Some important definitions:
Parametric Differentiation
Key result to learn!! [pic]
Example: If [pic], then:
[pic]
So: [pic]
Example 2: A curve, C, is given by
[pic].
A is the point (1, 3). Line l is the tangent to C at A. The line l also intersects the curve at B.
a) Find the equation of l.
b) Find the value of t at B.
a) [pic]
We need to know the value of t at A. The x value is given by 2t + 3, so 2t + 3 = 1 i.e. t = -1.
Therefore the gradient at A is: [pic]
So equation of tangent is [pic]
Substitute in x = 1, y = 3: [pic]
Therefore the equation of l is [pic]
b) To see where the tangent intersects the curve, we substitute [pic] into the equation of l:
[pic]
So: [pic]
This factorises: [pic]
So t = 2 is the required value. |Trigonometric differentiation:
Remember these results!!
[pic] [pic]
[pic]
[pic]
[pic]
[pic]
Note: When differentiating trigonometric functions, the angle measure is radians. |Implicit Differentiation
A function is given in implicit form if it hasn’t been written in the form y = … .
Note that: [pic]
Example: Find the equation of the normal to [pic] at the point (2, -1).
Solution: Differentiate term by term with respect to x:
[pic] [pic]
[pic] [pic]
Therefore: [pic]
So: [pic]
i.e. [pic]
When x = 2 and y = -1:
[pic]
The gradient of the normal is: [pic]
So equation is: [pic]
Put in x = 2, y = -1: [pic]
So equation is [pic] | |
| | | |
| |[pic] | |
| |Examples: | |
| |[pic]; [pic] | |
| | | |
| |Example: Find the location of the stationary points for the curve | |
| |[pic]. | |
| | | |
| |Solution: First we differentiate using the product rule: [pic] | |
| |Stationary points occur where [pic], i.e. where | |
| |[pic]. | |
| |[pic] | |
| |Therefore [pic] | |
| |So [pic] | |
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