Higher-order derivatives and power series

1. Higher-order derivatives and power series ACME General Store Sales receipt

2. Higher-order derivatives and power series 2nd derivative and curvature Power series 1 1/2 Power series for sine and cosine Approximation of p

3. 2nd derivative and curvature upward slant horizontal downward slant positive slope zero slope increasing slope negative slope positive second derivative 0 0 0

4. Higher-order derivatives and power series 2nd derivative and curvature Power series 1 1/2 Power series for sine and cosine Approximation of p

5. Constructing power series representations 0

6. Constructing power series representations Multiply coefficient and powers of binomial Subtract 0 Add terms

7. Construction of accurate power series representation can fail Too jagged 0 0 0 Too straight

8. Higher-order derivatives and power series 2nd derivative and curvature Power series 1 1/2 Power series for sine and cosine Approximation of p

9. Qualitative expectations for power series for sine 1 Expand around qE= 0 0 -1

10. Power series for sine Expand around qE= 0 1

11. Power series for sine Expand around qE= 0

12. Power series for sine is as expected 1 0 -1

13. Higher-order derivatives and power series 2nd derivative and curvature Power series 1 1/2 Power series for sine and cosine Approximation of p

14. Using a power series and iteration to approximate p 1 1/2

15. Higher-order derivatives and power series 2nd derivative and curvature Power series ACME General Store Sales receipt 1 1/2 Power series for sine and cosine Approximation of p