M119 2019 Fall Midterm

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What is the Fundamental Theorem of Calculus?

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The Fundamental Theorem of Calculus links the concept of differentiation and integration, stating that if a function is continuous on [a, b], then the integral of its derivative over that interval equals the difference in the values of the function at the endpoints: ∫_a^b f'(x) dx = f(b) - f(a).

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Define a limit in calculus.

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A limit is a value that a function approaches as the input approaches some value. It is fundamental in defining continuity, derivatives, and integrals.

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What is the derivative of a function?

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The derivative of a function at a point is the slope of the tangent line to the graph of the function at that point, representing the rate of change of the function with respect to its variable.

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Explain the concept of continuity in a function.

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A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. A function is continuous on an interval if it is continuous at every point in that interval.

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What is a critical point in calculus?

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A critical point of a function occurs where its derivative is zero or undefined. Critical points are important for finding local maxima and minima.

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Define an asymptote.

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An asymptote is a line that a graph approaches but never touches. There are vertical, horizontal, and oblique asymptotes.

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What is the difference between a definite and an indefinite integral?

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A definite integral computes the area under a curve between two specific points and results in a number, while an indefinite integral represents a family of functions and includes a constant of integration (C).

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What is the Chain Rule in differentiation?

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The Chain Rule is a formula for computing the derivative of the composition of two or more functions. If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

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Explain the concept of optimization in calculus.

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Optimization involves finding the maximum or minimum values of a function within a given domain, often using derivatives to identify critical points.

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What is the Mean Value Theorem?

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The Mean Value Theorem states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).