M119 2018 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 the interval [a, b], then the integral of its derivative over that interval is equal to 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 fundamental concept in calculus that describes the behavior of a function as its argument approaches a particular point. It is expressed as: lim (x -> c) f(x) = L, meaning that as x approaches c, f(x) approaches L.

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What is the derivative of a function?

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The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero: f'(x) = lim (h -> 0) [f(x+h) - f(x)] / h.

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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. Formally, f is continuous at x = c if: 1) f(c) is defined, 2) lim (x -> c) f(x) exists, and 3) lim (x -> c) f(x) = f(c).

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What is an asymptote?

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An asymptote is a line that a graph approaches but never touches. There are three types of asymptotes: vertical (x = a), horizontal (y = b), and oblique (slant) asymptotes.

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Define a critical point in calculus.

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

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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 numerical value, while an indefinite integral represents a family of functions and includes a constant of integration (C), representing the antiderivative of a function.

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Explain the concept of a function's concavity.

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Concavity refers to the direction in which a function curves. A function is concave up on an interval if its second derivative is positive (f''(x) > 0) and concave down if its second derivative is negative (f''(x) < 0).

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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 the 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).

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Define a Riemann sum.

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A Riemann sum is a method for approximating the total area under a curve by dividing the region into rectangles, calculating the area of each rectangle, and summing these areas. It is used to define the definite integral.