3.1 Exponential Functions
1a1b
§3.1 Exponential Functions
Form constant, variable
Warning! Do not confuse these with power functions of form .
Laws of Exponents:
1.
2.
3.
4.
Examples.
From 2. with
From 4.
definition of square root
by first example
by 4.
by 3.
by 3.
by first example
definition of square root
Graph and
Table x/ -2, -1 0, 1, 2 2x/ … (½)x/…
-, -,
For and
has domain
and range .
has domain and range
For any base , the graph of increases for and decreases for and passes through the point .
Limits at Infinity
for
and
for
and
for and
is a horizontal asymptote of
Example.
property of exponential functions
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Study of Derivatives
By definition
apply to the exponential function
simplify the difference quotient
factor out of numerator
Thus
const. multiple rule !
Let and use
Thus
the derivative of is proportional to !
-, - , ,
There is a number such that
Let . Then .
Since is an exponential function
Example
Let
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?? Find the line tangent to at .
One-to-one functions
A function is called one-to-one (1-1) if it never takes the same value twice.
whenever
Example. is not 1-1
- , -,
although
Horizontal Line Test
A function is one-to-one if and only if no horizontal line intersects its graph more than once.
Example. , domain , is a 1-1 function.
-,-, ,
§3.2 Inverse Functions and Logarithms
Inverse Functions
Let be a one-to-one function with domain and range B. Then its inverse function has domain and range and is defined by
for all
Function Machine Picture
Cancellation Equations
for each
for each
Example. , domain .
The inverse function is
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Warning! is a number but is a function.
and do not mean the same thing.
Example. Let
Then
but ■
Graphs of and
If then is on the graph of
If then is on the graph of
Example. , domain .
points are on the graph of
-, -,
The graphs of and are symmetric about .
In general, the graphs of functions and are symmetric about the line . ■
How to find the inverse of a one-to-one function algebraically?
1. write
2.solve for in terms of (if possible) to get
3.exchange roles of and
Example , domain
1.
2.Take the positive square root
3.exchange roles of and
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Example
1.
2.
3. ■
Derivatives and Tangent Functions
-, -, ,
is the angle that the tangent line to at makes with the -axis.
Derivatives of Inverse Function by Geometry
-, -, ,,,
add ,
Thus
Derivatives of Inverse Functions by Calculus
Cancellation Equation
has the form
where
differentiate wrt using the chain rule
solve for
or
suppose and
suppose
In Leibniz Notation
Let , then
the relationship between derivatives looks like
Example. with domain
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Logarithms
Recall exponential functions
,
The logarithm base is the inverse of this function
reverse the role of and
Example. ■
Laws of Logarithms
1.
2.
3.
Example. Find the exact value of .
By law #1
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Natural Logarithms
It is simplest to use base in calculus. The logarithm base is the natural logarithm.
Notation
Recall the exponential function base
The logarithm base is its inverse
reverse roles of and
in particular
Cancellation Equations
Example. Solve for :
■
Compare the graphs of and .
-, -, , ,
Change of base formula
recall
take natural logarithm of equation on right
or
Example. Evaluate correct to 4 significant figures.
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§3.3 Derivatives of Logarithmic and Exponential Functions
?? Example. Let . Find .
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?? Example Let . Find
Write as a composition.
Apply the chain rule.
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Combine the natural exponential function with the chain rule.
Let . Find .
Write as a composition.
,
Apply the chain rule.
=
we have
Examples.
??
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Example. Suppose . Find .
Write as a composition.
Apply the chain rule.
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Derivative of exponentials with base .
Strategy: transform into . Then use the chain rule to differentiate .
then
Example. ■
Example. Let . Find .
Write as a composition
Apply the chain rule.
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?? Class Practice
Derivatives of natural logarithmic functions
The natural logarithm is the inverse of the natural exponential function.
Differentiate the right hand wrt , using the chain rule.
Example. Let . Find .
Generalized power rule:
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Combine the natural logarithm and the chain rule
Let
Write as a composition.
Apply the chain rule.
Example. Let . Find .
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?? Class Practice
Very Important Example.
two cases:
(1)
(2)
write as a composition
chain rule
we have for both and :
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Derivative of Logarithm with base .
Recall the change of base formula
Example. Let . Find .
Write as a composition
Apply the chain rule
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?? Class Practice
Simplification before differentiation
Recall laws of logarithms
Example. Let . Find .
First simplify using the laws of logarithms
Compute using the chain rule:
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?? Class Practice
Logarithmic Differentiation
Recall
Problem is to differentiate , where is a complicated expression involving products, quotients and/or powers.
- Take logarithms of both sides and simplify using the laws of logarithms.
- Differentiate to get
- Solve for .
Example. Let . Find .
Step 1.
Step 2.
Step 3.
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?? Example. Let . Find .
Differentiatingexpressions with variable base and exponent
Use logarithmic differentiation
Example. Let . Find .
§3.6 Hyperbolic Functions
Combinations of exponential functions that are analogous to trig functions
Graph
-, -, , , , even
Graph
-, -, , , , odd
Graph
??
?? symmetry
??
By odd symmetry
-, -, , ,
Most important hyperbolic identity
analogous to
Proof
Notice that
then
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Derivatives of Hyperbolic Functions
Hyperbolic / TrigonometricExample. Show that
Example. Let Find .
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?? Example. Let . Find .
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§3.5 Inverse Trigonometric Functions
Let be a one-to-one function with domain and range . Then its inverse has domain and range and is defined by
Inverse sine
is not 1-1. It fails the horizontal line test.
-, -,
The restricted sine function is 1-1
domain
range emphasize
Its inverse is or
domain
range [ add
For the restricted sine function
Example. Evaluate
Recall 30°-60°-90° triangle sketch
since is an odd function
it follows
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Example. Evaluate
If then triangle
To find the derivative of consider
differentiate the equation on the right wrt .
Triangle for triangle
we have obtained
Example. Let . Find .
Recall
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Example. Let Find .
write as a composition
apply the chain rule
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Inverse Tangent
-,-,vert. asymptotes,
is not 1-1. It fails the horizontal line test.
The restricted tangent function is 1-1. emphasize
domain range
Its inverse is or add
domain range
and are horizontal asymptotes of
For the restricted tangent
Example. Evaluate
45°-45°-90° triangle
is odd
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Example. Evaluate
Triangle for triangle
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To find the derivative of consider
differentiate the equation on the right wrt
Example. Let . Find .
Recall
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?? Example. Let . Find .
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Example. Let . Find .
Write as a composition
Apply the chain rule
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?? Example. Let . Find .
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§3.7 Indeterminate forms and L’Hospital’s Rule
recall the quotient law for limits
Suppose and exist and . Then
.
Sometimes we cannot apply this rule
Example. Find
both and as .
An indeterminate form of type is a limit of form
(1)
where both and as .
L’Hospital’s Rule. Suppose (1) is of form . Then
if the limit on the right exists or is or .
Notes:
A. L’Hospital’s rule usually gives the wrong answer if eq. (1) is not indeterminate!
B. L’Hospital’s rule is valid for one-sided limits or limits at infinity.
Example. Evaluate .
This is a type indeterminate form!
Method 1. Divide out a common factor
if .
Method 2. Use L’Hospital’s rule.
.
Warning! We cannot apply L’Hospital’s rule again!
this is because the middle expression is not an indeterminate form. ■
Example. Evaluate .
This is a type indeterminate form.
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Example. Evaluate
This is a type indeterminate form.
Still type .
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Example (One-sided Limit)
Evaluate
This is a type indeterminate form.
simplify
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An indeterminate form of type is a limit of form
(2)
where both (or and (or ) as .
L’Hospital’s Rule. Suppose (2) is of form . Then
if the limit on the right exists (or is or ).
Notes:
A. L’Hospital’s rule usually gives the wrong answer if eq. (1) is not indeterminate!
B. L’Hospital’s rule is valid for one-sided limits or limits at infinity.
Example. (Limit at infinity)
Evaluate
This is a type indeterminate form.
We still have an indeterminate form.
. ■ STOP
An indeterminate form of type is a limit of form
where and (or ) as .
Method. Put product in quotient form
and then apply L’Hospital’s rule.
Example. Find
is of form . Write product as a quotient.
We kept in the numerator so that differentiation simplifies it.
is now of form . Apply L’Hospital’s rule.
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Indeterminate powers have form
Three cases type
1. as .
2. as .
3. as .
Method. Take the logarithm , use the previous techniques, and then exponentiate your result.
Example. Find
is of type .
Let .
Consider
type
type
simplify
We have
Then
where
*Since the exponential function is continuous, the limit symbol passes into the argument of the exponential(see theorem 7 in section 1.5)
This is an important result. ■