1/31/2007 pre-calculus chapter 6 review due 5/21 chapter 6 review due 5/21 # 2 – 22 even # 53 –...

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

1/31/2007

5 2,( 1) 33,2

PQ vuuur

2 23 2 9 4 13

2

slope3

P

Q

Pre-Calculus

1/31/2007

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

1/31/2007

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