Pages 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 8777 result(s) returned

1.1.1 Try some yourself

1 Evaluate the following:

  • (a) 62

  • (b) 0.52

  • (c) 1.52

Answer<
Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

Learning outcomes

By the end of this unit you should be able to:

  • evaluate the squares, cubes and other powers of positive and negative numbers with or without your calculator;

  • estimate square roots and calculate them using your calculator;

  • describe the power notation for expressing numbers;

  • use your calculator to find powers of numbers;

  • multiply and divide powers of the same number;

  • understand and apply negative powers, t
    Author(s): The Open University

    License information
    Related content

    Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

3.3.1 Try some yourself

1 Look at the diagram below and answer the following questions:

  • (a) Write down the coordinates of the points P, Q, R, S and T.

  • (b) On this diagram,
    Author(s): The Open University

    License information
    Related content

    Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

3.1.1 Try some yourself

1 Write down the coordinates of A and B.


Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

2.3.1 Try some yourself

1 This table categorises Tom's activities for the day.

ActivityTime/hours
Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

2.3 Pie charts

Pie charts are representations that make it easy to compare proportions: in particular, they allow quick identification of very large proportions and very small proportions. They are generally based on large sets of data.

The pie chart below summarises the average weekly expenditure by a sample of families on food and drink. The whole circle represents 100% of the expenditure. The circle is then divided into ‘segments’, and the area of each segment represents a fraction or pe
Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

2.2.1 Try some yourself

1 Consider the table about household sizes.


    Author(s): The Open University

    License information
    Related content

    Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

2.2 Tables and percentages

Tables often give information in percentages. The table below indicates how the size of households in Great Britain changed over a period of nearly 30 years.

Number of people in household1961 (%)1971 (%)1981 (%)1991 (%)
1<
Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

2.1.1 Try some yourself

1 The table below indicates the cooling rate of tea in a teapot.

Time/mins
Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

2.1 Tables

Experiments or surveys usually generate a lot of information from which it is possible to draw conclusions. Such information is called data. Data are often presented in newspapers or books.

One convenient way to present data is in a table. For instance, the nutrition panel on the back of a food packet:

Nutrition Information

Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

1.1.1 Try some yourself

1 On the plan of the bathroom in Example 1, what is the width of the window and
Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

1.1 Understanding scale diagrams

Plans of houses and instructions for assembling shelves, etc., often come in the form of scale diagrams. Each length on the diagram represents a length relating to the real house, the real shelves, etc. Often a scale is given on the diagram so that you can see which length on the diagram represents a standard length, such as a metre, on the real object. This length always represents the same standard length, wherever it is on the diagram and in whatever direction.


Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

1 Modelling with first order differential equations

The main teaching text of this unit is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook. When prompted after exercise 2.2 to watch the video for this unit, return to this page and watch the four clips below. After you've watched the clips, return to the workbook.

Click 'View document' to open the workbook (PDF, 1.0 MB).


Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

Learning outcomes

After studying this unit you should be able to:

  • understand and use the basic terms for the description of the motion of particles: position, velocity and acceleration;

  • understand, use and differentiate vector functions;

  • understand the fundamental laws of Newtonian mechanics;

  • solve mechanics problems in one dimension by drawing a sketch, choosing a suitable x-axis and origin, drawing a force diagram, applying Newton’s second law, tak
    Author(s): The Open University

    License information
    Related content

    Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

Introduction

This unit extends the ideas introduced in the unit on first-order differential equations to a particular type of second-order differential equation which has a variety of applications. The unit assumes that you have previously had a basic grounding in calculus, know something about first-order differential equations and have some familiarity with complex numbers.

This unit is an adapted extract from the course Author(s): The Open University

1 Modelling with Fourier series

The main teaching text of this unit is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook.

Click 'View document' to open the workbook (PDF, 0.6 MB).

1.5 Exercises

Exercise 1

A vector a has magnitude |a| = 7 and direction θ = −70°. Calculate the component form of a, giving the components correct to two decimal places.

<
Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

1.4.2 Displacements and bearings

The displacement from a point P to a point Q is the change of position between the two points, as described by the displacement vector

If P and Q represent places on the ground, then it is natural to use a bearing to describe the direct
Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

1.1: Converting to component form

In some applications of vectors there is a need to move backwards and forwards between geometric form and component form; we deal here with how to achieve this.

To start with, we recall definitions of cosine and sine. If P is a point on the unit circle, and the line segment OP makes an angle θ measured anticlockwise from the positive x-axis, then cos θ is the x-coordinate of P and sin θ is the y-coordinate of P (
Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

Bibliography

Ahmed, A. (1987) Better Mathematics, London, HMSO.

DfEE (2001) Key Stage 3 National Strategy: Framework for Teaching Mathematics: Years 7, 8 and 9, London, DfEE.

NCTM (1989) Curriculum and Evaluation Standards for School Mathematics Reston VA, National Council of Teachers of Mathematics.


Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

Pages 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439