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

Figure: 01-18 -1

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 Correspondence between degree and radian

Trigonometric Identities Some values of and Trigonometric Identities

Graphs of the Trigonometric Functions

Inverse 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.

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

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 .

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