雙曲函數(shù)間的關(guān)系

1、雙曲函數(shù)間的關(guān)系雙曲函數(shù)及其性質(zhì)雙曲正弦雙曲余弦雙曲正切2xxeeshx2xxeechxxxxxshxeethxchxeehyperbolic sinehyperbolic cosinehyperbolic tangent雙曲正弦函數(shù)名雙曲正弦符號shx表達式2xxee(,) 定義域奇偶性奇函數(shù)單調(diào)性單調(diào)遞增極限+lim2xxxee +-lim2xxxee - 雙曲正弦函數(shù)圖像shx雙曲正弦值域(,) 導數(shù)()2xxee()shx2xxeechx雙曲正弦的反函數(shù)2xxeey2yyeexyue12uxu2210uxu 22442xxu0,( )yue 舍去2= +1yexx 2ln(1)yxx2

2、1xx令(,)x (-+ ),(-+ )xy ，(-+ ),(-+ )xy ，雙曲正弦與反雙曲正弦的圖像shxarshx反雙曲正弦函數(shù)名反雙曲正弦符號arshx表達式(,) 定義域奇偶性奇函數(shù)單調(diào)性單調(diào)遞增2ln(1)yxx反雙曲正弦值域(,) 導數(shù)2(ln(1)xx()arshx22112(1)211xxxx211x雙曲余弦函數(shù)名雙曲余弦符號chx表達式2xxee(,) 定義域奇偶性偶函數(shù)雙曲余弦的單調(diào)性221122xxxxeeee因為函數(shù)為偶函數(shù)，所以只需討論單調(diào)遞增0,)(,0單調(diào)遞減0,)上的情況.設(shè)120,xx21211=2xxxxeeee（(）（）2121111=2xxxxeeee

3、（(）（）1221121-=2xxxxxxeeeee e（(）2112+11=2xxxxeee(）(1-）則0雙曲余弦函數(shù)圖像chx雙曲余弦的反函數(shù)2xxeey2yyeexyue12uxu2210uxu 22442xxu1,( )yue 舍去2= +-1yexx2ln( +-1)yxx21xx令1,)x(0,1)xy(10)xy，雙曲余弦與反雙曲余弦的圖像chxarchx反雙曲余弦函數(shù)名反雙曲余弦符號archx表達式1,)定義域單調(diào)性單調(diào)遞增2ln(1)yxx奇偶性無反雙曲余弦值域0,)導數(shù)2(ln(1)xx()archx22112(1)211xxxx211x雙曲正切函數(shù)名雙曲正切符號thx表

4、達式xxxxeeee(,) 定義域奇偶性奇函數(shù)雙曲正切單調(diào)性單調(diào)遞增xxxxeeee21xxxeee 2211xe 極限+limxxxxxeeee 1-limxxxxxeeee -1雙曲正切函數(shù)圖像thx雙曲正切值域( 1,1)導數(shù)()thx()shxchx2()()shx chxshx chxch x222ch xsh xch x21ch x雙曲正切的反函數(shù)-xxxxe eyeeyue1=1uuxuu2(1)1x ux令( 1,1)x (-+ ),(-11)xy ，(-11),(-+ )xy ，-yyyye exee221=1uxu211xux11xux11ln21xx1ln1xyx雙曲正切

5、與反雙曲正切的圖像thxarthx反雙曲正切函數(shù)名反雙曲正切符號arthx表達式( 1,1)定義域奇偶性奇函數(shù)單調(diào)性單調(diào)遞增11ln21xyx反雙曲正切值域(,) 導數(shù)11( ln)21xx()arthx111()1211xxxx211(1) (1)(1)(1)21(1)xxxxxxx211(1)(1)21(1)xxxxx211x雙曲函數(shù)的圖像thxchxxeshxxe雙曲函數(shù)的圖像thxchx2xeshx2xe雙曲函數(shù)間的關(guān)系()sh xyshxchychxshy()sh xyshxchychxshy()ch xychxchyshxshy()ch xychxchyshxshy221ch xsh x22sh xshxchx222ch xch xsh xsin()sincoscos sinxyxyxysin()sincoscos sinxyxyxycos()cos cos