Download Math 122 Lecture 1: Antiderivatives

Definition of an Antiderivative
Definition of an Antiderivative
For a function f , an antiderivative of f is a function F for which
F 0 (x ) = f (x )
Clint Lee
Math 122 Lecture 1: Antiderivatives
2/11
Antiderivatives Are Not Unique
Suppose that F and G are both antiderivatives of the same function f . Then
G 0 (x ) = F 0 (x ) = f (x )
So that
0
G 0 (x ) − F 0 (x ) = 0 ⇒ [G (x ) − F (x )] = 0
By the Mean Value Theorem the only differentiable functions whose first
derivative is zero for all x in their domains, are constant functions. Hence,
there is a constant C for which
G (x ) − F (x ) = C ⇒ G (x ) = F (x ) + C
Clint Lee
Math 122 Lecture 1: Antiderivatives
3/11
The Most General Antiderivative
The Most General Antiderivative
I
If F is an antiderivative of a function f , then any other antiderivative of
f is of the form
G (x ) = F (x ) + C
for some constant C.
I
Hence, if F is any antiderivative of a function f , then the most general
antiderivative of the function f is
F (x ) + C
where C is an arbitrary constant.
Clint Lee
Math 122 Lecture 1: Antiderivatives
4/11
The Indefinite Integral
The Indefinite Integral
If F is an antiderivative of a function f , then the indefinite integral of f is
Z
f (x ) dx = F (x ) + C
The operator
Z
. . . dx
is an instruction to find any antiderivative of whatever is between the
(integral) sign and the dx, and then add an arbitrary constant.
Clint Lee
Math 122 Lecture 1: Antiderivatives
R
5/11
Some Antiderivative (Integration) Formulas
Every differentiation formula gives a corresponding antiderivative
(integration) formula.
Some Integration Formulas
Z
Z
Z
x n dx =
eax
1
Z
x n +1 + C , n 6 = − 1
n+1
1 ax
dx = e + C
a
Z
sec2 x dx = tan x + C
Z
x −1 dx =
Z
1
dx = ln |x | + C
x
sin x dx = − cos x + C
1
1 + x2
dx = arctan x + C
Verify any of these formulas by differentiating the righthand side to obtain
the function under the integral sign, the integrand.
Clint Lee
Math 122 Lecture 1: Antiderivatives
6/11
The General Solution to a Differential Equation
An antiderivative of a function f is a solution to the differential equation
dy
= f (x )
dx
If F is an antiderivative of the function f , then one possible solution is
y = F (x ). The general solution is the most general antiderivative of the
function f , that is,
y = F (x ) + C
where C is an arbitrary constant.
Clint Lee
Math 122 Lecture 1: Antiderivatives
7/11
A Family of Functions
Suppose that the graph of one
solution to the differential equation
y
dy
= f (x )
dx
looks like this. The graph of any other
solution is the graph of F shifted
vertically by the constant C. Further,
for any x the slopes of the curves are
equal. In either case, the curves are
parallel.
Clint Lee
F(x) + 1
F(x
y
F(x)
F
x
x
Math 122 Lecture 1: Antiderivatives
8/11
A Family of Functions
The graphs of all possible solutions to
the differential equation
y
dy
= f (x )
dx
F(x) + 1
F(x
y
F(x)
F
x
x
form a family of functions, all of
whose graphs are parallel. At each x
the slopes of all the graphs are equal.
Clint Lee
Math 122 Lecture 1: Antiderivatives
9/11
A Particular Solution
The graphs in the family of functions
representing the general solution to
the differential equation
y
F(x) + 1
F(x
y
F(x)
dy
= f (x )
dx
F
y0
are all parallel. So they do not
intersect. Hence, to select a
particular solution it is only necessary
to specify one point on the graph, say
(x0 , y0 ). This information is sufficient
to determine a value for the arbitrary
constant C in the general solution.
Clint Lee
x
Math 122 Lecture 1: Antiderivatives
x0 x
10/11
The Initial Value Problem
The Initial Value Problem
For a function f , the differential equation
dy
= F 0 (x ) = f (x )
dx
together with the initial condition F (x0 ) = y0 determines exactly one, a
unique, particular solution.
The problem
F 0 (x ) = f (x ) where F (x0 ) = y0
is an initial value problem.
Clint Lee
Math 122 Lecture 1: Antiderivatives
11/11