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

Derivatives

Point Slope Concept x y

Solve it Out 0

Derivative Rules

Derivatives and Operations

Applications Position Speed/Velocity Acceleration

Maxima and Minima x f(x) f’(x)=0

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

Sample Problem

Integrals and Antiderivatives

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

Some Notation

Antiderivative Rules and Operations

U substitution Replace to visualize

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

Taylor Series

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

Taylor’s Formula

Practicing Taylor

Parametric Curves

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

Parametric Conversion