# 2 2 cos 1 2 2 cos cos 2 2 cos 2 2 cos cos 2 2 cos 2

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𝑦𝑦 ) = 2 𝑥𝑥 − 2 𝑦𝑦 cos( 𝑥𝑥 + 𝑦𝑦 ) [1 + 𝑦𝑦 ] = 2 2 𝑦𝑦 cos( 𝑥𝑥 + 𝑦𝑦 ) + 𝑦𝑦 cos( 𝑥𝑥 + 𝑦𝑦 ) = 2 2 𝑦𝑦 𝑦𝑦 cos( 𝑥𝑥 + 𝑦𝑦 ) + 2 𝑦𝑦 = 2 cos( 𝑥𝑥 + 𝑦𝑦 ) 𝑦𝑦 [cos( 𝑥𝑥 + 𝑦𝑦 ) + 2] = 2 cos( 𝑥𝑥 + 𝑦𝑦 ) 𝑦𝑦 = 2 cos( 𝑥𝑥 + 𝑦𝑦 ) cos( 𝑥𝑥 + 𝑦𝑦 ) + 2 𝑚𝑚𝑛𝑛𝑛𝑛𝑚𝑚𝐿𝐿𝐿𝐿𝐿𝐿𝑛𝑛𝐿𝐿𝑒𝑒 𝑥𝑥 = 𝜋𝜋 , 𝑦𝑦 = 𝜋𝜋 𝑦𝑦 = 2 − 𝑐𝑐𝐿𝐿𝑚𝑚 2 𝜋𝜋 𝑐𝑐𝐿𝐿𝑚𝑚 2 𝜋𝜋 + 2 = 1 3 Equation y- 𝜋𝜋 = 1 3 ( 𝑥𝑥 − 𝜋𝜋 ) P10. 𝑥𝑥 = 𝑦𝑦 2 4 𝑦𝑦 crosses y-axis, so x=0 Derivative 1 = 2 𝑦𝑦𝑦𝑦 4 𝑦𝑦 solve for 𝑦𝑦 ...common factor of 𝑦𝑦 𝑦𝑦 (2 𝑦𝑦 − 4) = 1 𝑦𝑦 = 1 2 𝑦𝑦 − 4
Sub x=0 into original... 0 = 𝑦𝑦 2 4 𝑦𝑦 𝑦𝑦 ( 𝑦𝑦 − 4) = 0 𝑦𝑦 = 0, 4 If y=0, 𝑥𝑥 = 𝑦𝑦 2 4 𝑦𝑦 = 0 … 𝑚𝑚𝐿𝐿 (0,0) is one point If y=4, 𝑥𝑥 = 𝑦𝑦 2 4 𝑦𝑦 = 16 16 = 0, 𝑚𝑚𝐿𝐿 (0,4) 𝐿𝐿𝑚𝑚 𝐿𝐿𝑙𝑙𝐿𝐿𝐿𝐿ℎ𝑒𝑒𝐿𝐿 𝑝𝑝𝐿𝐿𝐿𝐿𝑙𝑙𝐿𝐿 At (0,0) Subst. y=0 into derivative 𝑦𝑦 = 1 2 𝑦𝑦 − 4 = 1 4 At (0,4) Subst. y=4 into derivative 𝑦𝑦 = 1 2𝑦𝑦−4 = 1 4 P11. 𝑦𝑦 = −𝑥𝑥 2 5 𝑥𝑥 − 6 𝑦𝑦 = 2 𝑥𝑥 − 5 Intersect the x-axis, so y=0...multiply by -1 0 = 𝑥𝑥 2 + 5 𝑥𝑥 + 6 0 = ( 𝑥𝑥 + 2)( 𝑥𝑥 + 3) X= -3, -2 Subst. X= -3, -2 into 𝑦𝑦 At x=-3, 𝑦𝑦 = 2( 3) 5 = 1
𝐴𝐴𝐿𝐿 𝑥𝑥 = 2 𝑦𝑦 = 2 𝑥𝑥 − 5 = -2(-2) – 5 = -1 Q. 𝑆𝑆𝑒𝑒𝑐𝑐𝐿𝐿𝑙𝑙𝑢𝑢 𝐷𝐷𝑒𝑒𝐿𝐿𝐿𝐿𝑛𝑛𝐿𝐿𝐿𝐿𝐿𝐿𝑛𝑛𝑒𝑒𝑚𝑚 Q 1. 𝐿𝐿 ) 𝑦𝑦 = 3 𝑥𝑥 3 + 6 𝑥𝑥 𝑦𝑦 = 9 𝑥𝑥 2 + 6 𝑦𝑦 " = 18 𝑥𝑥 𝑛𝑛 ) 𝑓𝑓 ( 𝑥𝑥 ) = (2x 1) 6 𝑓𝑓 ( 𝑥𝑥 ) = 6 (2x 1) 5 (2) = 12(2x 1) 5 𝑓𝑓 ′′ ( 𝑥𝑥 ) = 12(5) (2x 1) 4 (2) 𝑓𝑓 ′′ ( 𝑥𝑥 ) = 120(2x 1) 4 𝑐𝑐 ) 𝑓𝑓 ( 𝑥𝑥 ) = 𝑥𝑥+1 3𝑥𝑥−1 𝑓𝑓 ( 𝑥𝑥 ) = ( 1 )( 3𝑥𝑥−1 ) −3 ( 𝑥𝑥+1 ) ( 3𝑥𝑥−1 ) 2 = −4 ( 3𝑥𝑥−1 ) 2 = 4(3 𝑥𝑥 − 1) −2 𝑓𝑓 "( 𝑥𝑥 ) = 8(3 𝑥𝑥 − 1) −3 (3) = 24 ( 3𝑥𝑥−1 ) 3 𝑢𝑢 ) 𝑓𝑓 ( 𝑥𝑥 ) = 3 𝑥𝑥 3 5 + 1 3 𝑥𝑥 −1 𝑓𝑓 ( 𝑥𝑥 ) = 3 3 5 𝑥𝑥 −2 5 + 1 3 ( 1 𝑥𝑥 −2 ) 𝑓𝑓 ( 𝑥𝑥 ) = 9 5 𝑥𝑥 −2 5 1 3 𝑥𝑥 −2 𝑓𝑓 "( 𝑥𝑥 ) = 9 5 ( −2 5 𝑥𝑥 −7 5 ) 1 3 ( 2 𝑥𝑥 −3 ) 𝑓𝑓 "( 𝑥𝑥 ) = −18 25 𝑥𝑥 −7 5 + 2 3 𝑥𝑥 −3
Q 2. 𝑓𝑓 ( 𝑥𝑥 ) = 𝑥𝑥 3 + 2 𝑥𝑥 2 + 6 𝑓𝑓 ( 𝑥𝑥 ) = 3 𝑥𝑥 2 + 4 𝑥𝑥 𝑓𝑓 "( 𝑥𝑥 ) = 6 𝑥𝑥 + 4 𝑓𝑓 3 ( 𝑥𝑥 ) = 6 𝑓𝑓 4 ( 𝑥𝑥 ) = 0 Q3. f(x) = sinx 𝑓𝑓 ( 𝑥𝑥 ) = 𝑐𝑐𝐿𝐿𝑚𝑚𝑥𝑥 𝑓𝑓 "( 𝑥𝑥 ) = −𝑚𝑚𝐿𝐿𝑙𝑙𝑥𝑥 𝑓𝑓 3 ( 𝑥𝑥 ) = −𝑐𝑐𝐿𝐿𝑚𝑚𝑥𝑥 𝑓𝑓 4 ( 𝑥𝑥 ) =
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