Unit 9 Calc BC

Question 1

derivative of parametric equations

Answer 1

dy/dx = (dy/dt) / (dx/dt)

Question 2

2nd derivative of parametric equations

Answer 2

d² y/dx² = d/dt[dy/dx] / (dx/dt)

Question 3

arc length of a parametric equation

Answer 3

b
s = ∫ √ [ (dx/dt)² + (dy/dt)²]dt
a

Question 4

4 equations for relationships between rectangular and polar coordinates

Answer 4

x = rcosθ
y = rsinθ
x ² + y ² = r ²
tanθ = y / x

Question 5

equation for slope in polar form

Answer 5

[dr/dθ(sinθ) + rcosθ] / [dr/dθ(cosθ) - rsinθ]

Question 6

expansions of cos2θ and cos²θ

Answer 6

cos2θ = 2cos²θ - 1
cos2θ = 1 - 2sin²θ
cos²θ = (1 + cos2θ) / 2

Question 7

expansion of sin²θ

Answer 7

sin²θ = (1 - cos2θ) / 2

Question 8

arc length of a polar equation

Answer 8

b
s = ∫ √ [ (dr/dθ)² + (r)²]dθ
a

Question 9

domain of a vector valued function

Answer 9

the intersection of the domain of the component functions

Question 10

continuity of a vector valued function

Answer 10

exists at point a if limit of r(t) exists (for both components) AND lim r(t) = r(a)

Question 11

velocity vector equation

Answer 11

v(t) = r’(t) = <x’(t), y’(t)>

Question 12

speed of a vector equation

Answer 12

speed is the magnitude of the velocity
||r’(t)|| = √ [x’(t)]² + [y’(t)]²

Question 13

acceleration of a vector equation

Answer 13

a(t) = r’‘(t) = <x”(t), y”(t)>

Question 14

displacement of velocity vector on
[a, b]

Answer 14

b b b
∫v(t) dt =< ∫ x’(t)dt, ∫ y’(t)dt >
a a a

Question 15

position of an object at an unknown time (and what you use to find general position vector)

Answer 15

(x(b), y(b)) = (x(a), y(a)) + < ∫ x’(t)dt, ∫ y’(t)dt >

Question 16

total distance traveled by an object represented by a vector on [a, b]

Answer 16

b b
∫||v(t)||dt = ∫√ [x’(t)]² + [y’(t)]² dt
a a

Question 17

area in polar coordinates

Answer 17

b
A = 1/2 ∫ r² dθ
a