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General Calculus Questions : Definition of a Limit and Derivative, Product Rule, Tangent Line Approximation, Taylor Polynomial, Newton's Method, L'Hopital's Rule, MVT, IVT, Fundamental Theorem

1. Give the definition of limit in three forms: ε?δ , graphical, and in your own words.
2. Define the derivative. List what you consider to be the five most useful rules concerning derivatives.
3. Give an argument for the product rule.
4. What is the tangent line approximation to a function?
5. What is the Taylor polynomial approximation to a function? Explain how to use the tangent line approximation of the derivative of a function to obtain a quadratic approximation to the function.
6. What does Newton's method solve and how does it solve it? What is the underlying idea behind the method? Is it guaranteed to work?
7. What is l'Hopital's rule? Be careful to explain when you can use it. Give some examples where alternative methods are quicker to use.
8. If you want to find the maximum value of a function, how do you do it using calculus? Explain the idea behind the method.
9. What is the Intermediate Value Theorem? How do you use it?
10. What is the Mean Value Theorem? How do you use it?
11. What is the Fundamental Theorem of Calculus? How do you use it?

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1. Give the definition of limit in three forms: ; graphical, and in your own words.
Consider
Definition 1. " "
For every , there exists so that whenever , then

Definition 2. graphical
For every , there exists so that whenever , the graph of f(x) located in the strip . Shown in the following blue part.

Definition 3. In your own words;

Given a function y=f(x), when x approaches to a, then f(x) tends to A.

2. Definition of derivative

Let y = f(x) be a function. The derivative of f is the function whose value at x is the limit

provided this limit exists.

Five useful rules.
(1)
(2)
(3)
(4)
(5)

3. Give an argument for the product rule.
Product rule is
Proof. By definition, we have

since ,
4. What's the tangent line approximation to a function?

See above graph. If we want to compute the value of f(x) at x. Then we can use the value to approximate it. How to do that? We make a tangent line at point . Then we can compute the equation of this line which is

i.e.,
So, we can use to approximate the value of f(x), namely,

Note: Only when x is very close to , the approximation is good.
5. What is the Taylor polynomial approximation to a function? Explain how to use tangent line approximation of the derivative of a function to get a quadratic approximation to the ...

Solution Summary

The Definition of a Limit and Derivative, Product Rule, Tangent Line Approximation, Taylor Polynomial Approximation, Newton's Method, L'Hopital's Rule, Maximum Value of a Function, Intermediate Value Theorem, Mean Value Theorem and Fundamental Theorem of Calculus are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.

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