Calculus Volume 2 · free preview
Calculus of the Hyperbolic Functions
Hyperbolic cosine and sine combine exponentials into a new pair of functions whose derivatives swap like the trigonometric pair, whose identity cosh squared minus sinh squared equals one describes a hyperbola, and whose inverses integrate the square root of sums.
1. Definitions and identities
Definitions and identities
2. Differentiation, integration, and the catenary
Differentiation, integration, and the catenary
3. Inverse hyperbolics and the sum-of-squares integral
Inverse hyperbolics and the sum-of-squares integral
4. The hyperbola parametrization
The hyperbola parametrization
5. Hyperbolic identity
Which identity do the hyperbolic functions satisfy, the analog of the Pythagorean identity?
6. Derivative swap sign
How do the derivatives of cosh and sinh differ from the derivatives of cos and sin?
7. Catenary curve
A hanging chain assumes the shape of which curve?
8. Arcsinh logarithmic form
Which logarithmic expression equals arcsinh x?
9. Which integral needs arcsinh
Which integral resolves to the inverse hyperbolic sine, completing the sum-of-squares pattern?
10. Cosh of zero
The hyperbolic cosine at zero equals the average of e^0 and e^(-0), which is what number? Answer with a number.
11. Even and odd identification
Which statement correctly identifies the parity of cosh x and sinh x?
12. Sinh integral
The integral of cosh x dx equals which hyperbolic function, plus C? Answer with the function name like sinh x.
13. Identity rearrangement
The identity cosh^2 x - sinh^2 x = 1 rearranges to cosh^2 x = 1 plus what? Answer like sinh^2 x.
14. Tanh definition
tanh x is the ratio of two hyperbolic functions. As a fraction with sinh over cosh, what is the numerator? Answer like sinh x.
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