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Utilizzare webquest per presentare tasks in modo più accattivante?

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A resource page for Educational Technology

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Video sui limiti

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Limits: To Infinity and Beyond!: Teacher Process

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This is a long term WebQuest. It will last the entire chapter on Limits. Before the first day of the chapter, assign the students to complete "Phase 1 - Background Information" in the WebQuest. This could be the HW assignment the night before this chapter is to start.

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Characteristics of Graphs of Rational Expressions

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This video is about Math Discontinuities Video

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Chapter 1 - Limits and Continuity Chapter 2 - Introduction to the Derivative Chapter 3 - Derivative Rules

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Areas to be covered
Functions
Inverse Functions
Limits
Limits of Trigonometric Functions
Asymptotes
Continuity
Discontinuity
Derivatives
Functions
Definition:
A function is an equation if for any x in the domain of the equation (the domain is the entire x's that can be plugged into the equation) the equation will yield exactly one value of y.
Ways to describe functions
A function is given a name (such as , , and ) and a formula for the function is also given. For example, describes a function. We refer to the input as the argument of the function, and to the output as the value of the function at the given argument.

A function is described using an equation and two variables. One variable is for the input of the function and one is for the output of the function. The variable for the input is called the independent variable. The variable for the output is called the dependent variable. For example, describes a function. The dependent variable appears by itself on the left hand side of equal sign.

Domain & Range
The domain of a function is all points that represent values of x which the function can accept as input. The range of a function is the set of all values which it attains (i.e. the y-values).
Types of Functions
Addition, Subtraction, Multiplication and Division of Functions
Composite Functions
Inverse Functions
Limits
A limit is a value that a function approaches as the variable within that function approaches a particular value. Limits are the core tool that we build upon for calculus. Many times, a function can be undefined at a point, but we can think about what the function "approaches" as it gets closer and closer to that point (this is the "limit"). Other times, the function may be defined at a point, but it may approach a different limit. There are many, many times where the function value is the same as the limit at a point.
How can a limit fail to exist?
1. The function attempts to approach multiple values as x approaches c
2. LH Limit is not equal to the RH Limit
Limit approaching infinity
Rational Function- Functions that can be written as proper functions

1. Divide each term by the variable with the highest power if the variable tends to infinity and if x tends to negative infinity we divide each term by the variable with the lowest power
2. If the function is a proper fraction (the highest power of the variable is in the denominator) then as x approaches infinity the function tends to 0 (Zero)
3. If the function is an improper fraction when x approaches infinity the function tends to infinity
4. If the function has the same variable then the coefficient

An asymptote is a line that a graph approaches without touching.
If a graph has a horizontal asymptote of
y = k , then part of the graph approaches the line y = k without touching it. y is almost equal to k , but y is never exactly equal to k . The following graph has a horizontal asymptote of y = 3

Asymptote
If a graph has a vertical asymptote of x = h , then part of the graph approaches the line x = h without touching it-- x is almost equal to h , but x is never exactly equal to h . The following graph has a vertical asymptote of x = 3 :
Asymptotes
One reason vertical asymptotes occur is due to a zero in the denominator of a rational function. For example, if f (x) = , then x cannot equal 5, but x can equal values very close to 5 (4.99, for example).
Similarly, horizontal asymptotes occur because y can come close to a value, but can never equal that value. In the previous graph, there is no value of x for which y= 0 ( ≠ 0 ), but as x gets very large or very small, y comes close to 0. Thus, f (x) = 1/(x-5) has a horizontal asymptote at y = 0 .

NB- A graph of a function may have several asymptotes
Limits of Trigonometric Functions
Continuity
A function is said to be continuous within a given interval if the graph of the function does not have any breaks or holes in it within that interval.
Derivatives
The derivative is the a key tool in calculus. The derivative is the slope of the tangent line to the graph of the function at that point. When given a function f, we can define a derivative function f' to take on the value of the derivative of f at each point in the domain.
Tangent
At the point at which it is tangent to the curve, the tangent line touches the curve, but does not "cross" it.

A tangent line has the same slope as the point of the graph it touches. The slope of the tangent line matches the slope of the curve at the point of tangency.

Slope of a curve at a point
The "slope" is the change in y divided by change in x. To calculate the slope of a line, we pick any two points on that line and divide the difference in their y-values by the difference in x values.

Any two points can be used in determining the slope of a line, since the slope is constant throughout.

Difference Quotient
This is called the difference quotient. It does not represent the slope of the graph at (x, f(x), but, it represents the slope of the secant line that goes through the points (x, f(x)) and (x+h, f(x+h))
Differentiability
A function f is said to be differentiable at x = a if f'(a) exists. In other words, a function is differentiable at x = a if the limit as h approaches zero of [f(a+h)-f(a)]/h exists. For a function to be differentiable, it needs to be both continuous and "smooth". What is meant by "smooth" is that there are no sharp turns in the graph. Tangent lines can only be drawn to graphs in places where they are both continuous and smooth, as shown below:

Note: Any differentiable function must also be continuous, since it is impossible to have a defined slope at a point of discontinuity, but, not all continuous functions are differentiable.
The Derivative Function
Since the derivative of a function f at any value x 0 where the function is differentiable is just a number, f'(x 0) , we may define the derivative function, denoted f'(x) or (x) , to be the function that maps a value x where f is differentiable to the derivative of f at x . Since f' is a function in its own right, we can graph it, just like any other function.
The Second Derivative
Given a function f , we have defined its derivative f' . This allows us to now define f'' , the second derivative of f , to be the derivative of f' . In terms of the graph of f , the second derivative tells how quickly the slope of the graph is changing near a particular point. Alternately, the second derivative tells how quickly the rate of change of a function is changing.
Since f'' is again a function, we can define f''' = f (3) to be the derivative of f'' . Continuing in this way, we can define the nth derivative of f , f(n) . The derivative of a differentiable function need not be differentiable everywhere (or even anywhere), so in general the domains of the successive derivatives f ( n) shrink as n increases.

Derivatives of Polynomial Functions
Important Note:

The derivative of a sum of two functions is equal to the sum of the derivatives of the two functions.
Derivative of Power Functions
Derivatives of power functions have the formula:

F'(x)=Anx^(n-1)
Basic Properties of Derivatives
There are two simple properties of differentiation that make the calculation of derivatives much easier. Let f(x) , g(x) be two functions, and let c be a constant. Then:
(dy/dx)[cf (x)] = cf'(x)
(f + g)'(x) = f'(x) + g'(x)

In words, these properties say that the derivative of a constant times a function is that constant times the derivative of the function, and the derivative of a sum of functions is the sum of the derivatives of the functions.

Implicit Differentiation
Definition: the process of finding the derivative of a dependent variable in an implicit function by differentiating each term separately, by expressing the derivative of the dependent variable as a symbol, and by solving the resulting expression for the symbol.
Derivative of Inverse Functions
Criteria for a function to be continuous
If a function is continuous at a point F(c) then:
1. f(c) must exist
2. The limit of f(x) as x approaches c must exist
3. f(c) must be equal to the limit of f(x) as x approaches c.
Discontinuity
A discontinuous function refers to one which has intervals or gaps in it. If a function is discontinuous, one will be unable to trace the function without lifting their pencil.
Types of Discontinuity
Jump Discontinuity- occurs at a point if the left hand limit is not equal to the right hand limit at that point
Removable Discontinuity- a discontinuity which you can remove by redefining the function at the point of discontinuity
Non-Removable Discontinuity- A discontinuity is called non-removable if f cannot be made continuous at x=c by defining of redefining the function at x=c
Approaches to solving limits
1. Numerical Approach- Construct a table of values
2. Graphical Approach- Draw graphs by hand or using technology
3. Analytical Approach- Use algebra or calculus
Common Types of Behaviour Associated with non existence of a limit:
1. f(x) approaches a different number from the right side of c than it approaches from the left side
2. f(x) increases or decreases without bound as x approaches c
3. f(x) oscillates between two fixed values as x approaches c
The Chain Rule
Suppose a function h is a composition of two other functions, that is, h(x) = f(g(x)) . We would like to express the derivative of h in terms of the derivatives of f and g . To do so, follow the chain rule, given below:
h'(x) = f'(g(x))g'(x)

The Chain Rule Formula
Formula:

n[f(x)]^(n-1) * f'(x)
Derivative of Trigonometric Functions

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Derivative of Power Functions Derivatives

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AP Calculus Prezi- Functions, Limits, Continuity and Derivatives

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Areas to be coveredFunctionsInverse FunctionsLimitsLimits of Trigonometric FunctionsAsymptotesContinuityDiscontinuityDerivativesFunctionsDefinition: A function is an equation if for any x in the domain of the equation (the domain is the entire x's that can be plugged into the equation) the equation will yield exactly one value of y.Ways to describe functionsA function is given a name (such as , , and ) and a formula for the function is also given.