C885 Advanced Calculus
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Free C885 Advanced Calculus Questions
The Egorov theorem requires almost-everywhere convergence and finite measure. Why does the “finite measure” condition cannot be dropped even if the dominating function is integrable over the whole space?
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The theorem relies on the σ-finiteness of the exceptional set, which fails globally
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Almost-uniform convergence fails when the space contains subsets of positive measure at arbitrary distance
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Without finite measure, sets of arbitrarily small measure can have unbounded integral of the dominator
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Egorov’s proof uses the fact that the tails of an integrable function go to zero uniformly
Explanation
Explanation:
Egorov’s theorem gives a strong (almost-uniform) form of convergence by controlling the size (measure) of the exceptional set where convergence is not yet close. The standard proof repeatedly trims off small-measure sets and uses the fact that the whole space has finite measure so those trimmed sets’ measures can be made arbitrarily small in a globally meaningful way; on an infinite-measure space one can place the “bad” behavior on disjoint regions drifting off to infinity so that each piece has tiny measure yet the convergence is not uniform on any co-finite set. Having a dominator integrable over the whole space does not prevent constructing such pathological arrangements (the dominator can be large on tiny sets so integrability alone doesn’t force uniform control), so the finite-measure hypothesis is essential.
Correct Answer:
Without finite measure, sets of arbitrarily small measure can have unbounded integral of the dominator
Explain the relationship between the derivative of sin(x) and the graph of sin(x). Specifically, what does the derivative tell us about the original function?
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The derivative, cos(x), represents the concavity of the sin(x) function at any given point.
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The derivative, cos(x), represents the y-intercept of the sin(x) function.
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The derivative, cos(x), represents the area under the sin(x) function.
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The derivative, cos(x), represents the slope of the tangent line to the sin(x) function at any given point.
Explanation
Explanation:
The derivative of sin(x) is cos(x), which gives the instantaneous rate of change of sin(x) at any point. Graphically, this corresponds to the slope of the tangent line to the sin(x) curve at each x-value. Positive values of cos(x) indicate that sin(x) is increasing, negative values indicate that sin(x) is decreasing, and zero indicates that the slope of the tangent line is flat, corresponding to local maxima or minima of sin(x).
Correct Answer:
The derivative, cos(x), represents the slope of the tangent line to the sin(x) function at any given point.
A particle's velocity is given by v(t) = 3/(1+t2). What is the total displacement of the particle from t = 0 to t = 1?
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arctan(3)
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arctan(1/3)
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3arctan(0)
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3arctan(1)
Explanation
Explanation:
Total displacement is found by integrating the velocity function over the given time interval. Here, the integral of v(t) = 3/(1+t2) from 0 to 1 is ∫₀¹ 3/(1+t²) dt. The antiderivative of 1/(1+t²) is arctan(t), so the integral becomes 3·arctan(t) evaluated from 0 to 1. Substituting the limits gives 3·(arctan(1) - arctan(0)) = 3·arctan(1). Since arctan(1) = π/4, the displacement can also be expressed as 3·π/4 if desired.
Correct Answer:
3arctan(1)
In Lebesgue’s differentiation theorem, almost-everywhere differentiability holds for absolutely continuous functions. Why does mere Lipschitz continuity not suffice for everywhere differentiability?
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Lipschitz functions can have derivative failing to exist on a set of measure zero (e.g., distance to a fat Cantor set)
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Lipschitz continuity implies bounded variation but not absolute continuity
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Rademacher’s theorem guarantees differentiability almost everywhere, but not everywhere
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All of the above are correct reasons
Explanation
Explanation:
Lipschitz continuity ensures that a function is bounded in its rate of change, which guarantees differentiability almost everywhere by Rademacher’s theorem. However, it does not guarantee differentiability at every point. Lipschitz functions can have derivative failures on sets of measure zero—for example, functions like the distance to a fat Cantor set are Lipschitz but nondifferentiable on the Cantor set. Moreover, while Lipschitz continuity implies bounded variation, it does not imply absolute continuity, which is necessary for differentiability everywhere under Lebesgue’s differentiation theorem. Hence, all the listed reasons collectively explain why mere Lipschitz continuity is insufficient.
Correct Answer:
All of the above are correct reasons
When finding the area between two curves, f(x) and g(x), what mathematical operation is used to determine the x-values that serve as the integral's bounds when they are not explicitly provided?
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Set f(x) = g(x) and solve for x.
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Calculate the derivative of f(x) and g(x).
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Find the average of f(x) and g(x).
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Use the given bounds, if any.
Explanation
Explanation:
To determine the limits of integration when finding the area between two curves, we need the points where the curves intersect. These intersection points are the x-values at which the curves are equal. Therefore, we set f(x) = g(x) and solve for x. These x-values then serve as the bounds of the definite integral, ensuring that the area is computed over the correct interval.
Correct Answer:
Set f(x) = g(x) and solve for x.
Explain the relationship between the derivatives of sin(x) and cos(x).
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The derivative of sin(x) is cos(x), and the derivative of cos(x) is −sin(x).
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The derivative of sin(x) is sec(x), and the derivative of cos(x) is csc(x).
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The derivative of sin(x) is tan(x), and the derivative of cos(x) is −cot(x).
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The derivative of sin(x) is −cos(x), and the derivative of cos(x) is sin(x).
Explanation
Explanation:
In calculus, the derivatives of the sine and cosine functions are closely related. The derivative of sin(x) is cos(x), which represents the instantaneous rate of change of the sine function. Conversely, the derivative of cos(x) is −sin(x), reflecting that the cosine function decreases where the sine function increases and vice versa. This relationship illustrates the phase shift and periodic behavior of these two fundamental trigonometric functions.
Correct Answer:
The derivative of sin(x) is cos(x), and the derivative of cos(x) is −sin(x).
Explain why we integrate the absolute value of the velocity function, ∣v(t)|, rather than the velocity function v((t) itself, when calculating the total distance traveled over an interval [A,B].
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Integrating|∣v(t)∣ is only necessary when the velocity is constantly negative
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Integrating ∣v(t)∣ accounts for changes in direction, ensuring all movement contributes positively to the total distance.
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Integrating v(t) directly always yields a positive value, equivalent to total distance.
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Integrating |v(t)| simplifies the calculation by removing the need for limits of integration.
Explanation
Explanation:
The velocity function v(t) can be positive or negative depending on the direction of motion, so integrating it directly would compute displacement—net change in position—allowing forward and backward movement to cancel each other out. To measure the total distance traveled, all motion must be counted positively regardless of direction. Taking the absolute value of velocity, ∣v(t)∣, converts all movement to a positive rate, ensuring that the integral represents the full path traveled rather than net position change.
Correct Answer:
Integrating∣v(t)∣ accounts for changes in direction, ensuring all movement contributes positively to the total distance.
According to standard calculus formulas, what is the derivative with respect to x of the inverse tangent function, arctan(x)?
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-1/(1+x²)
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-1/(1−x²)
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1/(1+x²)
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1/(1−x²)
Explanation
Explanation:
The derivative of the inverse tangent function follows a well-known calculus rule:
d/dx[arctan(x)] = 1/(1+x2).
This comes from differentiating the inverse trigonometric functions, where each has a distinct algebraic form involving square roots or rational expressions. For arctan(x), the derivative expresses how the angle whose tangent is x changes with respect to x, and this always gives a positive value since the denominator is always positive.
Correct Answer:
1/(1+x²)
According to standard integral formulas, what constant factor is multiplied by the arctangent function in the indefinite integral of 
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1/a
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a
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ln∣a∣
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ea
Explanation
Explain in your own words why the product rule is necessary for finding the derivative of f(x)g(x) and not simply taking the product of the individual derivatives.
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The product rule is just a convention and doesn't have a mathematical basis.
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Multiplying derivatives gives the same result as the product rule, but the product rule is simpler.
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The product rule accounts for how the rate of change of each function affects the other when they are multiplied together; simply multiplying derivatives would ignore this interaction.
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The product rule is only necessary when one of the functions is a constant.
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The product rule is used because the chain rule cannot be applied.
Explanation
Explanation:
The product rule is necessary because when two functions are multiplied, both functions change with respect to x, and their combined rate of change depends on how each function influences the other. Simply multiplying the derivatives assumes each function changes independently, which is not true. The product rule correctly captures the full interaction by differentiating one function while holding the other constant, then switching roles. This ensures all changes in the product are accounted for.
Correct Answer:
The product rule accounts for how the rate of change of each function affects the other when they are multiplied together; simply multiplying derivatives would ignore this interaction.
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