Derivatives

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MAT.126 5.5

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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