The number e, growth and decay
See why (١ + ١n)n settles at e, lay the graph of y = a×ex on a given curve, and build growth and decay models.
- سوريا: الصف ١٢
- إيران: الصف ١١
- أفغانستان: الصف ١١
- أوكرانيا: الصف ١١
- روسيا: الصف ١١
- ألمانيا: الصف ١٠
- فرنسا: الصف ١١
- بريطانيا: السنة ١٢
لم يُترجَم هذا المختبر إلى هذه اللغة بعد.
الدليل
Suppose £1 earns 100% interest a year. Paid once, it becomes £2. If the year is split into n parts with interest each time, the end-of-year amount is (١ + ١n)n. As n grows this number rises, but it never goes past about ٢٫٧١٨. We call this number e.
The graph of y = ex is the التابع الأسي with base e. In y = a×ex, a is the height of the curve on the y-axis, because e٠ = ١.
In the model P = A×bt, b > 1 means growth and 0 < b < 1 means decay. A fall of 20% a year multiplies by ٠٫٨ each year, not by ٠٫٢.
Common slips: rounding e to 3; thinking the number goes to infinity as n grows; adding instead of multiplying in an exponential model.