Stuent Stuy Session Solutions We have intentionally inclue more material than can be covere in most Stuent Stuy Sessions to account for groups that are able to answer the questions at a faster rate. Use your own jugment, base on the group of stuents, to etermine the orer an selection of questions to work in the session. Be sure to inclue a variety of types of questions (multiple choice, free response, calculator, an non-calculator) in the time allotte. General Rules u u u Bases Other than e: ( a ) a (ln u) Definition of Derivative: f ( ) f ( ) lim h0 f ( h) f () h Trigonometric Functions: Sum an Difference Rule: f ( ) g( ) f () g () Constant Multiple Rule: k f () k f () Prouct Rule: f ( ) g( ) Quotient Rule: f ( ) g( ) Chain Rule: f ( g( )) Power Rule: n Particular Rules n n Eponential functions: eu e u u f ()g() f () g () f ()g() f () g () [g()] f (g()) g () sin(u) u cos(u) cos(u) u sin(u) tan(u) sec (u) u csc(u) u csc(u)cot(u) sec(u) u sec(u)tan(u) cot(u) csc (u) u Logarithmic Functions: ln(u) u u Inverse Trigonometric Functions: sin (u) u u tan (u) u u Copyright 04 National Math + Science Initiative, Dallas, TX. All rights reserve. Visit us online at www.nms.org
Stuent Stuy Session Multiple Choice. A (99 AB4) f( ) ( ) ( ) 4 f (0) (0 0 ) ((0) ). E (99 AB8) y sec ( csc ) sec csc. E (985 AB) f( ) 4() () 8( ) f( ) 4() () 48( ) f( ) 96() () 9( ) f 4 ( ) 84 4. B (AP style) h( ) f( ) g( ) g( ) f( ) h(5) f(5) g(5) g(5) f(5) h(5) ()(5) ( )(4) 7 5. A (AP style) h( ) f( g( ))( g( )) h(4) f( g(4))( g(4)) h(4) f()( g(4)) h(4) ( 5)(9) 45 6. E (AP style) f() [ g()] g() f () (8)() 48 7. E (988 AB8) y sin y sin y cos cos Copyright 04 National Math + Science Initiative, Dallas, TX. All rights reserve. Visit us online at www.nms.org
8. D (988 BC0 appropriate for AB) Using the efinition of erivative, sin cos Stuent Stuy Session 9. B (988 BC appropriate for AB) f '( ) f( ) 0. E (969 AB5/BC5) Evaluate the equation for () () y y y y0 y Derivative 6 y y y() y 0 Evaluate y at (, ) 4 0; y oes not eist at (, ). C (969 AB4/BC4) y sin e ln(sin ) y ln(sin ) y is cos y y cot sin Copyright 04 National Math + Science Initiative, Dallas, TX. All rights reserve. Visit us online at www.nms.org
. D (969 AB9/BC9) Metho : Use substitution to create y in terms of : y tan ln ln. Determine y of the new equation. y sec ln ln (ln ) Evaluate when e y sec ln e sec (0) ln e e e(ln e) e Stuent Stuy Session Metho : Determine the erivative of each equation using implicit ifferentiation. y u u v v sec u ; ; v Evaluate when e v an vln e which prouces u e to be u e e y Finally u 0 so sec (0) e e. A (998 BC5 appropriate for AB) h( ) fg( ) g( ) h f g( ) g( ) g( ) f g( ) g( ) ( ) ( ) ( ) ( ) h f g g f g g 4. D (97 AB8) sin (u) u u () ( ) 4 5. E (97 AB40) y sec ( y) y() y y sec ( y) y sec ( y) cos ( y) y sec ( y) sec ( y) sec ( y) Copyright 04 National Math + Science Initiative, Dallas, TX. All rights reserve. Visit us online at www.nms.org
6. A (00 AB9/BC9) e f( ) 4 e (0) e f (0) (0) 04e 5 Stuent Stuy Session 7. D (008 AB) f( ) ( ) ( ) ( ) ( ) () f ( ) ( ) 6 ( ) ( ) f( ) ( ) (7 6 ) 8.A (008 AB 8) f(6), f () g() 6 Since g( ) ; g() f '( g( )) f( g()) f(6) 9. B (AP-like) This limit represents the alternative efinition of the erivative of evaluate at 9. ( ) 9 9 0. C (00 AB9/BC9) f () f (4) 4 ln 4 4ln4 ln4 m 4 f( ) ln4 Determine c where f() c ; c.64 Free Response. 00B AB6a (a) f ( ) f( ) f( ) f( ) f( ) 9 9 f () 5 : f ( ) eponential y (0) < - > prouct or chain rule error : value at Copyright 04 National Math + Science Initiative, Dallas, TX. All rights reserve. Visit us online at www.nms.org
. 000 AB5/BC5 y y (a) y y y 0 y y y y : implicit ifferentiation : verifies epression for y Stuent Stuy Session y y y y (b) When, y y 6 y y6 0 ( y)( y) 0 y, y y 9 9 At (, ), 0 6 Tangent line equation is y y 64 At (, ), 4 Tangent line equation is y ( ) (c) Tangent line is vertical when y 0 ( y ) 0 gives 0 or y There is no point on the cure with -coorinate 0. 5 When y, 6 4 5 6 4 5 4 4 : y y 6 : solves for y : tangent lines : sets enominator of y equal to 0 : substitutes y or y : solves for -coorinate Copyright 04 National Math + Science Initiative, Dallas, TX. All rights reserve. Visit us online at www.nms.org
. 007 AB () g() so g () g () gg () g() 5 An equation of the tangent line is y ( ) 5 : g () y (0) : g () : tangent line equation Stuent Stuy Session Copyright 04 National Math + Science Initiative, Dallas, TX. All rights reserve. Visit us online at www.nms.org
Stuent Stuy Session 4. 980 AB6/BC4 (in 980, each free response question was worth 5 points) a) 4 0y 0 yy6y y 0 Differentiation -: no prouct rule 4 0y 5y y = 6 -: other calculation error 0 y6y 5 y8y Solves for y 6 0 b) m y(,) 0 8 y y c) answers epen on answer from part (b) 5 for y case, ( a, a), a 0 for any other line from part (b), a a,, a0, a ( a, a), a 0 a ) for y case, a,, a 0 for any other line from part (b),, l( ) Linear equation 0 5 if not linear Correct points (0, 0) for any other line from part (b) receives 0 all other points Correct points 0 all other points Copyright 04 National Math + Science Initiative, Dallas, TX. All rights reserve. Visit us online at www.nms.org
Stuent Stuy Session 5. 978 AB b) h ( ) (ln ) ln No point values provie. (ln ) c) h( ) ln 6. 0 AB 6 (a) lim sin 0 4 lim e 0 f 0 So, lim f f 0 0. Therefore f is continuous at 0. : analysis cos for 0 f 4e for 0 cos for all values of 0. 4 4e when ln 0. 4 4 Therefore f for ln 4 4. (b) 4 : f : value of Copyright 04 National Math + Science Initiative, Dallas, TX. All rights reserve. Visit us online at www.nms.org