C646 Trigonometry and Precalculus
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Free C646 Trigonometry and Precalculus Questions
According to the power rule for integration, what is the result of ∫un du, assuming n ≠ −1?
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un+1 + C
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un−1/(n−1) + C
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nun−1 + C
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un+1/(n+1) + C
Explanation
Using the integral formula ∫ du/(u√(u²-a²)) = 1/a arcsec |u|/a + C, evaluate the definite integral ∫ (from 2 to 4) dx/(x√(x²-9)).
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1/3 (arcsin(4/3) - arcsin(2/3))
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1/3 arcsec(2/3)
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arcsec(4/3) - arcsec(2/3)
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1/3 (arcsec(4/3) - arcsec(2/3))
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1/3 arcsec(4/3)
Explanation
Explain in your own words why the formula 
gives the speed of a particle moving in the plane.
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The formula calculates the acceleration of the particle.
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The formula calculates the average velocity of the particle.
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The formula calculates the magnitude of the velocity vector, which represents the instantaneous rate of change of position with respect to time.
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The formula calculates the total distance traveled by the particle.
Explanation
Explain the significance of the natural logarithm, ln(a), in the derivative of ax.
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ln(a) represents the instantaneous rate of change of ax with respect to x, scaled by the value of the function itself.
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ln(a) is a constant that shifts the graph of ax vertically.
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ln(a) is the x-intercept of the function ax.
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ln(a) determines the concavity of the exponential function ax.
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ln(a) is only relevant when a is Euler's number, e
Explanation
Given the polar equations x = rcos(θ) and y = rsin(θ), and r = θ, determine the expression for dy/dx in terms of θ.
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1/θ
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(cos(θ) - θsin(θ))/(sin(θ) + θcos(θ))
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tan(θ)
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(sin(θ) + θcos(θ))/(cos(θ) - θsin(θ))
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θ
Explanation
To find dy/dx in polar coordinates, we use the formula dy/dx = (dy/dθ) / (dx/dθ). Substituting r = θ into x = θcos(θ) and y = θsin(θ), we differentiate: dx/dθ = cos(θ) - θsin(θ) and dy/dθ = sin(θ) + θcos(θ). Dividing dy/dθ by dx/dθ gives dy/dx = (sin(θ) + θcos(θ)) / (cos(θ) - θsin(θ)), which expresses the slope of the tangent line in terms of θ.
According to the limit rules for rational functions, what is the limit as x approaches infinity if the degree of the numerator is greater than the degree of the denominator?
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1
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0
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Infinite or Does Not Exist (DNE)
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Cannot be determined
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The ratio of the leading coefficients
Explanation
For a rational function where the numerator has a higher degree than the denominator, the numerator grows much faster than the denominator as x approaches infinity. This causes the value of the function to increase without bound. Therefore, the limit does not approach a finite number and is considered infinite or does not exist (DNE).
A solid has a circular base of radius 2. Its cross-sections perpendicular to the x-axis are squares. Which integral represents the volume of this solid?
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-
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Explanation
Given 
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cos(x)
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e(sin(x)) * cos(x)
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e(sin(x)) * sin(x)
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e(cos(x))
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e(sin(x))
Explanation
If sin(alpha) = 12/13, and cos(alpha) = 5/13, then tan(alpha) = ?
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7/13
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12/5
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5/12
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13/12
Explanation
To find tan(alpha), we use the identity tan(alpha) = sin(alpha) / cos(alpha). Substituting the given values sin(alpha) = 12/13 and cos(alpha) = 5/13, the fractions divide cleanly because the denominators are the same. Therefore, tan(alpha) = (12/13) ÷ (5/13) = 12/5. This follows directly from the trigonometric relationship between sine, cosine, and tangent.
Explain in your own words what it means for a series (\Sigma a_n) to converge absolutely, and why this is a stronger condition than just convergence.
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Absolute convergence means that both Σa_n and Σ|a_n| diverge, implying the series is unstable.
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Absolute convergence means that the terms a_n approach zero very slowly
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Absolute convergence means that the series Σa_n converges, but Σ|a_n| diverges, indicating conditional convergence.
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Absolute convergence means that the series Σ|a_n| converges, implying that the original series Σa_n also converges, and is more robust to rearrangement of terms.
Explanation
A series Σan converges absolutely if the series of absolute values Σ|an| converges. This is a stronger condition than ordinary convergence because absolute convergence guarantees that the original series Σan converges regardless of the signs of its terms. Moreover, absolutely convergent series are stable under rearrangements, whereas conditionally convergent series can change their sum if terms are reordered.
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