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