1. Math 19a: Modeling and Differential Equations for the Life Sciences Calculus Review Danny Kramer Fall 2013

2. Derivatives

3. Point Slope Concept x y

4. Solve it Out 0

5. Derivative Rules

6. Derivatives and Operations

7. Applications Position Speed/Velocity Acceleration

8. Maxima and Minima x f(x) f’(x)=0

9. Some Vocabulary Continuous- no holes or jumps in the graph Differentiable- continuous graph with a derivative at each point…no “cusps” ✓ X X ✓ X X

10. Sample Problem

11. Integrals and Antiderivatives

12. Area Concept It is area under a curve, but think of it more generally as multiplying a changing rate by the elapsed time over which the rate occurs, giving you the change in quantity that the rate is measuring. x y

13. Some Notation

14. Antiderivative Rules and Operations

15. U substitution Replace to visualize

16. Integration by Parts Opposite of product rule. Test it out! What Becomes u? Log Inverse Trig (the arcs) Algebra Trig Exponential

17. Taylor Series

18. Approximating Polynomial Curves x f(x) x = a f(a)

19. Taylor’s Formula

20. Practicing Taylor

22. Parametric Curves

23. Dimensions of Measurement x(t), y(t)  x(y) / y(x) ? Match up x and y at any given time t. t x, y x y 5 10 x y 5 5 t0t0 tftf t0t0 tftf

24. Parametric Conversion