Chapter 6 Review Due 5/21

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 magnitude (size) force acceleration velocity Starts at (0, 0) and goes to (x, y)

v B A AB v = equivalent

Vector addition Vector multiplication (multiplying a vector by a scalar or real number) sum terminal initial point point parallelogram law

unit vector unit vector direction

direction angle

25o

6.2 Dot Product of Vectors Day # 1

dot product work done vectors scalar (real number)

Theorem: Angles Between Vectors If θ is the angle between the nonzero vectors u and v, then

Prove that the vectors are orthagonal: Proving Vectors are Orthagonal

Prove that the vectors are parallel: Proving Vectors are Parallel The vectors u and v are parallel if and only if: u = kv for some constant k

Show that the vectors are neither: Proving Vectors are Neither If 2 vectors u and v are not orthagonal or parallel: then they are NEITHER

vector projection

Unit Circle

6.4 Polar Equations Day # 1

P r θ O polar axis pole polar coordinate system polar axis ( r, θ ) polar coordinates directed distance polar axis directed angle line OP

P(r, θ) r y θ x Cartesian (rectangular) Polar pole origin polar axis positive x – axis x = r cos θ y = r sin θ

P(x, y) r y θ x so so

Helpful Hints Polar to Rectangular multiply cos or sin by r so you can convert to x or y r2 = x2 + y2 re-write sec and csc as complete the square as necessary Rectangular to Polar replace x and y with rcos and rsin when given a “squared binomial”, multiply it out x2 + y2 = r2 (x – a)2 + (y – b)2 = c2 Where the center of the circle is (a, b) and the radius is c

6.5 Graphs of Polar Equations Day # 1

General Form: r = a cos n θ r = a sin n θ Petals: n: odd  n petals n: even  2n petals n: even n: odd cos  one petal on pos. x-axis cos  petals on each side of each axis sin  one petal on half of y-axis sin  no petals on axes

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) when , it touches the origin; “cardioid” (#6) when , it’s called a “dimpled limacon” (#7) when , it is a “convex limacon” (#8)

ANALYZING POLAR GRAPHS • 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.

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) • 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

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) • 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

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.

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.

Chapter 6 Review Due 5/21