a Steve Kangas 18 éve
1650
Derivatives
Derivatives
What does f' say about f?
Antiderivatives
f'' > 0 means f is concave up. f'' < 0 means f is concave down.
f' > 0 means f is increasing. f'< 0 means f is decreasing.
The derivative as function
Higher derivatives (derivatives of derivatives)
Differentiable functions are continuous
The other notation: dy/dx
Derivatives
Finding the derivative using the definition
The derivative is the instantaneous rate of change
The derivative is the slope of the tangent line
Definition
Tangents, velocities, rates of change
Tangents, instantaneous velocity, and instantaneous rates of change are all the same problem
Infinite limits
The trick of dividing top and bottom by the highest power that appears in the denominator
Horizontal asymtotes
Limits as x approaches infinity
Vertical asymptotes
Limits where f(x) goes to infinity and minus infinity
Continuity
Intermediate value theorem
Compositions of continuous functions are continuous
All elementary functions are continuous on their domains
Polynomials are continuous everywhere
Sums, differences, products, and quotients of continuous functions are continuous
Continuous from the right and the left
Definition: as we approach a, the limit of f(x) is f(a)
Limit Laws
Squeeze theorem
Add, subtract, multiply, divide: the limit laws are what you expect
Limits
One-sided limits
Using tables to guess limits. This is a risky way to calculate limits.
Definition: We can make f(x) as close to L as we like by making x close to a
Tangent & Velocity
The tangent problem is the same as finding the instantaneous velocity
