Transcript Review on Math110

Classification of Functions
We may classify functions by their formula as follows:
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Polynomials
Linear Functions, Quadratic Functions. Cubic Functions.
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Piecewise Defined Functions
Absolute Value Functions, Step Functions
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Rational Functions
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Algebraic Functions
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Trigonometric and Inverse trigonometric Functions
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Exponential Functions
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Logarithmic Functions
Function’s Properties
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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:
The Trigonometric Functions
Correspondence between degree and radian
Some values of
sin  and cos
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
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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
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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?
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In general, suppose an object moves along a straight line
according to an equation of motion s  f (t ), where s is the
displacement (directed distance) of the object from the origin at
time t . The function f that describes the motion is called the
position function of the object. In the time interval from t  a to
t  a  h the change in position is f (a  h)  f (a) . The
average velocity over this time interval is
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Now suppose we compute the average velocities over shorter
and shorter time intervals [a, a  h] . In other words, we let h
approach 0. We define the velocity or instantaneous velocity
at time t  a to be the limit of these average velocities:
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This means that the velocity at time t  a is equal to the slope of
the tangent line at P .
The Derivative of a Function
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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