equations are far easier to describe in parametric form.Example: a circle around the originCartesian form:Parametric form:
8How 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.
9How to get Cartesian from parametric Example 1:Let:So that:and
10How 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.
11How to get Cartesian from parametric Substituting in t:Which expands to:
12How to get Cartesian from parametric Example 2:Let:So that:and
13How to get Cartesian from parametric To change this we can see that:And
14How to get Cartesian from parametric And as we know thatWe can see that:
15How to get Cartesian from parametric Which equals:This is the Cartesian equation for an ellipse.
16ExampleExample 3: let:Be the Cartesian equation of a circle at the point (a,b).Change this into parametric form.
17ExampleIf we set:And:Then we can solve this using the fact that:
36Extended parametric example Therefore the gradient is:
37ConclusionParametric 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.
38Conclusion Standard parametric manipulation of polar co- ordinates is: x=rcos(t)Y=rsin(t)