C646 Trigonometry and Precalculus
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Free C646 Trigonometry and Precalculus Questions
What is the derivative of the secant function,
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sec x tan x
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sec2 x
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csc x cot x
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tan x
Explanation
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
Which of the following method returns the sine of 90 degrees?
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Math.sin(Math.toRadian(90))
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Math.sin(90)
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Math.sine(90)
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Math.sin(PI)
Explanation
In programming languages like Java, JavaScript, and others following similar math libraries, trigonometric functions expect input angles in radians, not degrees. Therefore, to compute sin(90∘), we must first convert 90 degrees to radians using a method such as Math.toRadian(90). After conversion, applying Math.sin() returns the correct value of 1.
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.
In polar coordinate conversions, what trigonometric function is associated with the x-coordinate?
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Tangent
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Sine
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Secant
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Cosine
Explanation
In polar coordinates, a point is represented as (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. The Cartesian x-coordinate is given by (x = r \cos(\theta)), directly associating the x-coordinate with the cosine function. This comes from projecting the point onto the horizontal axis using the angle θ.
Explain why ln(1) equals 0 in terms of exponential functions.
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ln(1) = 0 because e^0 = 1, where 'e' is the base of the natural logarithm.
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ln(1) = 0 because the integral of 1/x from 1 to 1 is 0.
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ln(1) = 0 because the derivative of ln(x) at x=1 is 0.
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ln(1) = 0 because 1/e = 0.
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ln(1) = 0 because e^1 = 1.
Explanation
The natural logarithm ln(x) is defined as the exponent to which the base e must be raised to produce x. Since e^0 = 1, the exponent that produces 1 is 0, meaning ln(1) = 0. This interpretation follows directly from the inverse relationship between the natural logarithm and the exponential function.
Which of the following statements best describes the convergence behavior of the power series representation of cos(x)?
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The power series converges only for x values between -1 and 1.
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The power series diverges for all x values except x = 0.
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The power series converges for all real numbers
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The power series converges only for x = 0.
Explanation
The power series for cos(x) is given by the infinite series ∑ (-1)ⁿ x^(2n) / (2n)!. Because the factorial in the denominator grows much faster than the powers of x in the numerator, the series converges for all real numbers. This property is a result of the ratio test, which shows that the limit of the ratio of successive terms approaches zero for any finite x. Therefore, the cosine series converges everywhere on the real line.
A particle's position is given by s(t)=t3 − 6t2 + 9t. Determine the time(s) when the particle's acceleration is zero.
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t = 1
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t = 2
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t = 0
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t = 3
Explanation
Consider the series 
what can you definitively conclude about the convergence or divergence of the series?
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Nothing can be definitively concluded; further tests are needed.
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The series diverges.
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The series converges.
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The series converges absolutely
Explanation
Explain the significance of '+ C' in the indefinite integral ∫ u^n du = u^(n+1)/(n+1) + C.
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'+ C' represents the complex conjugate of the function
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'+ C' represents the initial condition of the function.
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'+ C' represents the constant of integration, accounting for the fact that the derivative of a constant is zero, thus any constant could be part of the original function.
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'+ C' represents the error term in the integration.
Explanation
When evaluating an indefinite integral, the result represents a family of functions that all differ by a constant because the derivative of any constant is zero. This means many different functions could have produced the same derivative. Therefore, we include '+ C' to represent this entire family of possible original functions and to indicate that the antiderivative is not unique. Without '+ C', the solution would be incomplete.
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