Polar, Parametric and Vector

Question 1

x(t) =

When initial velocity is given

Answer 1

V(0)cos(theta)t + x(0)

Question 2

y(t) =

When initial velocity is given

Answer 2

0.5gt^2+ v(0)sin(theta)t + y(0)

g = -9.8 m/s^2 or -32ft/s^2

Question 3

dy/dx =

Given x(t) and y(t)

Answer 3

(dy/dt)/(dx/dt)

Same thing with theta instead of t

Question 4

d^2y/dx^2

Given x(t) and y(t)

Answer 4

((dy/dx)’)/(dx/dt)

Question 5

Arc length from t1 to t2

Given x(t) and y(t)

Answer 5

integral from t1 to t2 of sqrt((dx/dt)^2 + (dy/dt)^2)

Question 6

What does this derivative tell you?

dx/dt

Answer 6

The gives the horizontal velocity of the — at time t

Question 7

What does this derivative tell you?

dy/dt

Answer 7

This gives the vertical velocity of the — at time t

Question 8

What does this derivative tell you?

dy/dx

Answer 8

This gives the slope of the curve/path the — takes/travels on

Question 9

Speed

When you have a vector

Answer 9

abs(sqrt((x’(t))^2 + (y(t))^2)

This is just the length of the velocity vector

Question 10

Area under a parametric equation

From t1 to t2

Answer 10

integral from t1 to t2 of (y(t) * (dx/dt))

Question 11

Period of a circle

Answer 11

pi

Question 12

Period of a limaçon

Answer 12

2pi

Question 13

Period of a rose curve

Answer 13

2pi/n for odd petals
pi/n for even petals

n = number of petals

Question 14

Convert Cartesian to Polar

x=

Answer 14

rcos(theta)

Question 15

Convert Cartesian to Polar

y=

Answer 15

rsin(theta)

Question 16

Convert Polar to Cartesian

r=

Answer 16

sqrt(x^2 + y^2)

Question 17

Convert Polar to Cartesian

theta =

Answer 17

arctan(y/x)

Question 18

What is dy/dx equal to at the pole?

Answer 18

tan(theta)

Question 19

How to find tangent lines at the pole?

Answer 19

1. find when r=0
2. Check that dr/dtheta does not equal zero
3. Find dy/dx
4. Give answer as equation

Question 20

Polar

What does x’ equal?

Answer 20

-rsin(theta) + r’cos(theta)

Question 21

Polar

What does y’ equal?

Answer 21

rcos(theta) + r’sin(theta)

Question 22

Polar

What does dy/dx equal?

Answer 22

(rcos(theta) + r’sin(theta))/(-rsin(theta) + r’cos(theta))

Question 23

Steps to find polar graph’s tangent lines (or points furthest from the origin)

Answer 23

1. dy/dx equals either 0 or undifined
2. Not a shart point (dr/dtheta exists)
3. Theta must be in domain

Question 24

Area under a polar curve

From a radians to b radians

Answer 24

0.5 * integral from a to b of (r(theta))^2 dtheta

You only do this for one petal or section at a time

Question 25

Polar

Arc length from a to b

Answer 25

integral from a to b of sqrt((r^2) + (dr/dtheta)^2)dtheta