| Introduction |
|
1 | (4) |
|
|
|
1 | (1) |
|
|
|
2 | (1) |
|
|
|
3 | (1) |
|
|
|
3 | (1) |
|
|
|
4 | (1) |
| Part I: Getting Started |
|
5 | (66) |
|
Chapter 1 Moving towards Mathematical Mastery |
|
|
7 | (12) |
|
|
|
7 | (3) |
|
Setting up for study success |
|
|
8 | (1) |
|
|
|
8 | (1) |
|
Grabbing graphs by the horns |
|
|
9 | (1) |
|
Taming triangles and other shapes |
|
|
9 | (1) |
|
Attacking Advanced Algebra |
|
|
10 | (2) |
|
Picking over powers and surds |
|
|
10 | (1) |
|
Sorting out sequences and series |
|
|
11 | (1) |
|
|
|
11 | (1) |
|
|
|
12 | (1) |
|
Getting to Grips with Geometry |
|
|
12 | (3) |
|
Conquering coordinate geometry |
|
|
13 | (1) |
|
Setting up circles and triangles |
|
|
13 | (1) |
|
Taking trigonometry further |
|
|
14 | (1) |
|
|
|
14 | (1) |
|
|
|
15 | (4) |
|
Dashing off differentiation |
|
|
16 | (1) |
|
Inspiring yourself to integrate |
|
|
16 | (1) |
|
|
|
17 | (2) |
|
Chapter 2 Setting Yourself Up for Study Success |
|
|
19 | (12) |
|
|
|
20 | (2) |
|
|
|
20 | (1) |
|
|
|
21 | (1) |
|
|
|
21 | (1) |
|
Getting Your Head On Straight |
|
|
22 | (3) |
|
Sorting out your attitude |
|
|
23 | (1) |
|
|
|
23 | (1) |
|
Coping when things go wrong |
|
|
24 | (1) |
|
|
|
25 | (3) |
|
|
|
26 | (1) |
|
|
|
26 | (1) |
|
|
|
27 | (1) |
|
|
|
27 | (1) |
|
Quick-Fire Revision Techniques |
|
|
28 | (3) |
|
|
|
29 | (1) |
|
|
|
29 | (1) |
|
|
|
30 | (1) |
|
|
|
30 | (1) |
|
Chapter 3 All the Algebra You Missed |
|
|
31 | (22) |
|
The Brilliance of Boodles: Understanding the Order of Operations |
|
|
31 | (2) |
|
Practising Your Power Laws |
|
|
33 | (2) |
|
|
|
33 | (1) |
|
Knowing your squares, cubes and powers |
|
|
34 | (1) |
|
Handling nasty fractional powers |
|
|
35 | (1) |
|
Expanding Brackets and Simplifying |
|
|
35 | (3) |
|
|
|
35 | (1) |
|
|
|
36 | (1) |
|
Expanding several brackets |
|
|
37 | (1) |
|
Fiddling About with Fractions |
|
|
38 | (5) |
|
Manipulating fractions with numbers |
|
|
38 | (5) |
|
|
|
43 | (1) |
|
|
|
43 | (5) |
|
|
|
44 | (1) |
|
Dealing with simple fractions |
|
|
44 | (1) |
|
Doing rougher rearrangement |
|
|
45 | (1) |
|
|
|
45 | (3) |
|
Solving Simultaneous Equations |
|
|
48 | (5) |
|
Linear simultaneous equations |
|
|
48 | (2) |
|
Nonlinear simultaneous equations |
|
|
50 | (3) |
|
Chapter 4 Shaping Up to Graphs and Shapes |
|
|
53 | (18) |
|
|
|
53 | (3) |
|
|
|
54 | (1) |
|
|
|
54 | (2) |
|
|
|
56 | (1) |
|
|
|
56 | (2) |
|
|
|
57 | (1) |
|
|
|
57 | (1) |
|
|
|
58 | (2) |
|
|
|
60 | (9) |
|
|
|
62 | (1) |
|
|
|
62 | (1) |
|
|
|
63 | (2) |
|
|
|
65 | (1) |
|
|
|
66 | (3) |
|
|
|
69 | (4) |
|
|
|
69 | (1) |
|
Sectors, arcs and segments |
|
|
70 | (1) |
| Part II: Arithmetic and Algebra |
|
71 | (106) |
|
Chapter 5 With Great Power Comes |
|
|
73 | (20) |
|
|
|
73 | (2) |
|
|
|
74 | (1) |
|
Rationalising simple denominators |
|
|
74 | (1) |
|
Rationalising harder denominators |
|
|
75 | (1) |
|
|
|
75 | (1) |
|
Learning to Love the Logarithm |
|
|
76 | (9) |
|
|
|
76 | (1) |
|
Turning powers into logs (and vice versa) |
|
|
77 | (1) |
|
Combining and splitting logarithms |
|
|
78 | (2) |
|
Logs and numbers together |
|
|
80 | (2) |
|
Logarithmic simultaneous equations |
|
|
82 | (1) |
|
|
|
83 | (1) |
|
Solving tricky logs questions |
|
|
83 | (2) |
|
Making Sense of Euler's constant, e |
|
|
85 | (8) |
|
Understanding that e is just a number (but a special one) |
|
|
86 | (1) |
|
Converting between powers |
|
|
86 | (1) |
|
Solving things with e in them |
|
|
87 | (3) |
|
Watching out for booby-traps |
|
|
90 | (3) |
|
Chapter 6 Playing with Polynomials |
|
|
93 | (18) |
|
|
|
94 | (3) |
|
Following the basic method |
|
|
94 | (1) |
|
Solving a quadratic by completing the square |
|
|
95 | (1) |
|
|
|
96 | (1) |
|
Understanding where the quadratic equation comes from |
|
|
96 | (1) |
|
Factorising and Solving Simple Polynomials |
|
|
97 | (2) |
|
|
|
97 | (1) |
|
Taking off a quadratic's disguise |
|
|
98 | (1) |
|
|
|
99 | (5) |
|
Dealing with discriminants |
|
|
100 | (2) |
|
Finding the numbers of solutions |
|
|
102 | (1) |
|
|
|
103 | (1) |
|
|
|
104 | (7) |
|
|
|
105 | (1) |
|
Quadratic (and higher-order) inequalities |
|
|
106 | (2) |
|
|
|
108 | (1) |
|
Discriminant-related inequalities |
|
|
109 | (2) |
|
Chapter 7 Factors, Remainders and Fractions |
|
|
111 | (16) |
|
|
|
112 | (3) |
|
|
|
112 | (1) |
|
Using trial and improvement |
|
|
113 | (1) |
|
The prime directive: Getting the FACTs |
|
|
114 | (1) |
|
|
|
115 | (1) |
|
|
|
116 | (4) |
|
Long (and tedious) division |
|
|
116 | (2) |
|
|
|
118 | (1) |
|
Finishing off the question |
|
|
119 | (1) |
|
|
|
119 | (1) |
|
Putting the Factor and Remainder Theorems Together |
|
|
120 | (1) |
|
|
|
120 | (1) |
|
|
|
121 | (1) |
|
|
|
121 | (1) |
|
|
|
121 | (3) |
|
Cancelling first to get lowest common denominators |
|
|
122 | (1) |
|
Understanding simplest form |
|
|
123 | (1) |
|
Piecing Together Partial Fractions |
|
|
124 | (3) |
|
|
|
124 | (2) |
|
|
|
126 | (1) |
|
Chapter 8 Getting Serious about Series |
|
|
127 | (26) |
|
Explicit and Recursive Definitions |
|
|
128 | (2) |
|
Explaining explicitly defined sequences |
|
|
128 | (1) |
|
Getting your head around recursive sequences |
|
|
129 | (1) |
|
Series Stuff: Summing Up Sigma Notation |
|
|
130 | (1) |
|
Analysing Arithmetic Sequences |
|
|
131 | (5) |
|
|
|
132 | (1) |
|
|
|
133 | (1) |
|
|
|
134 | (1) |
|
|
|
135 | (1) |
|
Generating Geometric Sequences |
|
|
136 | (6) |
|
|
|
136 | (1) |
|
|
|
137 | (2) |
|
|
|
139 | (1) |
|
|
|
140 | (2) |
|
|
|
142 | (2) |
|
Proving the arithmetic series sum |
|
|
142 | (1) |
|
Proving the geometric series sum |
|
|
143 | (1) |
|
Breaking Down the Binomial Expansion |
|
|
144 | (6) |
|
|
|
144 | (3) |
|
|
|
147 | (3) |
|
Estimating with the Binomial Expansion |
|
|
150 | (3) |
|
Chapter 9 Fiddling About with Functions |
|
|
153 | (24) |
|
Putting the Tun' in Functions |
|
|
154 | (4) |
|
Nailing down the notation |
|
|
154 | (1) |
|
|
|
155 | (1) |
|
|
|
156 | (2) |
|
Composing and Inverting Functions |
|
|
158 | (6) |
|
Composition: A chain of machines |
|
|
159 | (1) |
|
Inverses: Running the machines backwards |
|
|
160 | (4) |
|
Making Sense of the Modulus |
|
|
164 | (2) |
|
|
|
165 | (1) |
|
Handling inequalities with a modulus |
|
|
165 | (1) |
|
|
|
166 | (3) |
|
Solving for composed functions |
|
|
166 | (1) |
|
Combining inverses and functions |
|
|
167 | (1) |
|
|
|
168 | (1) |
|
|
|
169 | (10) |
|
|
|
169 | (2) |
|
|
|
171 | (1) |
|
|
|
172 | (5) |
| Part II: Geometry |
|
177 | (92) |
|
Chapter 10 Coordinating Your Geometry |
|
|
179 | (26) |
|
The Many Equations of a Line |
|
|
179 | (4) |
|
|
|
180 | (2) |
|
The one you know: y = mx + c |
|
|
182 | (1) |
|
The better one: (y — y0) = m(x — x0) |
|
|
182 | (1) |
|
|
|
183 | (10) |
|
Starter kit: Getting the basic shapes right |
|
|
184 | (4) |
|
Trickier shapes: Sketching the advanced graphs |
|
|
188 | (2) |
|
Intercepts: Crossing your xs and ys |
|
|
190 | (2) |
|
|
|
192 | (1) |
|
Sketching with the DATAS method |
|
|
192 | (1) |
|
|
|
193 | (7) |
|
Looking out for Bad Guy x |
|
|
193 | (1) |
|
Making friends with Good Guy y |
|
|
194 | (1) |
|
The madness of the modulus |
|
|
195 | (1) |
|
Combining transformations |
|
|
195 | (5) |
|
Investigating Intersections |
|
|
200 | (5) |
|
|
|
200 | (1) |
|
|
|
201 | (1) |
|
Where curves meet each other |
|
|
202 | (3) |
|
Chapter 11 Making Sense of Circles and Triangles |
|
|
205 | (16) |
|
|
|
205 | (6) |
|
Where the circle equation comes from |
|
|
206 | (1) |
|
Rearranging circle equations |
|
|
206 | (2) |
|
|
|
208 | (1) |
|
|
|
208 | (2) |
|
|
|
210 | (1) |
|
|
|
211 | (4) |
|
Converting between radians and degrees |
|
|
212 | (1) |
|
|
|
213 | (1) |
|
|
|
214 | (1) |
|
Taking Care of Triangles and Segments |
|
|
215 | (6) |
|
|
|
215 | (1) |
|
|
|
216 | (1) |
|
|
|
217 | (4) |
|
Chapter 12 Taking Trigonometry Further |
|
|
221 | (24) |
|
|
|
221 | (1) |
|
Identifying Trig Identities |
|
|
222 | (6) |
|
Relating trig functions with the basic triangle |
|
|
223 | (1) |
|
Relating the minor trig functions |
|
|
223 | (1) |
|
Relating sine, cosine and tangent |
|
|
224 | (4) |
|
Taming Trigonometric Proofs |
|
|
228 | (2) |
|
|
|
228 | (1) |
|
Perfecting your proof techniques |
|
|
229 | (1) |
|
|
|
230 | (1) |
|
Clearing Up Compound Angles |
|
|
230 | (6) |
|
|
|
231 | (1) |
|
|
|
232 | (1) |
|
R sin (x + a)-type questions |
|
|
233 | (2) |
|
The formulas that don't come up but I have to mention anyway |
|
|
235 | (1) |
|
Solutions: Gotta Catch 'Em All |
|
|
236 | (3) |
|
Why there are multiple solutions |
|
|
237 | (2) |
|
The Dirty Tricks of Trigonometry |
|
|
239 | (6) |
|
|
|
239 | (3) |
|
|
|
242 | (3) |
|
Chapter 13 Making Vectors as Simple as i, j, k |
|
|
245 | (24) |
|
|
|
245 | (3) |
|
Opening your i's, j's and k's |
|
|
246 | (1) |
|
|
|
247 | (1) |
|
|
|
247 | (1) |
|
Lines in 3D: Writing Equations in Vector Form |
|
|
248 | (6) |
|
Writing the equation of a line through two points |
|
|
249 | (2) |
|
Finding the equation of a line in a given direction |
|
|
251 | (1) |
|
Showing that a point is on a given line |
|
|
251 | (1) |
|
Determining where (and whether) lines cross |
|
|
252 | (2) |
|
Dot Products: Multiplying Vectors |
|
|
254 | (2) |
|
Showing vectors are perpendicular |
|
|
255 | (1) |
|
Finding angles between vectors |
|
|
256 | (1) |
|
Answering Evil Vector Questions |
|
|
256 | (6) |
|
Points on perpendicular lines and shortest distances |
|
|
257 | (1) |
|
Points at a given distance |
|
|
258 | (1) |
|
|
|
259 | (1) |
|
|
|
259 | (3) |
|
|
|
262 | (1) |
|
Back to Normal: Picking Out Planes |
|
|
262 | (9) |
|
|
|
262 | (3) |
|
Plane intersections with lines |
|
|
265 | (4) |
| Part IV: Calculus |
|
269 | (94) |
|
Chapter 14 Climbing Slippery Slopes |
|
|
271 | (10) |
|
Taking Slope to the Limit: What Differentiation Is |
|
|
271 | (2) |
|
Designating Derivatives: A Note on Notation |
|
|
273 | (1) |
|
|
|
274 | (2) |
|
Taking care of special cases |
|
|
274 | (1) |
|
Lining up linear combinations |
|
|
275 | (1) |
|
Differentiating Functions |
|
|
276 | (2) |
|
|
|
276 | (1) |
|
Looking at logarithms and exponentials |
|
|
277 | (1) |
|
Finding the Gradient at a Point (and Vice Versa) |
|
|
278 | (3) |
|
Chapter 15 Touching on Tangents and Turning Points |
|
|
281 | (16) |
|
Finding Tangents and Normals |
|
|
282 | (4) |
|
Working out the equation of a tangent |
|
|
282 | (1) |
|
Getting the equation of a normal |
|
|
283 | (1) |
|
Working backwards: Finding points, given a gradient |
|
|
284 | (1) |
|
Finding a tangent to a circle |
|
|
284 | (1) |
|
|
|
285 | (1) |
|
Stationary Points: Turning Curves Around |
|
|
286 | (6) |
|
Finding stationary points with the first derivative |
|
|
287 | (1) |
|
Classifying stationary points with the second derivative |
|
|
288 | (2) |
|
Sketching the turning points of trig functions |
|
|
290 | (2) |
|
Increasing and Decreasing Functions |
|
|
292 | (1) |
|
Turning Points in the 'Real World' |
|
|
293 | (4) |
|
|
|
294 | (2) |
|
Handling 'real world' situations |
|
|
296 | (1) |
|
Chapter 16 Integrating in Style |
|
|
297 | (18) |
|
Opposite Day: What Integration Is |
|
|
297 | (2) |
|
Taking Care of Constants and Powers of x |
|
|
299 | (2) |
|
Accounting for mystery constants |
|
|
299 | (1) |
|
|
|
299 | (1) |
|
Finding the equation of a curve |
|
|
300 | (1) |
|
Finding Your Way with Other Functions |
|
|
301 | (1) |
|
Looking Things Up: Integrals from the Book |
|
|
302 | (1) |
|
|
|
302 | (10) |
|
|
|
303 | (1) |
|
|
|
303 | (2) |
|
|
|
305 | (1) |
|
Taking away and adding on shapes |
|
|
306 | (3) |
|
|
|
309 | (3) |
|
|
|
312 | (3) |
|
Chapter 17 When to Reach for the Rules |
|
|
315 | (24) |
|
Calculus with Linear Expressions |
|
|
315 | (2) |
|
Differentiating nested linear expressions |
|
|
316 | (1) |
|
Integrating nested linear expressions |
|
|
316 | (1) |
|
Differentiating with the Chain, Product and Quotient Rules |
|
|
317 | (8) |
|
|
|
317 | (2) |
|
|
|
319 | (1) |
|
|
|
320 | (2) |
|
|
|
322 | (1) |
|
|
|
323 | (1) |
|
Putting the rules together |
|
|
324 | (1) |
|
Exponentials: Tricky Derivatives |
|
|
325 | (2) |
|
|
|
326 | (1) |
|
|
|
326 | (1) |
|
Integrating by Substitution |
|
|
327 | (3) |
|
Substituting: The basic idea |
|
|
327 | (1) |
|
|
|
328 | (2) |
|
|
|
330 | (4) |
|
|
|
330 | (1) |
|
Deciding which bit is which |
|
|
331 | (1) |
|
|
|
331 | (1) |
|
|
|
332 | (2) |
|
Integration by Trig Identity |
|
|
334 | (3) |
|
|
|
334 | (1) |
|
|
|
335 | (1) |
|
Products of sine and cosine |
|
|
335 | (2) |
|
Deciding Which Integration Rule to Use |
|
|
337 | (2) |
|
Chapter 18 Overcoming Evil Questions |
|
|
339 | (24) |
|
|
|
339 | (8) |
|
|
|
341 | (1) |
|
|
|
341 | (2) |
|
Converting to Cartesian form |
|
|
343 | (3) |
|
Integrating parametrically |
|
|
346 | (1) |
|
|
|
347 | (4) |
|
Finding points on an implicit curve |
|
|
348 | (1) |
|
Differentiating implicitly |
|
|
348 | (3) |
|
|
|
351 | (5) |
|
|
|
353 | (1) |
|
|
|
354 | (1) |
|
|
|
355 | (1) |
|
Practising parametric volumes |
|
|
355 | (1) |
|
|
|
356 | (9) |
|
Generating general solutions |
|
|
357 | (2) |
|
Picking out particular solutions |
|
|
359 | (1) |
|
More-involved differential equations |
|
|
359 | (4) |
| Part V: The Part of Tens |
|
363 | (14) |
|
Chapter 19 Ten Classic Mistakes to Avoid |
|
|
365 | (6) |
|
|
|
365 | (1) |
|
|
|
366 | (1) |
|
|
|
366 | (1) |
|
|
|
367 | (1) |
|
|
|
367 | (1) |
|
Using the Wrong Angle Measure |
|
|
367 | (1) |
|
Falling into a Logarithmic Booby-Trap |
|
|
368 | (1) |
|
|
|
369 | (1) |
|
Mixing Up the Bits of a Vector Line |
|
|
369 | (1) |
|
Losing Track of the Letters |
|
|
370 | (1) |
|
Chapter 20 Ten Places to Start When You Don't Know Where to Start |
|
|
371 | (6) |
|
|
|
371 | (1) |
|
Sitting Up Straight and Breathing |
|
|
372 | (1) |
|
Making an Information Checklist |
|
|
372 | (1) |
|
Putting Information Together |
|
|
373 | (1) |
|
|
|
373 | (1) |
|
|
|
374 | (1) |
|
|
|
374 | (1) |
|
Starting with the Ugliest Thing |
|
|
375 | (1) |
|
|
|
375 | (1) |
|
Asking, 'What Would Colin Ask?' |
|
|
376 | (1) |
| Index |
|
377 | |
Lisainfo e-raamatute kohta