Final Review Guide - Final Exam Review Guide Math 1220...

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Final Exam Review Guide Math 1220 Spring 2012 A Note on the Exercises: There are A LOT of suggested exercises here. The exercises are subdivided in the book. It is unreasonable to have any sort of expectation of doing all of them; but try to do some exercises from each subdivision and from the nonbook ones. Chapter 12 Trigonometry 12.1 Measurement of Angles * Terms: angle, initial ray, terminal ray, vertex, standard position, degree, radian, unit circle, coterminal angles * Converting Degrees to Radians: f ( x ) = π 180 x radians where x is the number of degrees and f ( x ) is the number of radians. * Converting Radians to Degrees: g ( x ) = 180 π x degrees where x is the number of radians and g ( x ) is the number of degrees. * Know how to identify when two angles are coterminal. * Exercises: pp.763 1-26 12.2 The Trigonometric Functions * Terms: sine, cosine, periodic, trigonometric identities * cos( θ ) is the x -coordinate of the point on the unit circle θ radians away from the positive x -axis (p.765). * sin( θ ) is the y -coordinate of the point on the unit circle θ radians away from the positive x -axis (p.765). * tan = sin cos * Can also think of sin , cos , tan in terms of right triangles: (SOH)(CAH)(TOA). * sec x = 1 cos x , csc x = 1 sin x , cot x = 1 tan x * Know the values of these six functions at the common angles: 0 , π 6 , π 4 , π 3 , π 2 , 2 π 3 , 3 π 4 , 5 π 6 , π, 7 π 6 , 5 π 4 , 4 π 3 , 3 π 2 , 5 π 3 , 7 π 4 , 11 π 6 (from the formulas above, you only need to know these for sin and cos.) See table 2 on p. 766. * Identities: · sin( - θ ) = - sin θ · cos( - θ ) = cos θ · sin 2 θ + cos 2 θ = 1 · tan 2 θ + 1 = sec 2 θ · cot 2 θ + 1 = csc 2 θ · cos 2 θ = 1 2 (1 + cos 2 θ ) · sin 2 θ = 1 2 (1 - cos 2 θ ) · sin( A ± B ) = sin A cos B ± cos A sin B · cos( A ± B ) = cos A cos B sin A sin B · sin 2 A = 2 sin A cos A · cos 2 A = cos 2 A - sin 2 A 1
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· sin θ = cos ( π 2 - θ ) · cos θ = sin ( π 2 - θ ) * Exercises: pp.771-772 1-38 · Draw and label the Unit Circle · Verify the following identities. 1. 1 - 2 sin 2 θ = 2 cos 2 θ - 1 2. sin x = (sec x - cos x ) cot x 3. (1 + cot 2 x ) tan 2 x = sec 2 x 12.3 Differentiation of Trigonometric Functions * d dx (sin x ) = cos x * d dx (cos x ) = - sin x * d dx (tan x ) = sec 2 x * d dx (csc x ) = - (csc x )(cot x ) * d dx (sec x ) = (sec x )(tan x ) * d dx (cot x ) = - csc 2 x * These derivatives still adhere to the chain, product, and quotient rules. * Be prepared to use these facts in solving various ‘applications of the derivative’problems (e.g. relative extrema, increase/decrease, curve sketching, . . . ). * Exercises: pp.781-782 1-34, 48,51,53 · Find the first and second derivatives. 1. sin(cos(sin x )) 2. p tan(3 x ) cos( πx ) 3. e - 2 x sec x 4. csc( x ) ln(cos 2 ( x )) 5. e x 2 cos(2 x ) 6. cot 5 x 7. sec( p csc 3 (2 x )) 8. e - x cos( x x 2 - 4 ) 9. cos(3 x 2 - 4 x ) sin x + e x 12.4 Integration of Trigonometric Functions * Z sin xdx = - cos x + C * Z cos xdx = sin x + C * Z sec 2 xdx = tan x + C * Z csc 2 xdx = - cot x + C * Z sec x tan xdx = sec x + C * Z csc x cot xdx = - csc x + C * Z tan xdx = - ln | cos x | + C * Z sec xdx = - ln | sec x + tan x | + C * Z csc xdx = ln | csc x - cot x | + C * Z cot xdx = ln | sin x | + C 2
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* Be comfortable with using techniques like u -substitution, integration by parts . . . .
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