chapter 6 review due 5/21
DESCRIPTION
Chapter 6 Review Due 5/21. # 2 – 22 even # 53 – 59 odd # 62 – 70 even # 74, 81, 86 (p. 537). Vector Formulas. Unit Vectors:. Horizontal/Vertical components:. Angle between Vectors:. Projections:. 6.1 Vectors in a Plane Day # 1. RS starts at R and goes to S. v =. direction. - PowerPoint PPT PresentationTRANSCRIPT
Pre-Calculus
1/31/2007
Chapter 6 ReviewDue 5/21
Chapter 6 ReviewDue 5/21
# 2 – 22 even# 53 – 59 odd# 62 – 70 even
# 74, 81, 86
(p. 537)
Pre-Calculus
1/31/2007
Vector FormulasVector Formulas
v 1
u or vv v
v cos v sin
gu v
cosu v
v 2
u vproj u v
v
g
Unit Vectors:Unit Vectors:
Horizontal/Verticalcomponents:
Horizontal/Verticalcomponents:
Angle between Vectors:Angle between Vectors:
Projections:Projections:
Pre-Calculus
1/31/2007
6.1 Vectors in a Plane
Day # 1
6.1 Vectors in a Plane
Day # 1
Pre-Calculus
1/31/2007
magnitude (size) direction
force acceleration velocity
RS starts at R and goes to S
v = 1 2v , v
Starts at (0, 0) and goes to (x, y)
Pre-Calculus
1/31/2007
A
B
v
7 3 4
1 ( 4) 3
AB
2 2d 3 4 25 5
v = 3,4
equivalent
Pre-Calculus
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5 2,( 1) 33,2
PQ vuuur
2 23 2 9 4 13
2
slope3
P
Q
Pre-Calculus
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Vector addition
Vector multiplication (multiplying a vector by a scalar or real number)
1 2 1 2 1 1 2 2u ,u v , v u v ,u v
sum
1 2 1 2ku k u ,u ku ,ku
initial point terminal
point
parallelogram law
Pre-Calculus
1/31/2007
unit vector
unit vector
v 1
u or vv v
direction v
Pre-Calculus
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direction angle
v cos v sin
Pre-Calculus
1/31/2007
25o
o o70 cos 25 ,70 sin 25
250 63.44,433.01 29.38
2 2v w ( 186.56) (462.59) 498.79mph
462.59
tan186.56
o65o25
63.44,29.58
186.56,462.59
o111.96
Pre-Calculus
1/31/2007
6.2 Dot Product of Vectors
Day # 1
6.2 Dot Product of Vectors
Day # 1
Pre-Calculus
1/31/2007
dot product
work done
vectors scalar (real number)
g 1 1 2 2u v u v u v
g gu v v u g 2u u u g0 u 0 u (v w ) u v u wg g g
g g g(cu) v u (cv) c(u v) g g g(u v) w u w v w +
Pre-Calculus
1/31/2007
gu v
cosu v
Theorem: Angles Between VectorsTheorem: Angles Between Vectors
If θ is the angle between the nonzero vectors u and v, then
g1 u vcos
u v
Pre-Calculus
1/31/2007
Proving Vectors are OrthagonalProving Vectors are Orthagonal
u 3,2
v 8,12
Prove that the vectors are orthagonal:
g ou v u v cos 90 0
gu v 0
Pre-Calculus
1/31/2007
Proving Vectors are ParallelProving Vectors are Parallel
u 3,2
v 6, 4
Prove that the vectors are parallel:
The vectors u and v are parallel if and only if:
u = kvfor some constant k
Pre-Calculus
1/31/2007
Proving Vectors are NeitherProving Vectors are Neither
u 3,2
v 4, 6
Show that the vectors are neither:
If 2 vectors u and v are not orthagonal or parallel:then they are NEITHER
Pre-Calculus
1/31/2007
vector projection
vproj u
v 2
u vproj u v
vg
vu proj u
Pre-Calculus
1/31/2007
Unit CircleUnit Circle
Pre-Calculus
1/31/2007
6.4 Polar Equations
Day # 1
6.4 Polar Equations
Day # 1
Pre-Calculus
1/31/2007
polar coordinate system pole polar axis
polar coordinates ( r, θ )
directed distance
directed angle polar axisline OP
rθ
O polar axis
P
Pre-Calculus
1/31/2007
3, 2 n
4
53, 2 n
4
2,75 360n
2,255 360n
Pre-Calculus
1/31/2007
Polar Cartesian (rectangular)pole origin polar axis
positive x – axis
y
xθ
rP(r, θ)
y = r sin θ
x = r cos θ
Pre-Calculus
1/31/2007
y
tanx
1 ytan
x
2 2 2r x y 2 2r x y
so
soy
xθ
rP(x, y)
Pre-Calculus
1/31/2007
Helpful HintsHelpful Hints
Polar to Rectangular1. multiply cos or sin
by r so you can convert to x or y
2. r2 = x2 + y2
3. re-write sec and csc as
4. complete the square as necessary
Rectangular to Polar1. replace x and y with
rcos and rsin2. when given a “squared
binomial”, multiply it out3. x2 + y2 = r2
1 1
andcos sin
(x – a)2 + (y – b)2 = c2
Where the center of the circle is (a, b) and the radius is c
(x – a)2 + (y – b)2 = c2
Where the center of the circle is (a, b) and the radius is c
Pre-Calculus
1/31/2007
6.5 Graphs of Polar Equations
Day # 1
6.5 Graphs of Polar Equations
Day # 1
Pre-Calculus
1/31/2007
General Form:
r = a cos n θ
r = a sin n θ
Petals:
n: odd n petals
n: even 2n petals
n: odd n: even
cos one petal on pos. x-axis
sin one petal on half of y-axis
cos petals on each side of each axis
sin no petals on axes
Pre-Calculus
1/31/2007
General Form:
r = a + b sin θ
r = a + b cos θ
Symmetry:
sin: about y – axis
cos: about x – axis
when , there is an “inner loop” (#5)a
1b
when , it touches the origin; “cardioid” (#6)a
1b
when , it’s called a “dimpled limacon” (#7) a
1 2b
when , it is a “convex limacon” (#8)a
2b
Pre-Calculus
1/31/2007
We analyze polar graphs much the same way we do graphs of rectangular equations. The domain is the set of possible inputs for . The range is the set of outputs for r. The domain and range can be read from the “trace” or “table” features on your calculator. We are also interested in the maximum value of . This is the maximum distance from the pole. This can be found using trace, or by knowing the range of the function.
Symmetry can be about the x-axis, y-axis, or origin, just as it was in rectangular equations.
Continuity, boundedness, and asymptotes are analyzed the same way they were for rectangular equations.
ANALYZING POLAR GRAPHSANALYZING POLAR GRAPHS
r
Pre-Calculus
1/31/2007
What happens in either type of equation when the constants are negative? Draw sketches to show the results.
Rose Curve when “a” is negative(“n” can’t be negative, by definition)
Rose Curve when “a” is negative(“n” can’t be negative, by definition)
• if n is even, picture doesn’t change…just the order that the points are plotted changes
•if n is odd, the graph is reflected over the x – axis
r asin(n ) r asin(n ) r 2sin(3 )
r 2sin(3 )
Pre-Calculus
1/31/2007
What happens in either type of equation when the constants are negative? Draw sketches to show the results.
Rose Curve when “a” is negative(“n” can’t be negative, by definition)
Rose Curve when “a” is negative(“n” can’t be negative, by definition)
• if n is even, picture doesn’t change…just the order that the points are plotted changes
•if n is odd, the graph is reflected over the y – axis
r acos(n ) r acos(n ) r 2cos(3 )
r 2cos(3 )
Pre-Calculus
1/31/2007
What happens in either type of equation when the constants are negative? Draw sketches to show the results.
Limacon Curve when “b” is negative (minus in front of the b)(“a” can’t be negative, by definition)
• when r = a + bsinθ, the majority of the curve is around the positive y – axis.
•when r = a – bsinθ, the curve flips over the x – axis.
r a bsin r a bsin r 1 2sin
r 1 2sin
Pre-Calculus
1/31/2007
What happens in either type of equation when the constants are negative? Draw sketches to show the results.
Limacon Curve when “b” is negative (minus in front of the b)(“a” can’t be negative, by definition)
• when r = a + bcos θ, the majority of the curve is around the positive x – axis.
•when r = a – bcos θ, the curve flips over the y – axis.
r a bcos r a bcos r 1 2cos
r 1 2cos