# Calculus Proofs Help: Get Answers Now!

• Swamifez
Therefore, there is no x in [-1,1] satisfying f(x) = x.In summary, we discussed proving the boundedness and supremum of S = { n−1 |n ∈ N}, using the Intermediate Value Theorem to show a polynomial has real roots, and proving differentiability of a squared function without quoting a theorem. We also analyzed two statements and provided counterexamples.
Swamifez
Hey, I need help with a couple of questions in my analysis calc and proof class. Thanks in advance!

Prove that S = { n−1 |n ∈ N} is bounded above and that its supremumn
is equal to 1.

Use the Intermediate Value Theorem to show that the polynomial x4 +
x3 − 9 has at least two real roots.

Suppose that the function f is diﬀerentiable at a. Prove (without
quoting a theorem) that f 2 is diﬀerentiable at a.

Suppose that the function f is diﬀerentiable at a. Prove (without
quoting a theorem) that f 2 is diﬀerentiable at a.

Decide whether the following statements are true or false. Justify your
(a) Every continuous function f : [0, 1) → R which is bounded
takes on its maximum
(b) There exists a function f : [−1, 1] → [−1, 1] with no x ∈ [−1, 1]
satisfying f (x) = x.

If anyone knows anything on any of these problems, its highly appreciated!

a) False. Counterexample: Let f(x) = -1 for all x in [0,1). This function is continuous and bounded, but it does not take on its maximum. b) False. Counterexample: Let f(x) = x^2. For all x in [-1,1], f(x) ≠ x.

## 1. What is a calculus proof?

A calculus proof is a mathematical argument that uses logic and reasoning to demonstrate the truth of a statement or theorem in calculus. It involves using axioms, definitions, and previously proven theorems to arrive at a conclusion.

## 2. Why are calculus proofs important?

Calculus proofs are important because they provide a rigorous and logical foundation for the concepts and techniques taught in calculus. They help to solidify understanding and build problem-solving skills that are essential for further studies in mathematics and other fields.

## 3. How do I approach a calculus proof?

When approaching a calculus proof, it is important to carefully read and understand the statement or theorem you are trying to prove. Then, review any relevant definitions and theorems that may be helpful in the proof. Next, use logical reasoning and mathematical techniques to connect these concepts and arrive at a conclusion. Finally, review and refine your proof to ensure its clarity and accuracy.

## 4. What are some common techniques used in calculus proofs?

Some common techniques used in calculus proofs include direct proof, proof by contradiction, proof by induction, and proof by contrapositive. These techniques involve using logical arguments, algebraic manipulation, and previously proven theorems to arrive at a conclusion.

## 5. Where can I find help with calculus proofs?

There are various resources available for help with calculus proofs, such as textbooks, online tutorials, and study groups. Your instructor or a math tutor can also provide guidance and assistance. Additionally, there are websites and forums where you can ask specific questions and get answers from experts in the field.

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