1/31/2007 Pre-Calculus 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)

1/31/2007 Pre-Calculus Vector Formulas Unit Vectors: Horizontal/Vertical components: Horizontal/Vertical components: Angle between Vectors: Projections:

1/31/2007 Pre-Calculus 6.1 Vectors in a Plane Day # Vectors in a Plane Day # 1

1/31/2007 Pre-Calculus magnitude (size) direction force acceleration velocity RS starts at R and goes to S v = Starts at (0, 0) and goes to (x, y)

1/31/2007 Pre-Calculus A B v AB v = equivalent

1/31/2007 Pre-Calculus P Q

1/31/2007 Pre-Calculus Vector addition Vector multiplication (multiplying a vector by a scalar or real number) sum initial point terminal point parallelogram law

1/31/2007 Pre-Calculus unit vector direction

1/31/2007 Pre-Calculus direction angle

1/31/2007 Pre-Calculus 25 o

1/31/2007 Pre-Calculus 6.2 Dot Product of Vectors Day # Dot Product of Vectors Day # 1

1/31/2007 Pre-Calculus dot product work done vectors scalar (real number)

1/31/2007 Pre-Calculus Theorem: Angles Between Vectors If θ is the angle between the nonzero vectors u and v, then

1/31/2007 Pre-Calculus Proving Vectors are Orthagonal Prove that the vectors are orthagonal:

1/31/2007 Pre-Calculus Proving Vectors are Parallel Prove that the vectors are parallel: The vectors u and v are parallel if and only if: u = kv for some constant k

1/31/2007 Pre-Calculus Proving Vectors are Neither Show that the vectors are neither: If 2 vectors u and v are not orthagonal or parallel: then they are NEITHER

1/31/2007 Pre-Calculus vector projection

1/31/2007 Pre-Calculus Unit Circle

1/31/2007 Pre-Calculus 6.4 Polar Equations Day # Polar Equations Day # 1

1/31/2007 Pre-Calculus polar coordinate system pole polar axis polar coordinates ( r, θ ) directed distance directed angle polar axis line OP r θ O polar axis P

1/31/2007 Pre-Calculus

1/31/2007 Pre-Calculus Polar Cartesian (rectangular) poleoriginpolar axis positive x – axis y x θ r P(r, θ ) y = r sin θ x = r cos θ

1/31/2007 Pre-Calculus so y x θ r P(x, y )

1/31/2007 Pre-Calculus Helpful Hints Polar to Rectangular 1.multiply cos  or sin  by r so you can convert to x or y 2.r 2 = x 2 + y 2 3.re-write sec  and csc  as 4.complete the square as necessary Rectangular to Polar 1.replace x and y with r  cos  and r  sin  2.when given a “squared binomial”, multiply it out 3.x 2 + y 2 = r 2 (x – a) 2 + (y – b) 2 = c 2 Where the center of the circle is (a, b) and the radius is c (x – a) 2 + (y – b) 2 = c 2 Where the center of the circle is (a, b) and the radius is c

1/31/2007 Pre-Calculus 6.5 Graphs of Polar Equations Day # Graphs of Polar Equations Day # 1

1/31/2007 Pre-Calculus 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

1/31/2007 Pre-Calculus 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)

1/31/2007 Pre-Calculus 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 GRAPHS

1/31/2007 Pre-Calculus 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

1/31/2007 Pre-Calculus 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

1/31/2007 Pre-Calculus 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.

1/31/2007 Pre-Calculus 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.