C646 Trigonometry and Precalculus
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Free C646 Trigonometry and Precalculus Questions
What is the value of the tangent function at π/4 radians?
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1
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0
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√2/2
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√3/2
Explanation
According to the relationship between velocity and acceleration, under what condition does the speed of an object decrease?
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When velocity and acceleration are both positive
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When velocity and acceleration have the same sign
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When velocity and acceleration have opposite signs
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When velocity and acceleration are both negative
Explanation
Explanation:
Speed decreases when the velocity and acceleration are in opposite directions. If the object is moving in a positive direction but acceleration is negative (or vice versa), the acceleration acts against the motion, reducing the speed over time.
Correct Answer:
When velocity and acceleration have opposite signs
Explain the rationale behind using critical numbers and endpoints in the Candidates Test for finding absolute extrema.
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Critical numbers represent potential local maxima or minima, while endpoints define the interval's boundaries where absolute extrema could occur.
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Critical numbers indicate points of inflection, and endpoints show where the function is undefined.
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Critical numbers represent points where the function crosses the x-axis, and endpoints are used to find the y-intercepts
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Critical numbers are where the second derivative is zero, and endpoints are used for concavity analysis.
Explanation
The Candidates Test states that to find the absolute maximum and minimum values of a continuous function on a closed interval, you evaluate the function at all critical numbers (where f'(x) = 0 or f'(x) does not exist) and at the endpoints of the interval. Critical numbers indicate where the function could attain local extrema, and endpoints must be checked because absolute extrema could occur at the boundaries.
How much does the tangent function have at 0 radians?
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Undefined
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1
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-1
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0
Explanation
For all x, cos(x) is equal to
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sin(-x)
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sin(x)
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cos(x + π)
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cos(-x)
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sin(x) + cos(x)
Explanation
Explain in your own words why the logarithmic identity ln(MN) = ln(M) + ln(N) holds true
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This identity reflects the property that the logarithm of a product is the sum of the logarithms of the individual factors, stemming from the exponential relationship where multiplication corresponds to addition of exponents.
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This identity is a consequence of the chain rule in calculus
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This identity is derived from the geometric series formula
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This identity is only valid when M and N are equal.
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This identity is true because logarithms convert division into subtraction.
Explanation
The identity ln(MN) = ln(M) + ln(N) comes from the way logarithms relate to exponents. Since the natural logarithm is the inverse of the exponential function, multiplying two numbers corresponds to adding their exponents. Logarithms convert that multiplicative relationship into an additive one. This is a fundamental property of logarithms and holds for all positive values of M and N.
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
Find the values of xxx and yyy that maximize f(x, y) = 2xy − 2x2 − y2 +20y
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x=10, y=15
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x=20, y=20
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None of these
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x=10, y=20
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x=5, y=30
Explanation
A pendulum's angular displacement from its resting position is modeled by the function θ(t) = Asin(ωt), where A is the amplitude and ω is the angular frequency. If at time t, ωt = π, what is the pendulum's displacement?
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-A
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0
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A
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A/2
Explanation
Explain why the integral of csc ( u ) cot ( u ) du results in −csc ( u ) +C, including the significance of the “+ C”.
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The integral of csc ( u ) cot ( u ) is a standard power rule application.
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The derivative of -csc ( u ) is csc (u) cot(u), and '+ C' represents the constant of integration, accounting for all possible antiderivatives
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The '+ C' is added because the derivative of a constant is always 1
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The '+ C' is added to ensure the function is continuous.
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The derivative of csc ( u ) is csc ( u ) cot ( u ), and '+ C' represents the constant of integration.
Explanation
To evaluate the integral ∫ csc ( u ) cot( u ) du, we recall that the derivative of csc ( u ) is −csc ( u ) cot ( u ). Therefore, the antiderivative of csc(u) cot(u) must be −csc ( u ) because differentiating −csc ( u ) produces exactly csc( u )cot( u ). The term “+ C” appears because indefinite integrals represent a family of functions whose derivatives are the same. Since the derivative of any constant is zero, adding C accounts for all possible antiderivatives.
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