![Page 1: 10.1 Parametric Functions. In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function](https://reader030.vdocuments.mx/reader030/viewer/2022032414/56649ef35503460f94c04ef1/html5/thumbnails/1.jpg)
10.1 Parametric Functions
![Page 2: 10.1 Parametric Functions. In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function](https://reader030.vdocuments.mx/reader030/viewer/2022032414/56649ef35503460f94c04ef1/html5/thumbnails/2.jpg)
In chapter 1, we talked about parametric equations.Parametric equations can be used to describe motion that is not a function.
x f t y g t
If f and g have derivatives at t, then the parametrized curve also has a derivative at t.
dx dy
f t g tdt dt
![Page 3: 10.1 Parametric Functions. In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function](https://reader030.vdocuments.mx/reader030/viewer/2022032414/56649ef35503460f94c04ef1/html5/thumbnails/3.jpg)
The formula for finding the slope of a parametrized curve is:
dy
dy dtdxdxdt
This makes sense if we think about canceling dt.
Since the derivatives of each parametric equation are:
dx dy
f t g tdt dt
![Page 4: 10.1 Parametric Functions. In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function](https://reader030.vdocuments.mx/reader030/viewer/2022032414/56649ef35503460f94c04ef1/html5/thumbnails/4.jpg)
Example 2 (page 514): The parametric curve below is given by the equations 2 3 and 1.5 1x t t y t t t Find the values of t for which the line tangent to this curve is
a) vertical
b) horizontal
21 3
1 2
dy t
dx t
When is this equal to 0?
1
2t
1
4x
3
8y
![Page 5: 10.1 Parametric Functions. In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function](https://reader030.vdocuments.mx/reader030/viewer/2022032414/56649ef35503460f94c04ef1/html5/thumbnails/5.jpg)
Example 2 (page 514): The parametric curve below is given by the equations
a) vertical
b) horizontal
Find the values of t for which the line tangent to this curve is
21 3
1 2
dy t
dx t
When is this equal to 0?
1
3t 3 1
3x
2
3 3y
3 1
3x
2
3 3y
2 3 and 1.5 1x t t y t t t
![Page 6: 10.1 Parametric Functions. In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function](https://reader030.vdocuments.mx/reader030/viewer/2022032414/56649ef35503460f94c04ef1/html5/thumbnails/6.jpg)
The formula for finding the slope of a parametrized curve is:
dy
dy dtdxdxdt
Remember since this is still a parametric function
that will be given in terms of tdx
dy
In the next slides, we will be using y in place of
for simplicity.dx
dy
![Page 7: 10.1 Parametric Functions. In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function](https://reader030.vdocuments.mx/reader030/viewer/2022032414/56649ef35503460f94c04ef1/html5/thumbnails/7.jpg)
To find the second derivative of a parametrized curve, we find the derivative of the first derivative:
dydtdxdt
2
2
d y
dx d
ydx
1. Find the first derivative (dy/dx).2. Find the derivative of dy/dx with respect to t.
3. Divide by dx/dt.
![Page 8: 10.1 Parametric Functions. In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function](https://reader030.vdocuments.mx/reader030/viewer/2022032414/56649ef35503460f94c04ef1/html5/thumbnails/8.jpg)
Example 2 (page 514):2
2 32
Find as a function of if and .d y
t x t t y t tdx
![Page 9: 10.1 Parametric Functions. In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function](https://reader030.vdocuments.mx/reader030/viewer/2022032414/56649ef35503460f94c04ef1/html5/thumbnails/9.jpg)
Example 2 (page 514):2
2 32
Find as a function of if and .d y
t x t t y t tdx
1. Find the first derivative (dy/dx).
dy
dy dtydxdxdt
21 3
1 2
t
t
![Page 10: 10.1 Parametric Functions. In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function](https://reader030.vdocuments.mx/reader030/viewer/2022032414/56649ef35503460f94c04ef1/html5/thumbnails/10.jpg)
2. Find the derivative of dy/dx with respect to t.
21 3
1 2
dy d t
dt dt t
2
2
2 6 6
1 2
t t
t
Quotient Rule
![Page 11: 10.1 Parametric Functions. In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function](https://reader030.vdocuments.mx/reader030/viewer/2022032414/56649ef35503460f94c04ef1/html5/thumbnails/11.jpg)
3. Divide by dx/dt.
2
2
d y
dx
dydtdxdt
2
2
2 6 6
1 2
1 2
t t
t
t
2
3
2 6 6
1 2
t t
t
![Page 12: 10.1 Parametric Functions. In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function](https://reader030.vdocuments.mx/reader030/viewer/2022032414/56649ef35503460f94c04ef1/html5/thumbnails/12.jpg)
2 2( ) ( )L dx dy
The length of a segment of a parametric curve can be approximated using the Pythagorean theorem:
L
dx
dy
![Page 13: 10.1 Parametric Functions. In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function](https://reader030.vdocuments.mx/reader030/viewer/2022032414/56649ef35503460f94c04ef1/html5/thumbnails/13.jpg)
2 2( ) ( )L dx dy
dt
dt
2 2dx dy
dt dt
2 22
1( ) ( )
( )dx dy
dt dt
dt
Multiply by dt/dt
After some algebra
As dt gets smaller, the approximation gets better. As dt goes to zero, we can determine the exact length of the curve.
![Page 14: 10.1 Parametric Functions. In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function](https://reader030.vdocuments.mx/reader030/viewer/2022032414/56649ef35503460f94c04ef1/html5/thumbnails/14.jpg)
This equation for the length of a parametrized curve is similar to our previous “length of curve” equation.
(Notice the use of the Pythagorean Theorem.)
2 2b
a
dy dxL dt
dt dt
If we wanted to add up many segments over an interval of t, we can add an infinite amount of infinitely small segments to get…An Integral:
![Page 15: 10.1 Parametric Functions. In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function](https://reader030.vdocuments.mx/reader030/viewer/2022032414/56649ef35503460f94c04ef1/html5/thumbnails/15.jpg)
Example 2 (page 514): The parametric curve below is given by the equations 2 3 and 1.5 1x t t y t t t Find the length of this curve over the given interval
2 21
1.5
dx dyL dt
dt dt
21 2 2
1.51 2 1 3L t t dt
5.69
![Page 16: 10.1 Parametric Functions. In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function](https://reader030.vdocuments.mx/reader030/viewer/2022032414/56649ef35503460f94c04ef1/html5/thumbnails/16.jpg)
Likewise, the equations for the surface area of a parametrized curve are similar to our previous “surface area” equations:
Revolution about the -axis 0x y 2 2
2b
a
dx dyS y dt
dt dt
Revolution about the -axis 0y x
2 2
2b
a
dx dyS x dt
dt dt
Revolution about the -axis 0x y 2 2
2b
a
dx dyS y dt
dt dt
![Page 17: 10.1 Parametric Functions. In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function](https://reader030.vdocuments.mx/reader030/viewer/2022032414/56649ef35503460f94c04ef1/html5/thumbnails/17.jpg)
By the way, this is an actual curve whose equations are:
sin 2
2cos 5
x t t t
y t t t