math 37 unit 1.6
TRANSCRIPT
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7/29/2019 MATH 37 UNIT 1.6
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1.6INVERSE
HYPERBOLICFUNCTIONS
1These lecture slides were created byProf. Babierra. Hyperbolic Sine FunctionHyperbolic Sine Function
Domain:
Range:
xsinhy =
( )+ ,
( )+ ,
One-to-oneover the
domain
2These lecture slides were created byProf. Babierra.
Inverse Hyperbolic Sine FunctionInverse Hyperbolic Sine Function
Let .ysinhx=
ysinhx=
2
yy eex
=
xsinhrgy = if and only if
y
y
e
e
2
12
=
0122 = yy xee
3These lecture slides were created byProf. Babierra.
By the quadratic formula,
2
4422
+=
xxey
0122
=yy xee
12
+= xx
Since is positive,ye
12
++= xxey
Inverse Hyperbolic Sine FunctionInverse Hyperbolic Sine Function4These lecture slides were created byProf. Babierra.
Hence, .
12 ++= xxey
12
++= xxlny
12
++= xxlnxsinhArg
Inverse Hyperbolic Sine FunctionInverse Hyperbolic Sine Function5These lecture slides were created byProf. Babierra. Hyperbolic Cosine FunctionHyperbolic Cosine Function
Restricted
domain:
Range:
xcoshy =
[ )+,0
One-to-oneover a
restricteddomain.
[ )+,16
These lecture slides were created byProf. Babierra.
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Inverse Hyperbolic FunctionsInverse Hyperbolic Functions
12
++= xxlnxsinhArg
1lncosh2
+= xxxArg
11
1
2
1
+= x,
x
xlnxcothArg
7These lecture slides were created byProf. Babierra.
1011
2
++= x
x
xhxArg
Inverse Hyperbolic FunctionsInverse Hyperbolic Functions8These lecture slides were created byProf. Babierra.
Derivative of inverse hyperbolic sineDerivative of inverse hyperbolic sine
( )xsinhArgDx
( )[ ]12 ++= xxlnDx
++
++=
12
21
1
1
22 x
x
xx
1
1
1
1
2
2
2+
++
++=
x
xx
xx
9These lecture slides were created byProf. Babierra.
( )xsinhArgDx1
1
2+
=x
( )xcoshArgDx 1
1
2 = x1
>x,
Derivative of inverse hyperbolic sineDerivative of inverse hyperbolic sine10These lecture slides were created byProf. Babierra.
( )xtanhArgDx 11
1
2
= x,
x
Derivative of inverse hyperbolic sineDerivative of inverse hyperbolic sine11These lecture slides were created byProf. Babierra.
( )hxsecArgDx
10
1
1
2
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EvaluateEvaluate
Answer:
( )( )xcostanhArgx
Example13These lecture slides were created byProf. Babierra.
EvaluateEvaluate
Answer:
xcosArcxcoshrglnDx
221
11
1
11
xxcosArcxxcoshArg
Example14These lecture slides were created byProf. Babierra.
Integrals yielding inverse hyperbolicIntegrals yielding inverse hyperbolic
Let u be a differentiablefunction ofx.
CusinhArgduu
+=+
1
1
2
CausinhArgdu
au+=
+ 221
15These lecture slides were created byProf. Babierra.
Let u be a differentiablefunction ofx.
CucoshArgduu
+=
1
1
2
Ca
u
coshArgduau +=22
1
0>> au,
Integrals yielding inverse hyperbolicIntegrals yielding inverse hyperbolic16These lecture slides were created byProf. Babierra.
Let u be a differentiablefunction ofx.
CutanhArgduu
+=
21
1
au,Ca
utanhArg
adu
ua
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Evaluate .Evaluate .
Solution:
+
dxx 259
1
2
Let , .xu 3= 5=a
+
dxx 259
1
2 +
=22
53
1
u
du
Cx
sinhArg +=5
3
3
1
Example19These lecture slides were created byProf. Babierra.
Evaluate .Evaluate .
Solution:
dxe
ex
x
216
Let , .xeu = 4=a
=22
43
1
u
du
Ce
tanhArgx
+=44
1
dxe
ex
x
216
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