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CHAPTER 6
TRANSCENDENTAL FUNCTIONS
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6.1 Natural Logarithm Function
• The natural logarith function, denoted by ln, is defined by
• The domaian of the natural logarithm function is the set of positive real numbers.
x
xdtt
x1
0,1
)ln(
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Geometric Meaning
• Consider the graph of f(x) = 1/x. The ln(x) represents the area under f(x) between 1 and x.
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Use First Fundamental Theorem to find the Derivative of the Natural
Logarithm Function
x
xx xx
xDdtt
D1
0,1
ln1
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Use u-substitution within natural logarithm function
• Example:
CxCuu
du
dxxduxulet
dxx
x
)12sin(lnln2
1
2
1
)12cos(2),12sin(
)12sin(
)12cos(
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Properties of natural logarithms
• If a and b are positive numbers and r is any rational number, then
• A) ln 1 = 0
• B) ln ab = ln a + ln b
• C) ln(a/b) = ln a – ln b• D) ara r lnln
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Logarithmic differentiation
• Using properties of logarithms, a complicated expression can be rewritten as a sum or difference of less complicated expressions. Then, the function can be differentiated more easily.
• Example next slide.
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Find g’(x)
x
xx
xxxx
xxg
xxxx
xxg
xg
xxx
x
xxxg
x
xxxg
ln
)23()1(
ln
1
23
15
1
8)('
ln
1
23
15
1
8)('
)(
1
)ln(ln)23ln(5)1ln(4
ln
)23()1(ln)](ln[
ln
)23()1()(
542
2
2
2
542
542
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6.2 Inverse Functions & Their Derivatives
• If f is strictly monotonic on its domain, f has an inverse.
)()(
))((1
1
xfyyfx
xxff
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Inverse Function Theorem
• Let f be differentiable and strictly monotonic on an interval I. If f’(x) does NOT euqal 0 at a certain x in I, then the inverse of f is differentiable at the corresponding point y = f(x) in the range of f and
dxdydy
dx
xfyf
1
)('
1)()'( 1
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Graphical interpretation
• The slope of the tangent to a curve at point (x,y) is the reciprocal of the slope of the tangent to the curve of the inverse function at (y,x).
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6.3 The Natural Exponential Function
• The inverse of ln is called the natural exponential function and is denoted by exp. Thus x = exp y, and y = ln x.
• The letter “e” denotest he unique positive real number such than ln e = 1.
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The natural exponential function and the natural logarithmic function
are inverses of each other.• Properties that apply to inverse functions
apply to these 2 functions.
yallforye
xxey
x
,)ln(
0ln
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The derivative of the natural exponential function is itself
xx
xxx
edxe
eeD )(
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Integrate the following:
Ce
Cuu
du
dxedueulet
dxe
e
x
xx
x
x
223
3
)2(2
1
2
1
,2
)2(
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6.4 General Exponential & Logarithmic Functions
1,ln
1
ln
ln)ln()ln( ln
ln
aCaa
dxa
aaaD
axea
ea
xx
xxx
axx
axx
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Logarithms could have a base other than e.
1,1
ln
1log
ln
lnlog
1
aCa
xdxx
axxD
a
xx
aa
ax
a
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Example:
CCdu
xdxduxulet
xdx
xuu
x
5ln
55
5ln
15
sec,tan
sec5
tan
2
2tan
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6.5 Exponential Growth & Decay
• Functions modeling exponential growth (or decay) are of this form:
0:,0:
kdecaykgrowth
eyy kto
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Compound InterestA = amount
r = interest raten = # times compounded
t=time in years
rto
nt
o
eAtAlycontinuousCompound
n
rAtA
)(:
1)(
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6.6 1st-Order Linear Differential Equations
• Sometimes it is not possible to separate an equation such that all expressions with x and dx are on one side and y and dy are on the other.
• General form of a first-order linear differential equation:
operatoridentityI
operatorderivativeD
xQIxPyD
x
yx
)()(
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Example: Solve the differential equation:xy’(x) – 2y(x) = 2
1)(
)(
2)]([
2)(2)('
)(
2)(
2)('
2
22
32
332
2lnln22
2
Cxxy
Cxxyx
xxyxdx
d
xxyxxyx
xeeebythroughmultiply
xxy
xxy
xxdx
x
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6.7 Approximations for Differential Equations
Slope fields: Consider a first-order differential equation of the form y’ = f(x,y)
At the point (x,y) the slope of a solution is given by f(x,y).
Example:y’ = 3xy, at (2,4), y’=24, at (-2,1), y’=-6; at (0,5), y’ = 0; at (2,0), y’=0, etc.
If all the slopes (y’) were graphed on a coordinate axes at those specific points, the resulting graph would be a “slope field”.
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Approximating solutions of a differential equation
• Euler’s Method: To approximate the folution of y’ = f(x,y) with initial condition y(x-not)=y-not, choose a step size ha nd repeat the following steps for n = 1,2,3,…
),(.2
.1
111
1
nnnn
nn
yxhfyySet
hxxSet
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Applying Euler’s Method
• Use your calculator and the table function to evaluate the function until the solution is found with the desired error.
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6.8 Inverse Trigonometric Functions & Their Derivatives
• If the domain of the trigonometric functions is restricted, a portion of the curve is monotonic and has an inverse.
xxyyx
xxyyx
xxyyx
xxyyx
0,secsec
22,tantan
0,coscos
22,sinsin
1
1
1
1
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Using triangles, some useful identities are established.
1,1
1,1)tan(sec
1)sec(tan
1)cos(sin
1)sin(cos
2
21
21
21
21
xx
xxx
xx
xx
xx
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Derivatives of 4 Inverse Trigonometric Functions
1,1
1sec
1
1tan
11,1
1cos
11,1
1sin
2
1
21
2
1
2
1
xxx
xD
xxD
xx
xD
xx
xD
x
x
x
x
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Corresponding integral formulas follow from these derivatives
Cxdxxx
Cxdxx
Cxdxx
1
2
12
1
2
sec1
1
tan1
1
sin1
1
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Example
CxCx
CxCuduu
xdxduxulet
dxx
x
)5(5
1
)5(tantan5
1tan5
1
1
1
5
1
sec5),5tan(
)5(tan1
)5(sec
112
2
2
2
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6.9 Hyperbolic Functions & Their Inverses
)sinh(
1)(csc,
)cosh(
1)(sec
)sinh(
)cosh()coth(,
)cosh(
)sinh()tanh(
2)cosh(,
2)sinh(
xxh
xxh
x
xx
x
xx
eex
eex
xxxx
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Derivatives of Hyperbolic Functions
)coth()(csc)(csc
)tanh()(sec)(sec
)(csc)coth(
)(sec)tanh(
)sinh()cosh(
)cosh()sinh(
2
2
xxhxhD
xxhxhD
xhxD
xhxD
xxD
xxD
x
x
x
x
x
x
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Example
CxCuu
du
xdxduxulet
dxx
xdxx
sinhlnln
cosh,sinhsinh
cosh)coth(