can someone please explain to me
1. When doing parametric curves how do you find concavity (concave up or concave down)
2. When doing polar coordinates how do you graph them and how to find different polar points that satisfy a condition like theta<0 or r<0 or r>0 or theta>0
Have an exam and these two sections are not clicking thank you in advance
when the curve is horizontal dy/dx = 0
at that point if d^2y/dx^2 is NEGATIVE, the slope gets smaller than zero as you move up in x
That means it starts down, sheds water, maximum
if d^2y/dx^2 is POSITIVE
then the slope gets greater than zero as you move up in x from that horizontal point.
That means it starts up, bottom of bowl, holds water, minimum
r < 0 does not make sense. r is the distance from the origin.
theta is the distance counterclockwise from the x axis.
theta = 0 is +x direction from the origin
theta = pi/2 or 90 deg is +y from origin
theta = pi or 180 deg is -x from origin
thera = 3 pi/2 or 270 deg is -y from origin
If r<0 you move in the direction opposite to θ
(1,π) = (-1,0)
for concavity in parametric equations, it gets a little more complicated finding d^2y/dx^2
y' = dy/dx = (dy/dt) / (dx/dt)
So you have to find all that as a messy rational function of t.
Then d^y/dx^2 = (dy'/dt) / (dx/dt) and you need the quotient rule to come up with dy'/dt.
Or, you may be able to eliminate t as a parameter, getting an equation in just x and y, and then find y" as usual, (though that can also be involved if you get stuck using implicit derivatives)
Sure! I can help you with both of these topics.
1. Finding the concavity of a parametric curve:
To determine whether a parametric curve is concave up or concave down, you can follow these steps:
a. Calculate the second derivative of each component of the parametric equations. Let's say we have x(t) and y(t) as the parametric equations.
b. Evaluate the second derivative for t-values that lie within the domain of the curve.
c. If both second derivatives are either positive or negative for all relevant t-values, then the curve is concave up. If they alternate signs, then the curve is concave down.
For example, given the parametric equations x(t) and y(t), find the second derivative of x and y and evaluate them for the relevant t-values. If both second derivatives are positive or negative, the curve is concave up. If they alternate signs, the curve is concave down.
2. Graphing polar coordinates and finding points that satisfy conditions:
To graph polar coordinates, you can follow these steps:
a. Convert the polar coordinates (r, θ) to Cartesian coordinates (x, y) using the formulas:
x = r * cos(θ)
y = r * sin(θ)
To find polar points that satisfy certain conditions, such as θ<0, r<0, r>0, or θ>0, you need to consider the properties of the coordinate plane.
a. If θ<0, it means the angle θ is measured in the clockwise direction from the positive x-axis. This corresponds to the lower half of the coordinate plane.
b. If r<0, it means the distance from the origin is negative, which represents the opposite direction from positive r, indicating points on the opposite side of the origin.
c. If r>0, it means the distance from the origin is positive, representing points in the usual direction from the origin.
d. If θ>0, it means the angle θ is measured in the counterclockwise direction from the positive x-axis.
Using these concepts, you can determine the quadrants or regions in the polar coordinate plane that satisfy the given conditions and plot the points accordingly.
I hope these explanations help you understand the concepts better. Good luck with your exam! If you have any more questions, feel free to ask.