Differentiation Rules
d
[ f ( x)] = f '( x)
dx
If the derivative of a function is its slope, then for a
constant function, the derivative must be zero.
d
(c ) = 0
dx
example:
y=3
y = 0
The derivative of a constant is zero.
If we find derivatives with the
Application of the Derivative
Related Rates
This is an application of implicit dierentiation or use
of the chain rule.
In dierential calculus related rates problems
involve nding a rate that a quantity changes by
Section 3.7
Using Calculus to Solve Optimization Problems
Applied Maximum and Minimum Problems
3.7
Procedures for Solving Applied Maximum and
Minimum Problems
Assign symbols to all given quantities and quantities to be determined.
If applicable, make a s
Newtons Method
Method to approximate the roots of
a func6on
Newtons Method
Newtons method is an itera6ve method for
genera6ng a sequence ( x1, x2, x3,.) of
approxima6ons to a solu6on xr of an equa6on in the
for
More Op(miza(on Problems
1. A printer need to make a poster that will have a total area of 200 in2 and
will have 1 inch margins on the sides, a 2 inch margin on the top and a 1.5
inch margin on the boBom.
Finding a Par*cular Solu*on to a
Dieren*al Equa*on
Ini*al Value Problem
Finding a Par*cular Solu*on to a
Dieren*al Equa*on
An equa*on that involves the deriva*ve of a
func*on is called a dieren*al equa*on.
The ge
8.2 Integration By Parts
Start with the product rule:
d
dv
du
(uv ) = u + v
dx
dx
dx
d (uv ) = u dv + v du
d (uv ) v du = u dv
u dv = d (uv ) v du
u dv = (d (uv ) v du )
u dv = (d (uv ) v du
u dv = uv v du
This is the Integration by Parts
formula.
u dv
Implicit Differentiation
An implicit function is a function in which the
dependent variable has not been given
explicitly in terms of the independent
variable. An explicitly defined function
provides a prescription for determining the
output value of a f
Differentials
Application of
Differentiations
Differentials
l
l
We have used dx as part of the derivative
symbol in d/dx or dy/dx. Now we will use dx,
for an increment in a variable x called a
differential of x (instead of x).
Differentials are used to es
The Chain Rule
The Chain Rule
If f is a dieren/able func/on of u and u is a
dieren/able func/on of x, then the composite
f (u) is a dieren/able func/on of x, and
d
du
[ f (u)] = f (u)
dx
dx
The deriva/ve of
Concavity
The Second Derivative Test
Concavity and the Second Derivative Test
n
n
Let f be differentiable on an open interval.
We say that the graph of f is concave upward
if f is increasing on the interval and concave
downward if f is decreasing on the i
A function f is said to be increasing on an
interval if for any 2 numbers x1 and x2 in
the interval x1<x2 implies that f(x1)< f(x2).
A function f is said to be decreasing on an
interval if for any 2 numbers x1 and x2 in
the interval x1<x2 implies that f
The Mean Value Theorem
The Extreme Value Theorem
and
Rolles Theorem
Rolles Theorem
l
l
The Extreme Value Theorem tells us that a
continuous function on a closed interval must have
both a maximum and a minimum. These values can
occur at the endpoints.
Roll
Determining Behavior of a Function
Let f be defined on an interval I containing c.
1. f(c) is the minimum of f on I if f(c) f(x) for
all x in I.
2. f(c) is the maximum of f on I if f(c) f(x) for
all x in I.
n The minimum and maximum of a function on
an in
Antiderivatives
4.1
Introduction
A physicist who knows the velocity of a particle
might wish to know its position at a given time.
A biologist who knows the rate at which a
bacteria population is increasing might want
to deduce what the size of the popu