9231 · 2.1
Hyperbolic functions flashcards
Revision flashcards for Cambridge 9231 Hyperbolic functions (syllabus 2.1). Flip, recall, then mark a real past-paper question.
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What is the definition of sinh(x)?
sinh(x) = (e^x - e^-x) / 2
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What is the definition of cosh(x)?
cosh(x) = (e^x + e^-x) / 2
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What is the definition of tanh(x)?
tanh(x) = sinh(x) / cosh(x) = (e^x - e^-x) / (e^x + e^-x)
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What is the fundamental hyperbolic identity, analogous to sin²(x) + cos²(x) = 1?
cosh²(x) - sinh²(x) = 1. Note the minus sign!
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What is the derivative of sinh(x)?
d/dx (sinh(x)) = cosh(x). (No sign change, unlike trigonometry!)
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What is the derivative of cosh(x)?
d/dx (cosh(x)) = sinh(x). (No sign change!)
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What is the derivative of tanh(x)?
d/dx (tanh(x)) = sech²(x)
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Describe the graph of y = cosh(x).
A 'catenary' curve, symmetric about the y-axis, with a minimum point at (0, 1). Its range is y ≥ 1.
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What is the range of y = tanh(x)?
The range is -1 < y < 1. The graph has horizontal asymptotes at y = 1 and y = -1.
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Common Trap: What is a frequent mistake when solving an equation like `cosh(x) = k`?
After forming a quadratic in u = e^x and finding two solutions for u, remember that e^x must be positive. Any negative solutions for u must be rejected.
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How is the identity involving tanh(x) and sech(x) derived?
Start with cosh²(x) - sinh²(x) = 1 and divide every term by cosh²(x).
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Is sinh(x) an odd or even function?
sinh(x) is an odd function because sinh(-x) = -sinh(x). Its graph has rotational symmetry about the origin.