Chapter 3 Application of Derivatives
3.1 Extreme Values of Functions
Absolute maxima or minima are also referred to as global maxima or minima.
Examples
Extreme Value Theorem
The requirements in Theorems 1 that the interval be closed and finite, and
That the function be continuous, are key ingredients.
Local (Relative) Extreme Values
Finding Extrema
Critical Point
Theorem 2 says that a function’s first derivative is always zero at an
interior point where the function has a local extreme value and the
derivative is defined.
Hence the only places where a function f can possibly have an extreme
value (local or global) are
1. Interior points where f’=0
2. Interior points where f’ is undefined,
3. Endpoints of the domain of f.
How to Find the Absolute Extrema
Thus the only domain points where a function can assume extreme
values are critical points and endpoints.
Example
Example. Find the absolute maximum and minimum values of
f(x)=10x(2-lnx) on the interval [1, ex].
Solution. We evaluate the function at the critical points and the
endpoints and take the largest and the smallest of the resulting
values.
The first derivative is
f’(x)=10(2-lnx)-10x(1/x)=10(1-lnx).
Let f’(x)=0, we have x=e. Then
Critical point value: f(e)=10e
Endpoint values: f(1)=20, and f(e2)=0.
So the function’s absolute maximum value
is10e at x=e. The absolute minimum value
is 0 and occurs at the right endpoint x=e2.
3.2 The Mean Value Theorem
Mathematical Consequences
3.3 Monotonic Functions and the First Derivative Test
A function that is increasing or decreasing on an interval is said to be
monotonic on the interval.
Graphs of functions
Each tangent line
Has positive slope.
Each tangent line
Has negative slope.
Each tangent line
Has zero slope.
Theorem
Example
Example: Find the critical points of f(x)=x3 -12x-5 and identify the intervals
on which f is increasing and on which f is decreasing.
First Derivative Test for Local Extrema
Example
Find the critical points of f(x)=(x2-3)ex. Identify the intervals on which f is
increasing and decreasing. Find the function’s local and absolute
extreme values.
Solution.
3.4 Concavity and Curve Sketching
Two ways to characterize the concavity of a differentiable function f on an
open interval:
• f is concave up on an open interval if its tangent lines have increasing
slopes on that interval and is concave down if they have decreasing slopes.
• f is concave up on an open interval if its graph lies above its tangent lines
on that interval and is concave down if it lies below its tangent lines
Concavity and the Second Derivative Test for Concavity
If y=f(x) is twice-differentiable, we will use f’’ and y’’ interchangeable
When denoting the second derivative.
Example
3
Example: Find the intervals on which f ( x)  x 12 is concave up and
the intervals on which it is concave down.
Solution:
Inflection Points
Second Derivative Test for Local Extrema
This test requires us to know f’’ only at c itself and not in an interval about c.
This makes the test easy to apply.
However, this test is inconclusive if f’’=0 or if ‘’ does not exist at x=c.
When this happens, use the First derivative Test for local extreme values.
Strategy for Graphing y=f(x)
Graphical Behavior of Functions from Derivatives
3.5 Parametrizations of Plane Curves
Parametric Formula for dy/dx
Example. Find dy/dx as a function of t if x = t - t2, y = t - t3.
Solution.
3.6 Applied Optimization
To optimize something often means to maximize or minimize some
aspect of it.
Example 1
An open-tup box is to be made by cutting small congruent squares from
the corners of a 12-in.–by-12-in. sheet of tin and bending up the
sides. How large should the squares cut from the corners be to
make the box hold as much as possible?
Example 2: You have been asked to design a 1-liter can shaped like a right
Circular cylinder. What dimensions will used the least material?
3.7 Indeterminate Forms and L’Hopital’s Rule
How to Use L’Hopital’ Rule
Example
sin 3 x
x 0
5x
Example: Find lim
Example
e3 x  1
Example: Find lim 3
x 0
x
Indeterminate Forms of Type /
Example
x2
Example: Find lim 5 x
x e
Example
Example: Find lim
x 
ln x
x
Sometimes when we try to evaluate a limit as xa by substituting x=a
we get an ambiguous expression like   0, or    . Sometimes
these forms can be handled by using algebra to convert them to a
o/o or  /  form.
For example: Find
(a) lim( x sin 1 )
x 
x
(b) lim(
x 0
1
1
 )
sin x x
Indeterminate Powers
Limits that lead to the indeterminate forms 1 ,00 , and o can sometims be
Handled by first taking the logarithm of the function. We use l’Hopital’s
Rule to find the limit of the logarithm expression and then exponentiate
the result to find the original function limit.
1/ x
lim
x
Example: Find x 
.
3.8 Newton’s Method (Optional)
Newton’s method is a technique to approximate the solution to an
equation f(x)=0. Essentially it uses tangent lines in place of the
graph of y=f(x) near the points where f is zero.
(A value of x where f is zero is a root of the function
f and a solution of the equation f(x)=0.)
To find a root r of the equation f(x)=0,
• select an initial approximation x1. If f(x1)=0, then r=x1. Otherwise,
use the root of the tangent line to the graph of f at x1 to approximate
r. Call this intercept x2 .
x2  x1 
f ( x1 )
f '( x1 )
• We can now treat x2 in the same way we did x1. If f(x2 )=0, then r=
x2 . Otherwise, we construct the tangent line to the graph of f at x2,
and take x3 to be the x-intercept of the tangent line.
f ( x2 )
x3  x2 
f '( x2 )
Continuing in this way, we can generate a succession of values x1,x2,,
x3,,,x4…that will usually approach r.
Example
Use Newton’s Method to approximate the real solutions of x3-x-1=0
Solution:
3.9 Hyperbolic Functions
Identities and Derivatives for Hyperbolic Functions
Inverses Hyperbolic Sine, Cosine, and Secant of x
Inverse Hyperbolic Tangent, Cotangent, and Cosecant of x
Identities and Derivatives of Inverse Hyperbolic Functions