Classification of Functions We may classify functions by their formula as follows: Polynomials...

Classification of Functions

We may classify functions by their formula as follows:• Polynomials Linear Functions, Quadratic Functions. Cubic

Functions.• Piecewise Defined Functions Absolute Value Functions, Step Functions• Rational Functions• Algebraic Functions• Trigonometric and Inverse trigonometric

Functions• Exponential Functions• Logarithmic Functions

Function’s Properties We may classify functions by some of their

properties as follows:• Injective (One to One) Functions• Surjective (Onto) Functions • Odd or Even Functions• Periodic Functions• Increasing and Decreasing

Functions• Continuous Functions• Differentiable Functions

Power Functions

Combinations of Functions

Composition of Functions

Inverse Functions

Exponential Functions

Logarithmic Functions

The logarithm with base e is called the natural logarithm and has a special notation:

Correspondence between degree and radian

The Trigonometric Functions

Some values of and

sin cos

Trigonometric Identities

Graphs of the Trigonometric Functions

When we try to find the inverse trigonometric functions, we have a slight difficulty. Because the trigonometric functions are not one-to-one, they don’t have inverse functions. The difficulty is overcome by restricting the domains of these functions so that hey become one-to-one.

Inverse Trigonometric Functions

The Limit of a Function

Calculating Limits Using the Limit Laws

Infinite Limits; Vertical Asymptotes

Limits at Infinity; Horizontal Asymptotes

Tangents• The word tangent is derived from the Latin word tangens, which

means “touching.”• Thus, a tangent to a curve is a line that touches the curve. In

other words, a tangent line should have the same direction as the curve at the point of contact. How can his idea be made precise?

• For a circle we could simply follow Euclid and say that a tangent is a line that intersects the circle once and only once. For more complicated curves this definition is inadequate.

Instantaneous Velocity; Average Velocity• If you watch the speedometer of a car as you travel in city

traffic, you see that the needle doesn’t stay still for very long; that is, the velocity of the car is not constant. We assume from watching the speedometer that the car has a definite velocity at each moment, but how is the “instantaneous” velocity defined?

• In general, suppose an object moves along a straight line according to an equation of motion , where is the displacement (directed distance) of the object from the origin at time . The function that describes the motion is called the position function of the object. In the time interval from to

the change in position is . The average velocity over this time interval is

)(tfs s

t f

at hat )()( afhaf

• Now suppose we compute the average velocities over shorter and shorter time intervals . In other words, we let approach . We define the velocity or instantaneous velocity at time to be the limit of these average velocities:

• This means that the velocity at time is equal to the slope of the tangent line at .

],[ haa h0

at

at P

The Derivative of a Function

1

Differentiable Functions

The Derivative as a Function

What Does the First Derivative Function Say about the Original Function?

What Does the Second Derivative Function Say about the Original Function?

Indeterminate Forms and L’Hospital’s Rule

Antiderivatives