All AP Calculus BC Resources
Example Questions
Example Question #1 : Parametric, Polar, And Vector
Rewrite as a Cartesian equation:
So
or
We are restricting
to values on , so is nonnegative; we choose.
Also,
So
or
We are restricting
to values on , so is nonpositive; we choose
or equivalently,
to make
nonpositive.
Then,
and
Example Question #2 : Parametric, Polar, And Vector Functions
Rewrite as a Cartesian equation:
, so
This makes the Cartesian equation
.
Example Question #3 : Parametric, Polar, And Vector Functions
If
and , what is in terms of (rectangular form)?
Given
and , we can find in terms of by isolating in both equations:
Since both of these transformations equal
, we can set them equal to each other:
Example Question #191 : Ap Calculus Bc
Given
and , what is the arc length between ?
In order to find the arc length, we must use the arc length formula for parametric curves:
.
Given
and , we can use using the Power Rulefor all , to derive
and
.
Plugging these values and our boundary values for into the arc length equation, we get:
Now, using the Power Rule for Integrals
for all ,
we can determine that:
Example Question #4 : Parametric, Polar, And Vector Functions
Given
and , what is the length of the arc from ?
In order to find the arc length, we must use the arc length formula for parametric curves:
.
Given
and , we can use using the Power Rulefor all , to derive
and
.
Plugging these values and our boundary values for into the arc length equation, we get:
Now, using the Power Rule for Integrals
for all ,
we can determine that:
Example Question #151 : Parametric, Polar, And Vector
Find the length of the following parametric curve
, , .
The length of a curve is found using the equation
We use the product rule,
, when and are functions of ,
the trigonometric rule,
and
and exponential rule,
to find and .
In this case
,
The length of this curve is
Using the identity
Using the identity
Using the trigonometric identity
where is a constant and
Using the exponential rule,
Using the exponential rule,
, gives us the final solution
Example Question #4 : Parametric, Polar, And Vector Functions
Find dy/dx at the point corresponding to the given value of the parameter without eliminating the parameter:
The formula for dy/dx for parametric equations is given as:
From the problem statement:
If we plug these into the above equation we end up with:
If we plug in our given value for t, we end up with:
This is one of the answer choices.
Example Question #7 : Parametric, Polar, And Vector Functions
Draw the graph of
from .Between
and , the radius approaches from .From
to the radius goes from to .Between
and , the curve is redrawn in the opposite quadrant, the first quadrant as the radius approaches .From
and , the curve is redrawn in the second quadrant as the radius approaches from .Example Question #1 : Parametric, Polar, And Vector Functions
Draw the graph of
where .Because this function has a period of
, the amplitude of the graph appear at a reference angle of (angles halfway between the angles of the axes).Between
and the radius approaches 1 from 0.Between
and , the radius approaches 0 from 1.From
to the radius approaches -1 from 0 and is drawn in the opposite quadrant, the fourth quadrant because it has a negative radius.Between
and , the radius approaches 0 from -1, and is also drawn in the fourth quadrant.From
and , the radius approaches 1 from 0. Between and , the radius approaches 0 from 1.Then between
and the radius approaches -1 from 0. Because it is a negative radius, it is drawn in the opposite quadrant, the second quadrant. Likewise, as the radius approaches 0 from -1. Between and , the curve is drawn in the second quadrant.Example Question #5 : Parametric, Polar, And Vector Functions
Graph
where .Taking the graph of
, we only want the areas in the positive first quadrant because the radius is squared and cannot be negative.This leaves us with the areas from
to , to , and to .Then, when we take the square root of the radius, we get both a positive and negative answer with a maximum and minimum radius of
.To draw the graph, the radius is 1 at
and traces to 0 at . As well, the negative part of the radius starts at -1 and traces to zero in the opposite quadrant, the third quadrant.From
to , the curves are traced from 0 to 1 and 0 to -1 in the fourth quadrant. Following this pattern, the graph is redrawn again from the areas included in to .