All Calculus 2 Resources
Example Questions
Example Question #321 : Parametric, Polar, And Vector
Given the Cartesian coordinate
, what is in the polar form ?
The formula to find theta in polar form is:
Plug in the Cartesian coordinates into the equation.
However, this angle is located in the fourth quadrant and is not in the right quadrant. Add
radians to get the correct angle since the Cartesian coordinate given is located in the second quadrant.
Example Question #1 : Polar Calculations
Tom is scaling a mathematical mountain. The mountain's profile can be described by
between and . Tom climbs from to the peak of the mountain. How far did he climb? You'll need an equation solver for certain parts of the problem. Round everything to the nearest hundredth.
This is a two step problem. First step is to maximize the function to find the peak of the mountain. The next step is to use the arc length formula to find the distance he climbed.
To maximize, we'll take a derivative and set it equal to zero.
.
Setting this equal to zero, we get
.The derivative is positive prior to this value and negative after, so it is a max. We now must take the arc length from
to .The formula for arc length is
.
For this case, the integral becomes
.
This will give us
. No units were given in the problem, so leaving the answer unitless is acceptable.Example Question #2 : Polar Calculations
Find the length of the polar valued function
from to .
Recall the formula for length in polar coordinates is given by
.
We were given the formula
.
In our case, this translates to
.
Example Question #1 : Polar Calculations
Convert
to Cartesian coordinates.
Write the formulas to convert from polar to Cartesian.
The
and values are known. Substitute both into each equation and solve for and .
The Cartesian coordinates are:
Example Question #324 : Parametric, Polar, And Vector
Convert
to Cartesian coordinates and find the coordinates of the center.
Write the conversion formulas.
Notice the
term. If we multiplied by on both sides of the equation, we will get:
Substitute this back into the first equation.
Add
on both sides.
Complete the square with the
terms.
This would then become:
This is a circle centered at
with a radius of 4.The answer is:
Example Question #9 : Polar Calculations
Convert
to Cartesian coordinates.
When converting from polar to Cartesian coordinates, we must use the formulas
the values
and are given, so we can calculate that
and
So the Cartesian coordinate form is
Example Question #10 : Polar Calculations
Determine the equation in polar coordinates of
can be immediately transformed into polar form by:
.
Dividing by
,
Dividing both sides by
Example Question #841 : Calculus Ii
Example Question #12 : Polar Calculations
Determine how many points of intersection exist for the curves
and
.
Solving the equations
and yields .Hence,
Therefore, the values of
between and that satisfy both equations are:
From this, it can be deduced that there are four points of intersection between the given curves:
However, an identical graph to
in polar coordinates is , since these two equations describe the same circle with a radius units long. Therefore, the equations and must also be solved to yield the remaining points of intersection:,
From this, it can be deduced that there are four other points of intersection between the given curves:
Hence, there are eight total points of intersection between the curves
and .Example Question #162 : Polar
Convert
to Cartesian coordinates
we are given
and we know that
we have r and 3cos(theta). Multiplying each side of the equation by r would give us
substitute out the parts we know from the formulas above
Certified Tutor
Certified Tutor
All Calculus 2 Resources
