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Math 151, Benjamin
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4.1 Exponential Functions
Exponential functions are those of the form f (x) = ax where a > 0, a 6= 1.
Case 1: f (x) = ax where a > 1
Case 2: f (x) = ax where 0 < a < 1
Increasing Exponential/Exponential Growth
Decreasing Exponential/Exponential Decay
Domain: (−∞, ∞)
Domain: (−∞, ∞)
Range: (0, ∞)
Range: (0, ∞)
y-intercept: (0, 1)
y-intercept: (0, 1)
VA: None
VA: None
HA: y = 0
HA: y = 0
lim f (x) =
x→∞
lim f (x) =
x→−∞
lim f (x) =
x→∞
lim f (x) =
x→−∞
Consider exponentials of the form f (x) = a−x .
f (x) = 2−x
f (x) =
−x
Consider f (x) = ex where the base is the natural number e.
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Math 151, Benjamin
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Recall the rules of exponents:
(1) ax ay = ax+y
(2)
ax
= ax−y
ay
(3) (ax )y = axy
(4) (ab)x = ax bx
NOTE: (a + b)x 6= ax + bx – DO NOT DO THIS!
Calculate the following limits:
lim 0.3−x
lim 0.3−x
x→∞
lim
x→−3+
3
2
x→−∞
x
x+3
=
lim
x→−3−
3
2
x
x+3
=
4 + 3ex
x→−∞ 3 + e−x
5 + (0.1)x
x→∞ 2 + e−x
3e2x + e−3x
x→∞ e2x − 4e−3x
3e2x + e−3x
x→−∞ e2x − 4e−3x
lim
lim
lim
lim
2
(5)
x
a
b
=
ax
bx
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Math 151, Benjamin
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Derivatives of Exponentials:
d x
e = ex
dx
d g(x)
e
= g′ (x)eg(x)
dx
Calculate the following derivatives:
f ′ (x) where f (x) = ex sin x + x2 e3x +
f ′ (x) where f (x) =
1
e
+ cos(ex ) + xe +
e−5x
1 + e−2x
f (n) (x) where f (x) = xex
3
√
e2x + 1
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Math 151, Benjamin
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√
Find an equation of the tangent line to the curve y = x2 e
x
when x = 4.
Find the slope of the tangent line to the curve 4exy − ex = y at the point (0, 3).
For what values of r does the function f (x) = e−rx satisfy the differential equation: 2y ′′ + 3y ′ − 2y = 0.
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Math 151, Benjamin
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4.2 Inverse Functions
A function is one-to-one if every element in the domain has a UNIQUE value in the range. In other words,
if f (x1 ) = f (x2 ), then it MUST be true that x1 = x2 if the function is one-to-one.
Show that the function f (x) =
x
x−2
is one-to-one.
Horizontal Line Test: Graphically, a function is one-to-one if no horizontal line intersects its graph more
than once.
If a function f is one-to-one, then it has an inverse f −1 defined by:
f −1 (y) = x ⇔ f (x) = y
If f has domain A and range B, then f −1 has domain B and range A.
If (a, b) is on the graph of f , then (b, a) is on the graph of f −1 .
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Math 151, Benjamin
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The graph of f −1 is obtained by reflecting the graph of f about the line y = x. If f is a continuous function
defined on an interval, then f −1 is also a continuous function.
Cancellation Equations: If x is in the domain of f and y is in the range of f , then
f −1 (f (x)) = x and f (f −1 (y)) = y
Find the inverse of the function f (x) =
√
Find the inverse of the function f (x) =
4x + 3
.
x−2
2x + 6 and state its domain and range.
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Math 151, Benjamin
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If f is a one-to-one differentiable function with inverse g = f −1 , then g is also differentiable and
g′ (a) =
1
f ′ (g(a))
provided that f ′ (g(a)) 6= 0.
Suppose f (x) = 3x5 + 2x3 − 3. Find g ′ (2) where g is the inverse of f .
Suppose f (x) = 4 + 2x + ex . Find g ′ (5) where g is the inverse of f .
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Math 151, Benjamin
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4.3 Logarithmic Functions
We’ve dealt with exponential functions and we know that the graph of an exponential function of the form
f (x) = ax is one-to-one, which means it must have an inverse. The inverse of the exponential function
f (x) = ax is the logarithmic function with base a.
loga x = y ⇔ ay = x
In words, loga x is the EXPONENT to which a must be raised to get x.
Evaluate the following:
(1) log2 32
(2) log3
(4)log a 1
1
81
(5) Find x such that log4 x = −3
f (x) = ax , a > 1
f (x) = loga x, a > 1
Domain: (−∞, ∞)
Domain:
Range: (0, ∞)
Range:
Asymptotes: y = 0
Asymptotes:
Intercepts: (0, 1)
Intercepts:
lim ax = ∞
x→∞
lim ax = 0
x→−∞
(3) log49 7
lim loga x =
x→∞
lim loga x =
x→0+
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Math 151, Benjamin
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Cancellation equations:
aloga x = x for x > 0
loga (ax ) = x for all x
The common logarithm is the logarithmic function with base 10. log10 x = log x
The natural logarithm is the logarithmic function with base e. It has special notation. loge x = ln x
ln x = y ⇔ ey = x
ln e = 1
ln(ex ) = x
eln x = x, x > 0
Calculate the following limits:
(1) lim+ log3 (x − 5)
x→5
(2) limπ ln(sin x)
x→ 2
1
)
(3) lim− log( 3−x
x→3
Properties of Logarithms:
(1) loga (xy) = loga x + loga y
(2) loga
x
y
= loga x − loga y
(3) loga (xy ) = y loga x
IMPORTANT: loga (x + y) 6= loga x + logb y – DO NOT DO THIS
Examples:
(1) (# 12, 4.4) Evaluate log5 10 + log5 20 − 3 log5 2
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Math 151, Benjamin
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(2) Express
1
2
ln x + b ln y − c ln z − ln(x2 + 1) as a single logarithm.
(3) Calculate lim [ln(3x2 + x) − ln(2x2 − x)]
x→∞
(4) Solve the equation 4log4 6 + ln(4x − 2) = 15 for x.
(5) Solve the equation log(x − 7) + log(x + 2) = 1 for x.
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Math 151, Benjamin
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(6) Solve the equation 32x−9 − 5 = 0 for x.
(7) Find the inverse of the function f (x) =
2e3x−1
1 + e3x−1
(8) Find the domain of y = log(x2 − 4).
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Math 151, Benjamin
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ln x
ln a
Example: Calculate log7 12 correct to 6 decimal places.
Change of Base: loga x =
4.4 Derivatives of Logarithmic Functions
•
d
1
ln x =
dx
x
•
d
1
ln |x| =
dx
x
• Chain Rule Version:
•
d
g′ (x)
ln(g(x)) =
dx
g(x)
1 1
1
d
loga x =
· =
dx
ln a x
x ln a
• Chain Rule Version:
d
g′ (x)
loga (g(x)) =
dx
g(x) ln a
Examples: Find the derivatives of the following functions.
√
(1) f (x) = ln(3x2 + x)
(2) g(x) = log2 (x4 − 5x)
(3) h(x) = x ln(cos x)
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Math 151, Benjamin
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(4) f (x) =
p
ln(x + ln x)
t2 − 4
t3 − 7t
(5) g(t) = log
!
Now we have the ability to differentiate exponential functions where the base is not e.
•
d x
a = ax ln a
dx
• Chain Rule Version:
d g(x)
a
= ag(x) (ln a)g′ (x)
dx
Examples: Differentiate the following functions.
(1) f (x) = x3 5x
(2) h(θ) = 4ln(θ
2 −7x
2 +1)
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Math 151, Benjamin
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Logarithmic Differentiation: Sometimes it is easier to differentiate a function by first taking the logarithm
of both sides, differentiating implicitly and then solving for y ′ . Use this method when:
(1) The function is a quotient or product of a lot of terms. – Log. Diff. recommended, but not necessary.
(2) The function is of the form y = f (x)g(x) . – Logarithmic Differentiation NECESSARY.
√
e2x x5 + 2
′
.
Find f (x) for f (x) =
(x − 1)4 (x2 + 9)2
Find f ′ (x) for f (x) = xcos x .
Find y ′ for y = (sin x)e .
x
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