Csc x cot x 2 2 2 csc xsec x tan x cot x csc xsec x 1

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csc x cot x) 2 2 2 csc x(sec x tan x cot x) csc x(sec x 1) csc x tan x 1 sin x tan x sec x tan x sin x cos x 3.b
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11 Slope and Equation of Tangent eg *1. Find the slope at 4 for y = xsinx y / = (1)sinx + x(cosx) / 2 2 y sin cos 4 4 4 2 4 2 2 2 4 2 2 2(4 ) or 2 8 8 8 2. Find the equation of the tangent at 1 , 6 4 for y = sin 2 x. y / = 2sinxcosx = sin2x y / = 3 sin 3 2 1 y 3 4 2 x 6 1 3 2y 3x 2 6 12y 3 6 3x 3 3 3 3 6 3x 12y 3 3 0 OR y x 2 12 Increasing and Decreasing Functions/Local Maximum and Minimum eg 1. Analyze y = 2sinx x for using y / for [0, 2 ]: y / = 2cosx 1 0 = 2cosx 1 cosx = 1 2 y / x = 5 , 3 3 increasing for 5 0, , ,2 3 3 and decreasing for 5 , 3 3 The critical points are: 3 3 , 3 3 is a local maximum, 5 3 3 5 , 3 3 is a local mimima Assignment Read 308 318 pg. 313 # 1 h, j, k, m, o, 2 a, b, 3 a, 4a, 5 b, 6 a pg. 319 # 1 b, d, e, h, n ( Read eg.6 p 319) 3.c X 3 5 3 0 2 + +
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12 Concavity and Points of Inflection eg 1. Find the intervals where y = 2cosx + x is concave up and concave down for [0, 2 ]. y / = 2sinx + 1 y // = 2cosx 0 = 2cosx x = 3 , 2 2 (both are points of inflection) concave up: 3 , 2 2 down: 0, 2 3 ,2 2 Vertical Asymptotes A vertical asymptote will usually exist at x values where a factor of the denominator equals 0. If this factor can cancel with a numerator factor a hole may result instead. Left and right hand limits can confirm the presence and direction of a hole or a V.A. eg 1. 5 y 2cosx 1 for [ , ] x x 3 3 5 5 5 lim , lim , 2cos x 1 small 2cos x 1 2cosx = 1 cosx = 1 2 V.A. at x = , and x 3 3 2. 2 2 1 sin x y 2cos x cos x for [0, 2 ] y = 2 2 1 sin x 2cos x cos x 2 cos x cos x cos x(2cos x 1) 2cos x 1 , cosx(2cosx + 1) = 0 cosx = 0 (cancels) cosx = 1 2 so x = 3 , 2 2 are holes. V.A. at x = 2 4 , 3 3 3. 1 cosx f(x) sin x for [0, 2 ] has potential V.A. at x 0, ,2 but 1 cosx = 0 at x = 0, 2 and 1 cosx = 2 at x = . V.A. at x = where 2 f (x) 0 and holes at x = 0, 2 where 0 f (x) 0 . x sin x 0 Also lim undefined 1 cos x 0 3.d X 2 3 2 0 2 +
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13 Newton's Method eg 1. sinx + x 2 = 0 (radians) n n 1 n / n f (x ) x x f (x ) y / = cosx + 1 start at x = 1 x = ans sin(ans) ans 2 cos(ans) 1 x = 1.106060158 Graphing (plot endpoints) eg 1. Given y = 2cosx + x for [0, 2 ], find intercepts, intervals of increase and decrease, critical points, concavity and points of inflection, and sketch the graph. y int: (0, 2) x int: (newton's method) x = ans 2cos(ans) ans 2sin(ans) 1 ( 1.029866529, 0) y / = 2sinx + 1 0 = 2sinx + 1 sinx = 1 2 x = 5 , 6 6 increase: 0, 6 5 ,2 6 decrease: 5 , 6 6 maximum: ,2.3 6 minimum: 5 ,0.9 6 y // = 2cosx 0 = 2cosx x = 3 , 2 2 concave up: 3 , 2 2 concave down: 0, 2 3 ,2 2 points of inflection: ,1.6 2 and 3 ,4.7 2 endpoints: (0, 2), (2 , 8.3) X 6 5 6 0 2 + + X 2 3 2 0 2 + -1 1 2 3 4 5 6 7 8 9 0 3.e
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14 Graphing (plot endpoints) 1. y = 2sinx x for [0, 2 ] x int: (0, 0), (1.895494267, 0) y int: (0, 0) increase: : 0, 3 5 ,2 3 decrease: 5 , 3 3 maximum: ,0.7 3 , minimum: 5 , 7.0 3 concave up: ( , 2 ) concave down: (0, ) points of inflection: (0, 0), ( , ), (2 , 2 ) *2. y = cosx + x for [ , ] x int: ( 0.74, 0) y int: (0, 1) y / = sinx + 1 sinx = 1 x = 2 increase: , 2 , 2 decrease: never critical point: , 2 2 y // = cosx 0 = cosx x = 2 , 2 concave up: , 2 , 2 concave down: , 2 2 points of inflection: , 2 2 , , 2 2 endpoints: ( , 1 ), ( , 1 + ) Assignment pg. p314 # 8 a pg. 325 # 1 a, 2 a (use cos2 = 2cos 2 1), 8 10, 14 pg320 # 2 a, 3 a, 5 a 5.a) y = cscx cotx = 1 cos x 1 cos x See#2 above sin x sin x sin x + + 2 2 + + -1π -4 -3 -2 -1 1 2 3 0 2 4.a -7 -6 -5 -4 -3 -2 -1 1 2 0
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15 Problems eg 1. A clock has a second hand that is travelling at 6.3 radians/minute.
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