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Calculus Fundamentals and Theorems

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  • What are the main numerical sets introduced in calculus?

    Natural numbers, integer numbers, and rational numbers are the main numerical sets introduced.

  • What is the axiom of separation in real numbers?

    Axiom of separation states that for any two distinct real numbers, there exists a number between them.

  • Define an open set in the topology of real numbers.

    An open set is a set where every point has a neighborhood fully contained within the set.

  • What is Bolzano’s theorem?

    Bolzano’s theorem states that a continuous function on a closed interval that changes sign has at least one root in that interval.

  • State the Intermediate Value Theorem.

    If a function is continuous on [a, b], it takes every value between f(a) and f(b) at some point in (a, b).

  • What is the Weierstrass theorem about maxima and minima?

    Weierstrass theorem states that a continuous function on a compact set attains its maximum and minimum values.

  • How is the derivative of a function defined?

    The derivative is the limit of the incremental ratio as the increment approaches zero.

  • List the basic rules of differentiation.

    Rules include sum, difference, product, quotient, reciprocal, composite functions, and inverse functions.

  • What does Fermat’s theorem state about derivatives?

    If a function has a local extremum at a point and is differentiable there, its derivative at that point is zero.

  • Explain Rolle’s theorem.

    If a function is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists c in (a, b) with f'(c) = 0.

  • What is Lagrange’s Mean Value Theorem?

    There exists c in (a, b) such that f'(c) equals the average rate of change (f(b)-f(a))/(b-a).

  • What is the purpose of de l'Hopital's rule?

    It helps evaluate limits of indeterminate forms like 0/0 or ∞/∞ by differentiating numerator and denominator.

  • State the fundamental theorem of integral calculus.

    It links differentiation and integration, stating that the integral of a function's derivative over an interval equals the function's net change.

  • What is a Riemann integrable function?

    A function is Riemann integrable if its integral can be approximated by sums of function values times subinterval lengths.

  • Describe integration by substitution.

    It changes variables in an integral to simplify the integrand and make integration easier.

  • What is integration by parts?

    It is a technique based on the product rule for derivatives to integrate products of functions.

  • Explain the Sandwich theorem for limits.

    If a function is squeezed between two functions with the same limit at a point, it shares that limit.

  • What are the properties of limits?

    Limits are unique, preserve sign, and can be combined using arithmetic operations.

  • What is the significance of compactness in continuous functions?

    Compactness ensures continuous functions attain maxima and minima and are bounded.

  • What is Taylor's formula used for?

    Taylor's formula approximates functions near a point using polynomials derived from derivatives at that point.