For the AB eam, students need to: determine antiderivatives of the basic functions calculate antiderivatives of functions using u-substitution use algebraic manipulation to rewrite the integrand prior to integrating use trigonometric definitions and properties of eponents and logarithms to rewrite solutions use geometric interpretations of the definite integral
Complete this worksheet. These rules should be memorized. Basic Integration kf( udu ) [ f ( u) g( u)] du du n u du du u u a du u e du Inverse Trigonometric du u du u Trigonometric Functions: sin( udu ) cos( u) du sec ( udu ) csc ( udu ) sec( u) tan udu csc( u)cot udu Helpful to know: sin u tan( udu ) du cosu cosu cot( udu ) du sin u
Multiple Choice. (calculator not allowed) d 0 (A) ln 4 6 6. (calculator not allowed) d (A) ln 8 ln ln 8 ln ln 8 ln ln. (calculator not allowed) 4 d (A) ln ln ln
4. (calculator not allowed) e d = (A) e e e e e e e e 5. (calculator not allowed) ( ) d = 0 (A) 0 6 5 7 5 6. (calculator not allowed) d e (A) ln e C e C e C ln e C e C
7. (calculator not allowed) sin( ) cos( ) d (A) cos( ) sin( ) C cos( ) sin( ) C cos( ) sin( ) C cos( ) sin( ) C cos( ) sin( ) C 8. (calculator not allowed) cos d (A) sin C sin C sin C sin C 4 sin C 4 9. (calculator not allowed) cos d sin 4 (A) ln ln 4 ln ln ln e
0. (calculator not allowed) dy If sin cos d and if y = 0 when, what is the value of y when = 0? (A) 0. (calculator not allowed) If the substitution (A) u du u 4 4 u du u u du u u du 4u u du u u is made, the integral 4 d. (calculator not allowed) k If ( k ) d 8, then k 0 (A) 9 9 8
. (calculator not allowed) If f( c) d5 where c is a constant, then c f ( ) d c (A) 5 c 5 5 c c 5 5 4. (calculator not allowed) b If f ( ) da b b, then ( ( ) 5) a f d a (A) a b 5 5b 5a 7b 4a 7b 5a 7b 6a 5. (calculator not allowed) If f is a linear function and 0 a b, then f ( ) d a (A) 0 ab b a b a b
6. (calculator not allowed) What are all values of k for which (A) 0 and, 0, and k d 0? 7, (calculator not allowed) (008 AB7) A particle moves along the -ais with velocity given by vt t 6tfor time t 0. If the particle is at position at time t 0, what is the position of the particle at time t? (A) 4 6 9 8. (calculator allowed) If f is a continuous function and if F( ) f( ) for all real numbers, then f ( ) d (A) F() F() () F () F F(6) F() F(6) F() F(6) F() 9. (calculator allowed) On the graph of y f( ), the slope at any point (, y ) is twice the value of. If f (), what is the value of f ()? (A) 6 7 8 9 0
Free Response 0. (calculator not allowed) The figure above shows the graph of f, the derivative of a function f. The domain of f is the set of all such that 0. (a) Write an epression for f ( ) in terms of. (b) Given that f () 0, write an epression for f ( ) in terms of. (c) Sketch the graph of y f( ).
. (calculator not allowed) Let f be a function such that f ( ) 6 8. (a) Find f () if the graph of f is tangent to the line y at the point (0, ). (b) Find the average value of f ( ) on the closed interval [,].. (calculator not allowed) Let f be a differentiable function, defined for all real numbers, with the following properties. (i) f ( ) a b (ii) f () 6 and f () 8. (iii) f( ) d 8 (a) Find f ( ). Show your work.
. (calculator not allowed) Let f ( ) 4 and g( ) k sin for k 0. k (a) Find the average value of f on [, 4]. (b) For what value of k will the average value of g on [0, ] k be equal to the average value of f on [, 4].