The lesson on Exponetnical function of y=e^x

Lesson on the number e

1. The Exponential Function y = e^x Understanding properties, graphs, and applications

2. Warm-Up: Indices Recap Simplify: 1) 2^3 × 2^4 2) (5^2)^3 3) 10^0

3. The Number e • e ≈ 2.718 • Appears in compound interest • Called the 'natural base'

4. The Function y = e^x Properties: • Passes through (0,1) • Always positive • As x→∞, y→∞ • As x→-∞, y→0

5. Graph of y = e^x Key Points: (-2,0.14), (-1,0.37), (0,1), (1,2.72), (2,7.39)

6. Examples 1) e^2 ≈ 7.389 2) e^-1 = 1/e ≈ 0.368 3) Solve e^x = 10 → x ≈ 2.303

7. Practice Questions 1) Work out: e^0, e^1, e^3 (2dp) 2) True/False: a) e^x can be negative b) e^x passes (0,1) c) As x→-∞, e^x→-∞

8. Sketch Practice Complete table: x: -2, -1, 0, 1, 2 y: ? Then sketch y = e^x

9. Applied Questions 1) Solve e^x = 50 2) A population grows as P=100e^(0.05t): a) Find P after 10 years b) Is growth linear or exponential?

10. Challenge Questions 1) Which is bigger: e^3 or 20? 2) Show e^-2 = 1/e^2 3) If y=e^x, what is y when x=-3? MCQs: • As x→∞, e^x→? (a)0 (b)1 (c)∞ • e^0 = ? (a)0 (b)1 (c)e

11. Summary • y=e^x is exponential growth • Always positive • Passes (0,1) • Grows faster as x increases