1
6.1 Average value of a function f (x)
• The average value of a function f (x) over an interval [a, b] is given by
Z b
1
f (x)dx.
b−a a
Rb
1
• Note that: Let c = b−a a f (x)dx be the average value of a function f (x) over the
interval [a, b]. The definite integral of the constant function c over the interval [a, b]
and the definite integral of f (x) over the interval [a, b] are the INDENTICAL, e.i,
Z b
Z b
f (x)dx.
c · dx =
c · (b − a) =
a
a
• If f (x) is the derivative of the function F (x), i.e., F ′(x) = f (x) then the average
value of f (x) over the interval [a, b] is also the average rate of change of F 9x) over
the interval [a, b] (the fundamental theorem of calculus), i.e.,
Z b
F (b) − F (a)
1
f (x)dx.
=
b−a
b−a a
2
6.3 Present and furture value.
• Recall: If P is the value of a deposit today, the present value,then B = P · ert is
the future value of the deposit t years later assuming a fixed interest rate r per year
compounded continuously. Thus the present value P expressed interms of the future
value B is P = B · e−rt.
• If S(t) is an income stream, i.e., the amount of $ deposit per year (the instantaneously
rate of deposit measured in $/year) at the time t after the initial deposit (t = 0), then
the present value P (M ) at time M years of this income (deposit) stream is given by
Z M
S(t)dt
(Present value)P (M ) =
0
• Of course as always
Future value = P (M ) · ert.
3
7.1 Definition, name and notation of antiderivatives.
• Let f (x) and F (x) be two functions. If F ′(x) = f (x) the function F (x) is called
an antiderivative to f (x). (Of course we already call f (x) the derivative function of
F ′(x)).
• If F (x) is an antiderivative of f (x) then the function F (x) + C, where C is a
constant, is ALSO an antidetivative of f (x). In fact all antiderivatives to f (x) are
obtained this way. Thus if you find one you can find them all by just adding any constant
C.
• Any antiderivatives of a function f (x) has a special name and notation. The name is
the indefinite integral and the notation is
Z
f (x)dx.
R
Thus f (x)dx = F (x) + C, where C is a constant and F ′(x) = f (x).
4
7.1 Special antiderivatives.
Note that we have (you can check by find the derivative of the right hand sides):
•
•
•
•
Z
k · dx = kx + C,
where k is a constant.
Z
xαdx =
Z
1
x
Z
xα+1
α+1
+ C, α 6= −1
dx = ln(|x|) + C
exdx = ex + C
5
7.1 Simple rules for antiderivatives.
• Let f (x) be a function and let k be a constant. Then
Z
Z
k · f (x) · dx = k · f (x)dx,
i.e., in words: the indefinite integral of a constant times a function is the constant times
the indefinite integral of the function.
• Let f (x) and g(x) be two functions
Z
Z
Z
(f (x) + g(x))dxdx =
f (x)dx + g(x)dx
i.e., in words: the indefinite integral of a sum of two functions is the sum of the indefinite
integral indefinite integrals of the two functions.
• By combining the two above rules we of course get
Z
Z
Z
(f (x) − g(x))dxdx =
f (x)dx − g(x)dx