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Example Questions
Example Question #31 : Functions, Graphs, And Limits
What is the derivative of ?
In order to find the derivative of a polar equation
, we must first find the derivative of
with respect to
as follows:
We can then swap the given values of and
into the equation of the derivative of an expression into polar form:
Using the trigonometric identity , we can deduce that
. Swapping this into the denominator, we get:
Example Question #32 : Functions, Graphs, And Limits
What is the derivative of ?
In order to find the derivative of a polar equation
, we must first find the derivative of
with respect to
as follows:
We can then swap the given values of and
into the equation of the derivative of an expression into polar form:
Using the trigonometric identity , we can deduce that
. Swapping this into the denominator, we get:
Example Question #1 : Vector Form
Find the vector form of to
.
When we are trying to find the vector form we need to remember the formula which states to take the difference between the ending and starting point.
Thus we would get:
Given and
In our case we have ending point at and our starting point at
.
Therefore we would set up the following and simplify.
Example Question #33 : Parametric, Polar, And Vector Functions
Given points and
, what is the vector form of the distance between the points?
In order to derive the vector form of the distance between two points, we must find the difference between the ,
, and
elements of the points.
That is, for any point
and
,
the distance is the vector
.
Subbing in our original points and
, we get:
Example Question #1 : Vector Form
Given points and
, what is the vector form of the distance between the points?
In order to derive the vector form of the distance between two points, we must find the difference between the ,
, and
elements of the points.
That is, for any point and
, the distance is the vector
.
Subbing in our original points and
, we get:
Example Question #2 : Vector Form
The graph of the vector function can also be represented by the graph of which of the following functions in rectangular form?
We can find the graph of in rectangular form by mapping the parametric coordinates to Cartesian coordinates
:
We can now use this value to solve for :
Example Question #34 : Parametric, Polar, And Vector Functions
The graph of the vector function can also be represented by the graph of which of the following functions in rectangular form?
We can find the graph of in rectangular form by mapping the parametric coordinates to Cartesian coordinates
:
We can now use this value to solve for :
Example Question #1 : Vector Form
In general:
If ,
then
Derivative rules that will be needed here:
- Taking a derivative on a term, or using the power rule, can be done by doing the following:
- Special rule when differentiating an exponential function:
, where k is a constant.
In this problem,
Put it all together to get
Example Question #92 : Vector Calculations
Calculate
Calculate the sum of vectors.
In general,
Solution:
Example Question #1 : Limits
Evaluate the following limit:
The limit does not exist
The limit we are given is one sided, meaning we are approaching our x value from one side; in this case, the negative sign exponent indicates that we are approaching 3 from the left side, or using values slightly less than three on approach.
This corresponds to the part of the piecewise function for values less than 3. When we substitute our x value being approached, we get
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