BUYS Workbook. Preparing to come to Durham. Instructions

Transcription

1 BUYS Workbook Preparing to come to Durham Instructions There are two steps we ask ou to take before ou arrive at Durham in October: STEP : Attempt the practice questions in this workbook. The questions are graded as follows; Questions from areas that should be familiar from the core A-level. Good knowledge of these topics will be assumed. More dicult questions from these areas. Questions from areas outside the core A-level that will have been seen b man. These topics will be covered in the lectures. More dicult questions from these areas. STEP : All students taking the H maths modules Calculus and Probabilit and Linear Algebra are required to take an online timed Diagnostic Test that we designed on the basis of the core mathematics material taught at A level accross the eamination boards. You can view the completion of the workbook as a good preparation for the diagnostic test. The diagnostic test is available online at the BUYS (Brush Up Your Skills) website and can be accessed through DUO at using our CIS username and password, which ou received after completing enrollment. Contact CIS at if ou haven't received our username and password et. After logging in to DUO, look for a frame called M Courses located on our duo Home page. In M Courses, click on Brush Up Your Skills in Maths (08). You can access the Diagnostic Test from the course Announcements or from the left menu. If ou were not automaticall enrolled in BUYS or ou can't nd a link to enrol on our DUO home screen, please follow this link The links for the Diagnostic Test and enrollment are also available at the BUYS Wiki page at If ou are having issues accessing the test, please contact Camila Caiado at c.c.d.s.caiado@durham.ac.uk. If ou are registered with Disabilit Support or are going to be registering with Disabilith Support and need the test in a dierent format, please contact maths.office@durham.ac.uk.

2 Wh a workbook? It is important to realise before starting our degree that Universities are not centres of teaching; the are centres of learning. In this respect the dier from Schools, where our learning has probabl been directed fairl closel b teaching sta. At Durham Universit we will epose ou to lecture courses that provide a framework on which to base our own stud. You will be epected to take control of our own degree, direct our own learning and above all be aware of our own strengths and weaknesses. The students who ourish are those who can independentl identif and correct gaps in their knowledge; this requires a degree of introspection and reection that ma be new to ou. You will have to adapt to this new environment during the rst term of stud. How well ou do so will to a great etent determine the outcome of our degree. (Those who believe that the can coast during the rst terms and catch up later invariabl do badl.) That process begins with this workbook, where ou can identif and deal with an areas of weakness in order to la down rm foundations for a successful period of stud at Durham. Mathematics is above all other disciplines one in which advanced knowledge builds upon more basic knowledge. It is essential therefore that before coming to Durham ou make sure that ou are familiar with the basic skills. The workbook presents a set of material which it would be useful to know before starting our degree to help ou do that. It is not necessar to know all this material, but it would make our rst term more comfortable if ou did know it. Rather than presenting a list of topics, the workbook contains questions on each topic that should be straightforward; b tackling these questions, ou will see how much knowledge is epected. In order to make it less daunting the questions are graded as above. We estimate that completing all the straightforward core questions (those marked with a ) will probabl take ou hours or more. These are areas with which ou should be ver familiar; the better ou know them the more comfortable our adjustment to the degree programme will be. If ou nd that some of the core material is unfamiliar, ou should take the time now to look it up. You ma wish to look up some of the non-core material too. Although this will be covered in lectures it will give ou a head start. All the material here is covered in the standard A-level tets. If a particular core area is reall unfamiliar to ou, it would be worth doing more eercises from a tet book to supplement those given here. You ma also like to ask our current director of studies for help while the are still available to ou. If ou have diculties with some questions, don't worr; ou will have oppportunities to cover the material when ou get to Durham. Most of ou will be ver familiar with some of the material covered here, especiall if ou have done Further Maths. However it is still worth sketching out a solution to these questions even if ou do not go through them in detail: it will be good revision and anwa ou never reall know that ou have understood a problem until ou do it. The answers are given at the end of the workbook. Note that none of the questions requires the use of a calculator. There is a secondar more specic reason for providing a workbook. It is important for us to keep a track of what our incoming students have done at A-level. This has become increasingl dicult in recent ears as the A-level sllabus has changed signicantl. A large part of an A-level sllabus is common to all eamining boards, but the rest varies between eamining boards and according to the choice of modules. The information we gather is passed on to our rst ear lecturers so that the can tailor their courses accordingl in order to provide as smooth a transition as possible from secondar education. Thank ou for helping us do this. Please comments or corrections to c.c.d.s.caiado@durham.ac.uk. September 7, 08

3 Functions of a real variable Practice Questions You should be familiar with basic functions of a real variable. You should also be familar with the denitions of domain and range and the notation (that for a function f of a real variable ) that f : R R : f().. Match the following functions to the graphs below (a) f () = log e (b) f () = arcsin (c) f 3 () = / (d) f () = 3 (e) f 5 () = (f) f () = sin ( + π) (g) f 7 () = cos + (h) f 8 () = (i) f 9 () = 3 (j) f 0 () = e (k) f () = (l) f () = tan Note You will be epected to be able to draw the graphs of elementar functions, and to perform basic manipulations of graphs (horizontal and vertical translation, smmetr w.r.t. the -ais, the -ais and the origin, dilatations and contractions) graph graph graph 3 graph

4 graph 5 graph graph 7 graph 8 8 graph 9 graph 0 graph graph

5 Algebra Although computers and even calculators are ver good at algebra, all mathematicians agree that it is important to be able to do routine algebra quickl and accuratel. You should be able to state elementar series epansions including binomial, sine and cosine, and ln series.. Factorization Factorize the following polnomials: (i) 3 + (ii) (iii) (iv) 3 (v) Notes In part (iii) ou will need the quadratic formula to nd the factors; part (iv) has one linear and one quadratic factor; for part (v) ou can use the factor theorem. 3. More factorization Find the values of for which 3 < Partial fractions Epress the following in partial fractions: (i) (ii) ( + )( ) 3 + (iii) (iv) + ( + ) ( ) 7 ( )( + ) Note It is best for these purposes not to use the `cover-up rule'; there are at least two other was which involve elementar mathematics, whereas the cover-up rule works for more sophisticated reasons and to most users is simpl a recipe (which does not alwas work). 5. Completing the square Find the smallest value (for real and ) of: (i) + (ii) (iii) sin + sin Note Of course, ou could nd the smallest value b calculus, but epressing the function as a perfect square plus a remainder term is a surprisingl useful technique for eample, when integrating a function with a quadratic denominator.. Eponentials and logs (i) The eponential function e can be dened b the series epansion e = + +! + 3 3! +. Use this denition to show that de d = e. (ii) The natural log function log e t can be dened (for t > 0) as the inverse of the eponential function, so that log e e =. Set t = e and use the relationship d / dt dt = to show that d d log e t dt (iii) Assuming that the eponential function has the propert e s e t = e s+t, prove that log e () = log e +log e. = t. (iv) The denition of a for an a is e log e a. Prove that a a = a + and a b = (ab). Note If ou have not thought of dening a in this wa, it is worth considering how else ou could give it a meaning when is not an integer.

6 7. Binomial epansions (i) Find the coecient of k (for 0 k 0) in the binomial epansion of ( + 3) 0. (ii) Use the binomial theorem to nd the epansion in powers of up to of ( + + ), b writing it in the form ( + ( + ) ). (iii) Use binomial epansions to nd the epansion in powers of up to of ( 3 ) ( ). (iv) Find the rst four terms in the binomial epansion of ( + ). 8. Series epansions of elementar functions Using onl the series epansions sin = 3 /3!+ 5 /5!+, cos = /! + /! +, e = + + /! + 3 /3! + and log e ( + ) = / + 3 /3 +, nd the series epansions of the following functions: (i) tan (up to the 5 term) (ii) sin cos (up to the 5 term) (iii) e + e (up to the 5 term) (iv) log e (e ) (up to the 3 term) (v) cos (up to the term). Notes Do part (ii) without using a trig. formula, and compare our answer with the epansion for sin(). The function in part (iii) is cosh, of which more later. The interesting thing about the function in part (v) is that the series shows it is `well-behaved' at = 0, despite appearances. 9. Proof b induction Prove b induction that the following results are valid. ( ) r (i) a + ar + ar + + ar n n = a. r n (ii) r 3 = n (n + ). r= Note Most students should have met mathematical induction ; if ou haven't, ou will probabl want to tr out this straightforward but important method of proof. 0. Arithmetic and geometric progressions (i) Find the sum of all the odd integers from to 99. (ii) Evaluate (iii) Find sin θ + sin 3 θ + sin 5 θ +. (What ranges of values of θ are allowed?). (iv) Estimate roughl the approimate number of times a piece of paper has to be torn in half, placing the results of each tearing in a stack and then doing the net tearing, for the stack of paper to reach the moon. Note The dots in (ii) and (iv) indicate that the series has an innite number of terms. For part (iii), recall that the epansion a( + r + r + ) onl converges if < r <. You ma nd the approimation 0 3 = 0 useful for part (iv). The distance from the Earth to the Moon is about 0 5 km.

7 Trigonometr It is not necessar to learn all the various trigonometrical formulae; but ou should certainl know what is available. The double-angle formulae (such as tan = tan /( tan )) are worth learning, as are the basic formulae for sin(a ± B), cos(a ± B) and tan(a ± B), the Pthagoras-tpe identities (such as sec = + tan ) and a few special values (such as sin π/ = / ) that can be deduced from right-angled triangles with sides (,, ) or (, 3, ).. Basic identities Starting from the identit sin(a + B) = sin A cos B + cos A sin B, use the basic properties of the trig. functions (such as sin( A) = sin A) to prove the following: (i) sin(a B) = sin A cos B cos A sin B (ii) cos(a + B) = cos A cos B sin A sin B tan A + tan B (iii) tan(a + B) = tan A tan B (iv) sin C + sin D = sin (C + D) cos (C D) ( ) a + b (v) tan a + tan b = tan. ab Notes. Trig. equations For part (ii), recall that cos A = sin(π/ A). You can use part (iii) to help with part (v). (i) Solve the following equations: (a) sin( + π ) + sin( π ) = 3 (b) cos + cos + cos 3 = 0 (ii) Write down the value of cot(π/) and use a double angle formula to show that cot(π/) satises the equation c 3c = 0. Deduce that cot(π/) = Trig. identities using Pthagoras Prove the following identities: (i) tan θ + cot θ = sec θ cosec θ (ii) (cot θ + cosec θ) = + cos θ cos θ (iii) cos θ = t t t ; sin θ = ; tan θ = + t + t t, where t = tan θ.

8 Comple numbers No prior knowledge of comple numbers is assumed in our rst ear course because some A-level sllabuses do not cover comple numbers. Nevertheless, the material is lectured quite fast, so if ou have not met comple numbers before, working through the following eamples with the help of a tetbook would make the rst few weeks of the course more comfortable. A comple number z can be written as + i, where is the real part, is the imaginar part and i =. The modulus of z (written z or r) is ( + ) and the argument (written arg z or θ) is dened b = r cos θ, = r sin θ and π < θ π. The comple conjugate of z (written z ) is i. The inverse, z, of z is the comple number that satises z z = (for z 0).. Algebra of comple numbers Use the denitions above with z = + i and z = + i to show: (i) z z = ( ) + i( + ) (ii) z = zz (iii) z = z z (iv) z z = z z (v) arg(z z ) = arg z + arg z (assume that 0 < arg z π/ and 0 < arg z π/). Give a sketch of the plane (called also the comple plane or the Argand diagram) showing the points representing the comple numbers z = i, z = 3 + i. Verif results (iii), (iv) and (v) for these numbers. 5. De Moivre's theorem (i) Show b means of series epansions that cos θ + i sin θ = e iθ and deduce that cos θ i sin θ = e iθ and that z = re iθ. Deduce also that cos θ = (eiθ + e iθ ) and sin θ = i (eiθ e iθ ). (ii) Use the above result to show that re iθ = (for real θ and r > 0) if and onl if r = and θ = nπ for some integer n. (iii) Use part (ii) to nd the three distinct roots of the equation z 3 =. Draw them on the comple plane and convert them from modulus-argument form to realimaginar form.

9 Hperbolic functions Prior knowledge of hperbolic functions is not assumed for our mathematics course, but it is worth getting to know the denitions and basic properties, which are given below. Hperbolic functions are ver similar to trigonometric functions, and man of their properties are direct analogues of the properties of trigonometric functions. The denitions are sinh = e e, cosh = e + e, tanh = e e e + e and cosech = (sinh ), sech = (cosh ), coth = (tanh ).. Basic properties Give a rough sketch of the graphs of the si hperbolic functions. Show from the above denitions that (i) cosh sinh = d(sinh ) d(cosh ) (ii) = cosh ; = sinh d d (iii) cosh( + ) = cosh cosh + sinh sinh (iv) cosh(i) = cos, sinh(i) = i sin, cos(i) = cosh(), sin(i) = i sinh. Notes You can use (iv) to prove (i) and (iii) using the corresponding trig. identities. In fact, (iv) is behind the rule which sas that an formula involving trig. functions becomes the corresponding formula involving hperbolic functions if ou change the sign of ever product of two odd functions (such as sin or tan ). 7. Further properties Use the denitions, and the results of the previous question, to show that (i) sech = tanh d(tanh ) (ii) = sech d (iii) d (sinh ) d = sinh (iv) sinh ( = log e + ( + ) ). Note For part (iv), note that sinh is the inverse function, not the reciprocal.

10 Dierentiation Dierentiation of standard functions, products, quotients, implicit function epressions, and functions of a function (using the chain rule) should be routine. 8. Direct dierentiation Dierentiate () with respect to in the following cases: (i) = (5 + ) (ii) = sin(3 + ) (iii) = e ( ) (iv) = 5 cos( + 3) (v) = cos 3 (vi) = log e ( 5). 9. More direct dierentiation Dierentiate () with respect to in the following cases: (i) = log e ( + ( + ) ) (ii) = a (iii) = (iv) = sin +. Notes Simplif our answer to part (i). For (ii), see the denition in question (iv). One wa to do (iv) is to take the sine on both sides and use the chain rule. Can ou see wh the answer to (iv) is surprisingl simple? 0. Parametric dierentiation Show that if = a cos θ, = b sin θ then d < 0 for > 0. d. More Chain and Product rule Using the chain and product rules etc, nd the derivatives of the following (where a is a positive constant) (i) = sin( ) (ii) = log e ( a + a ). Stationar points Find the stationar points of the function f() = + a, where a > 0, classifing them as either maimum or minimum. Sketch the curve (without using a calculator).

11 Integration You need to be able to recognise standard integrals (without having to leaf through a formula book) and evaluate them (referring to our formula book, if necessar). You need to be familiar with the techniques of integration b parts and b substitution. 3. Indenite integrals Calculate the following integrals: (i) d (vi) (ii) 3 sin d (vii) (iii) e d (viii) (iv) d (i) (v) 3 d () 5 d d d 3 cos d sec d.. More indenite integrals Calculate the following integrals: (i) a d (vi) cosec d (ii) + d (vii) sec d (iii) e a cos(b) d (viii) (c + m ) d (iv) e a e ib d (i) tan d (v) 3 d () 3 e d. Notes The dicult of these varies. For part (i), see question (iv). For part (ii), see question 5(i). Note that ou can obtain (iii) from (iv) b taking the real part. Use partial fractions for part (v). For (vi) and (vii), use the substitution t = tan(/) rather than the trick of multipling top and bottom b e.g. sec +tan. Tr also deriving (vii) from (vi) b means of the substitution = π/. For (viii), ou can substitute m = c sinh θ if ou are familar with hperbolic functions. Use integration b parts for (i) and (). 5. Denite integrals Calculate the following integrals: (i) (ii) 0 π/ 0 e d sin 3θ cos θ d (iii) (iv) 0 π/ dθ cos θ dθ Notes The dicult of these varies. For part (iv) use the substitution t = tan(θ/).

12 Answers. f (): graph 5; f () : graph 8; f 3 () : graph ; f () : graph f 5 () : graph 7; f () : graph ; f 7 () : graph ; f 8 () : graph ;f 9 () : graph 9; f 0() : graph 3; f () : graph ; f () : graph 0. (i) ( )( ) (ii) 3( )( + ) (iii) ( ( 5))( ( + 5)) (iv) ( )( + + ) (v) ( 3)( ) ( + ). 3. < or 0 < < 3.. (i) (iii) + ( + ) + + (ii) ( 3 + ) + (iv) (i) 5; (ii) ; (iii) 3 (smallest when sin = ). 7. (i) 3 k 0 k 0! k!(0 k)! (iii) Same as (ii) (iv) (ii) ( ). 8. (i) (ii) (iii) + + (iv) (v) (i) 75 (ii) (iii) sin θ cos θ ( π/ + nπ < θ < π/ + nπ) (iv) about times.. (ia) nπ + ( ) n π/ (ib) nπ/ + π/, nπ ± π/3.. z =, arg z = π/, z =, arg z = 5π/. 5. (iii), e iπ/3, e iπ/3 or, ( ± i 3)/. 8. (i) 0(5 + ) 3 (ii) 3 cos(3 + ) (iii) e ( ) (iv) 0 sin( + 3). (v) 3 sin cos (vi) (i) + (ii) a log e a (iii) ( + log e ) (iv) + (sin + = tan ).. (i) cos( ), (ii) a(a a ) a + a

13 . Maimum at ( a, ) ( ), minimum at a,. a a 3. (i) C (ii) 3 cos + C (iii) e + C (iv) + C (v) 3 + C (vi) 5 /log e + C (vii). (i) C (viii) log e +C (i) 3 sin + C () tan + C (iii) (v) log e a a + C (ii) a + b (a cos b + b sin b)ea + C (iv) log + + e ( ) + ( ) + tan + C 3 3 tan + C 5 5 a + ib e(a+ib) + C (vi) log e tan(/) + C (vii) log e ( + tan(/) tan(/) ) + C (viii) m ( log e (m) + m + c ) + C or m sinh (m/c) + C (i) tan log e( + ) + C () ( )e + C 5. (i) (ii) (iii) 3 log e 3 (iv) log e 3 Department of Mathematical Sciences, September 7, 08