Download Lectures 17 to 19

MATH 2413Fall2012, Lectures17-19
QingwenHu
DepartmentofMathematicalSciences
TheUniversityofTexasatDallas
Richardson, Texas
[email protected]
October, 2012
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QingwenHu (UTD)
Lectures17-19
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OutlineofSections3.3–3.5
1. Speciallimits
lim
h→0
sin h
cos h − 1
= 1, lim
= 0.
h→0
h
h
2. Derivativesoftrigonometricfunctions: e.g.,
d
d
(sin x) = cos x, (cos x) = − sin x,
dx
dx
d
1
(tan x) =
= sec2 x
dx
cos2 x
3. TheChainrule: If y = f (u) and u = g(x) aredifferentiable, then
d
′
′
dx f (g(x)) = f (g(x))g (x). i.e.,
dy
dy d u
=
dx d u dx
4. TheexponentialRule:
from
d x
dx a
= ax ln a, a > 0, whichcanbederived
ax = eln a = ex ln a .
x
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QingwenHu (UTD)
Lectures17-19
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October, 2012
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5. TheexponentialrulecombinedwiththeChainRule:
If y = f (x) isdifferentiable, then
d f (x)
e
= ef (x) f ′ (x).
dx
6. ApplyingtheChainrulemultipletimes: e.g., If y = f (u), u = g(v)
and v = h(x) aredifferentiable, then
d
f (g(h(x))) = f ′ (g(h(x)))g ′ (h(x))h′ (x).
dx
i.e.,
dy
dy du dv
=
.
dx d u dv d x
7. Implicitdifferentiation.
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QingwenHu (UTD)
Lectures17-19
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8. Derivativesofinversetrigonometricfunctions:
d
1
arcsin x = √
dx
1 − x2
d
1
arccos x = − √
dx
1 − x2
d
1
arctan x =
dx
1 + x2
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QingwenHu (UTD)
Lectures17-19
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Exercises:
1. Define f : R → R by
{
f (x) =
if
x ≤ 1,
ax + b if
x > 1.
x2
Findthevaluesof (a, b) sothat f isdifferentiableeverywhere. Isthe
valueof (a, b) unique?
2. Canyouseparatelyusethequotientruleandthepowerrule
combinedwiththechainruletoderive
d
1
?
dx cos2 x
3. Supposethatafunction y = f (x) satisfies x2 + (y − 1)2 = 1 forevery
x ∈ [−1, 1]. Wheredoes f ′ (x) failtoexist?
.
QingwenHu (UTD)
Lectures17-19
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October, 2012
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4/4