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Engineering Calculus Notes 153

# Engineering Calculus Notes 153 - 141 2.2 PARAMETRIZED...

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Unformatted text preview: 141 2.2. PARAMETRIZED CURVES can be parametrized as follows. The substitution x et ± e−t = a 2 et ∓ e−t y = b 2 yields x a 2 et e2t e−t ±2 4 2 2 2t −2t e 1e = ±+ 4 2 4 y b 2 e2t et e−t ∓2 4 2 2 −2t 2t 1e e ∓+ = 4 2 4 y b 2 1 1 =± − ∓ 2 2 = ±1. = + e−2t 4 + e−2t 4 and similarly = so x a 2 − The functions cosh t = sinh t = e t + e −t 2 e t − e −t 2 (2.19) are known, respectively, as the hyperbolic cosine and hyperbolic sine of t. Using Euler’s formula (Calculus Deconstructed, p. 475), they can be interpreted in terms of the sine and cosine of an imaginary multiple of t, and satisfy variants of the usual trigonometric identities (Exercise 6): cosh t = cos it sinh t = −i sin it. We see that → − (t) = (a cosh t, b sinh t) p −∞ < t < ∞ ...
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