What are the fundamental concepts needed for advanced calculus?

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In summary, the conversation is about a first-year student struggling in an advanced calculus class and seeking help with various concepts, such as proving uniqueness in inequalities, using induction to prove inequalities, and understanding the concepts of inf/sup/max/min, identity functions, and right/left inverses. They also mention a specific question involving proving a theorem using induction and defining addition and multiplication on a set. They ask for someone to explain the question and suggest finding a study partner since it is an honors elementary calculus class.
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Asmodeus
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Hello,

I'm a first-year student and I am in advanced calculus (specifically, Analysis I). I, however, switched into the class during the third week. So I am having quite a bit of difficulty, because I haven't learned any fundamental concepts.

So I am asking if someone can please discuss a couple of concepts/questions with me:

-How to prove uniqueness (specifically with inequalities)

-I know about induction, but I'm not sure how to use it to prove inequalities

-inf/sup/max/min

-identity function and right/left inverse

And for this question, I do not understand it at all:

"1a) let k be any natural number. use induction to prove: theorem

for any integer n, which is an integer, there is a unique q, which is an integer, such that 0 <= n+qk <k
(the resulting number n + qk {0,...,k-1} is somtines denoted "n mod k")

b)
for any n that is an integer (I have to be explicit, because I don't know how to type those math symbols) let [n] C Integers, be the subset.

[n] = {n+qk, q is any integer}

note that
[n + pk] = [n] for any n,p that are integers. hence, by part a) there are actually just k distinct subsets, [0],...,[k-1]. Let Z(subscript k) be the set of such subsets,

z(subscript k) = {[0],...,[k-1]}.

define an addition and a multiplication on z(subscript k) by

[n] + [m] = [n+m], [n][m]=[nm]

show that z(subscript 3) satifies the axioms (p1) to (p9), but z (subscript 4) does not. (don't have to write out the proof that these formulas for + and * are well-defined)."

the axioms are from my spivak textbook. can someone just discuss what is going on in this question.

Thanks for any help.
 
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  • #2
advice, find a friend to work with. this is honors elementary calculus.
 
  • #3


The fundamental concepts needed for advanced calculus include the following:

1. Limits and Continuity: These concepts are essential for understanding the behavior of functions and their graphs. The concept of limits helps us determine the behavior of a function as its input approaches a specific value, while continuity ensures that there are no abrupt changes in the function's behavior.

2. Derivatives: Derivatives are used to calculate the rate of change of a function at a specific point. They are also used to find the slope of a tangent line to a curve at a given point and can be used to optimize functions.

3. Integrals: Integrals are used to calculate the area under a curve and can also be used to find the volume of a solid with curved boundaries. They are also used to solve differential equations and can be used to find the average value of a function.

4. Sequences and Series: These concepts deal with the behavior of infinite lists of numbers and are crucial in understanding calculus. They are used to approximate functions and can also be used to solve certain types of differential equations.

5. Multivariable Calculus: This includes the study of functions of multiple variables and their derivatives and integrals. It also involves topics such as partial derivatives, gradient, and vector calculus.

6. Convergence and Divergence: These concepts are important in determining the behavior of infinite sequences and series. They are also crucial in understanding the convergence of integrals and the convergence of functions.

7. Continuity and Differentiability in Multiple Dimensions: This involves understanding how functions behave in multiple dimensions and how to calculate partial derivatives and gradients.

8. The Fundamental Theorem of Calculus: This theorem connects the concepts of derivatives and integrals and is crucial in understanding the relationship between the two.

9. Mathematical Induction: This is a powerful tool used to prove mathematical statements and is often used in advanced calculus to prove theorems.

10. Set Theory: This is the study of collections of objects and their properties. It is important in advanced calculus to understand the concepts of subsets, unions, and intersections.

Some of the specific concepts you mentioned, such as uniqueness, inequalities, and inf/sup/max/min, are related to the above fundamental concepts. For example, proving uniqueness often involves using limits and derivatives, while inequalities are important in determining the behavior of functions. Inf/sup/max/min are used to find the extreme values of functions.

As for the question about induction and the sets [n], [
 

1. What is the definition of a conceptual question?

A conceptual question is a question that requires the use of critical thinking skills and understanding of abstract concepts rather than a straightforward answer. It often requires the person to think outside the box and make connections between different ideas.

2. How are conceptual questions different from factual questions?

Conceptual questions focus on understanding and analyzing abstract concepts, while factual questions focus on retrieving specific information or details. Conceptual questions require more critical thinking and application of knowledge, while factual questions can often be answered with a simple fact or piece of information.

3. How can conceptual questions be used in scientific research?

Conceptual questions can be used to guide scientific research by helping researchers think critically about the underlying principles and theories behind their work. They can also be used to generate new ideas and hypotheses for further investigation.

4. How can one improve their ability to answer conceptual questions?

One can improve their ability to answer conceptual questions by practicing critical thinking skills and actively seeking to understand abstract concepts. Additionally, engaging in discussions and debates with others can help to improve one's ability to think critically and answer conceptual questions.

5. Can conceptual questions have multiple answers?

Yes, conceptual questions can have multiple answers as they often require the person to think creatively and make connections between different ideas. This can lead to a variety of possible interpretations and solutions.

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