Calculus Volume 1 · 免费预览
双曲函数的微积分
双曲余弦和双曲正弦把指数函数组合成一对新的函数：它们的导数像三角函数对那样互换，恒等式“双曲余弦的平方减去双曲正弦的平方等于一”描述一条双曲线，而它们的反函数则可对平方和的平方根进行积分。
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.