1 sin x cosx methods. (1 cosx) 2 Ba C3 Trig solves π Bb C3 Trig solves 5

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1 rre nt wo Drill Done BP Ready? A Assignment eta Cover Sheet Name: Question Topic Answers Aa e (sin cos ) Ab e (cos sin ) cos Ac sin sin ln Ad cos sin Ae cos Af sin cos ( cos) Ba C Trig solves π Bb C Trig solves 5, 6 6 Bc C Trig solves 4, Ca C Rcos 7, 5.0 Cb C Rcos, 0.95 Cc C Rcos 0,.89 Da C Integration by inspection c 6 Db C Integration by inspection 6 c Dc C Integration by inspection e + c Dd C Integration by inspection cos( ) c De C Integration by inspection ln c Df C Integration by inspection tan c TTA TTB TTC TTD C Inverse trig functions Check on google inc asymptotes C Inverse trig functions a C Creating an iterative a 5, b X:\Maths\TEAM - A\A Assignments 6-7\A Mechanics\MPM(7)eta6-7.doc Undated; 0/0/06

2 Ch 6 M Force diagram with S Practice Consolidation formula b C Creating an iterative formula a 5, b c C Creating an iterative a formula 4 d C Creating an iterative a, b formula e C Creating an iterative formula a 5 f C Creating an iterative a formula, b 4 C Numerical. root 5a C Numerical.4 5b C Numerical Discuss in class justify 7a friction C Functions- domain and range a) 0.8N b) 0.75 c) F ma =.94N, 6sin5 =.54N,.54<.94 so the weight remains in equilibrium (sketches, make sure they are one to one in order to have an inverse) 7b 7c C Functions- domain and 0 f ( ) ; f ( ) range f f C Functions- domain and ( ) 0; ( ) range f f 8 C Using d/dy + trig ids to PROOF find dy/d 9a C Trig proof PROOF 9b C Trig solve 5 0,,,, a C Trig solve π 0b C Trig solve 0,, C Trig proof PROOF all Challenge! / REDO C PAST PAPER IF <90% ( ) 5; ( ) REDO C PAST PAPER IF <90% X:\Maths\TEAM - A\A Assignments 6-7\A Mechanics\MPM(7)eta6-7.doc Undated; 0/0/06

3 A linguist would be shocked to learn that if a set is not closed this does not mean that it is open, or that E is dense in E does not mean the same thing as E is dense in itself. J E Littlewood A Maths with Mechanics Assignment eta (5 questions including drill plus redoing the C past paper June 005 if you achieved less 90%. Due in w/b 7/) Maths Trip: Maths In Action University Lectures in London. 0 a ticket (0 tickets available) 5 th November Maths Trip: Maths In Action University Lectures in London. 0 a ticket (0 tickets available) 4 th December Drill Part A Differentiate with respect to : e (a) e sin (b) cos (d) sin (e) sin cos (c) (f) sin sin cos ln Part B Solve these equations for 0 < θ < π c (a) sec (b) cot (c) cosec Part C By writing each of these functions in the form given, state the greatest value of each function and the smallest positive value of (in radians to dp) at which this occurs. (a) 8cos 5sin, Rcos( ) (b) 5sin cos, Rsin( ) (c) sin cos, Rsin( ) Part D Integrate the following with respect to :by considering the reverse of differentiation (a) 4 d (b) 5 (d) sin( ) d (e) TT FOCUS: A) B) d (c) e d d (f) sec d X:\Maths\TEAM - A\A Assignments 6-7\A Mechanics\MPM(7)eta6-7.doc Undated; 0/0/06

4 C) D) Current work : Inverse trig, Numerical y arcsin, y arccos, y arctan. Sketch the graph of, labelling your aes and aes crossing points clearly. Given that arctan( ), find the value of, Show the steps by which the following iterative formulae can be derived from the given equations. State the values of the constants a and b in each case: Equation to be solved (a) 5 0 Iterative formula n a n b (b) 5 0 n a b n (c) (d) (e) (f) 4ln e 9 0 ln 5 cos 0 n e a n n ln(a b n ) n e 4 Given that f () 5 4sin where is measured in radians, show that f ( ) 0 has a root in the interval (.,.5). Use the iterative formula n psin n q (where p and q are constants to be found) and 0. to find 4 to sf. a n n arccos(a b n ) 5 The root of the equation f() = 0, where f() ln 4 is to be estimated using the iterative formula n 4 ln n, with 0.4. (a) Showing your values of,,,, obtain the value, to decimal places, of the root. (b) By considering the change of sign of f() in a suitable interval, justify the accuracy of your answer to part (a). X:\Maths\TEAM - A\A Assignments 6-7\A Mechanics\MPM(7)eta6-7.doc Undated; 0/0/06

5 M Practice (Preparation for M) 6. A weight of 6N rests on a rough 5 o incline. The perpendicular reaction is measured to be 0N. A horizontal force H pushes the weight so that it is just on the point of slipping up the plane. a) Complete a force diagram and find force H b) Find μ, the coefficient of friction. Force H is now removed c) Showing all your calculations clearly, justify whether the 6N weight will slide down the plane, or remain in equilibrium. Consolidation 7. For each of the following functions, whose domain is the set of positive real numbers, sketch the function and hence state the range. For each function find its inverse (a) f ( ) (b) f ( ) (c) f ( ) Prove that if sec y, then dy d ( P Q) ( P Q) 9. Using the result that sin P sin Q cos sin (a) Show that sin 05 o sin 5 o = (b) Solve, for 0,sin 4 sin 0 0. Solve the following equations on the interval 0. Give eact answers: (a) cos(sin ) = (b) cot 6. Forming the factor formulae Use the formulae for sin(a+b) and sin( A B) ( P Q) ( P Q) sin P sin Q sin cos to derive the result that Challenge Question X:\Maths\TEAM - A\A Assignments 6-7\A Mechanics\MPM(7)eta6-7.doc Undated; 0/0/06

6 Triangle ABC has AB C = 90 and AC B = 0. If a point inside the triangle is chosen at random, what is the probability it is nearer AB than it is to AC? X:\Maths\TEAM - A\A Assignments 6-7\A Mechanics\MPM(7)eta6-7.doc Undated; 0/0/06