5. In This section we’re going to make sure that
you’re familiar with functions and function
notation. Both will appear in almost every
section in a Calculus class and so you will need
to be able to deal with them.
6. What exactly is a function? An equation will be
a function if for any x in the domain of the
equation (the domain is all the x ’ s that can be
plugged into the equation) the equation will
yield exactly one value of y.
This Topic needs an Example to understand
more. here is the example:
7. Example 1 Determine if each of the following are functions.
(a) y = x2 + 1
(b) y2 = x + 1
Solution
(a) This first one is a function. Given an x, there is only one
way to square it and then add 1 to the result. So, no matter
what value of x you put into the equation, there is only one
possible value of y.
y2=3+1=4
(b) The only difference between this equation and the first is
that we moved the exponent off the x and onto the y. This
small change is all that is required, in this case, to change the
equation from a function to something that isn’t a function.
8. To see that this isn’t a function is fairly simple. Choose a value of x,
say x=3 and plug this into the equation.
Now, there are two possible values of y that we could use here. We
could use y = 2 or
y = -2 . Since there are two possible values of y that we get from a
single x this equation isn’t a function.
9. Paradox- logical steps which gives an
illogical conclusion / contrary to
intuition
Method of Exhaustion - approximation technique
area of 0 ≈ area of n-gon
10. Infinite number of points
Finite length
∞
A B
Deals with infinity
- Jean Le Rond d’Alembert
11. -English Mathematician
-Born in 1642
-Differential approach
-Realized that in using infinite series in approximation gave the
exact value
12. Approximation ∞ exact
-Discovered the inverse relationship of the slope of the tangent to
a curve and the area under the curve
-Wrote in 1687. Philosophiae Naturalis Principia Mathematica
Principia Calculating m of tangent line to a curve, v c+1
-Find the velocity and calculation ac+1 function from position
function s c+1
-Calculation arc lengths and volume of solids
-Calculation relative and absolute extrema
-Calculating m of tangent line to a curve, v c+1
-Find the velocity and calculation ac+1 function from position
function s c+1
-Calculation arc lengths and volume of solids
-Calculation relative and absolute extrema
14. Counting Numbers, IN = {1,2,3…}
Whole Numbers, W= {0,1,2,3,…}
Integers, Z={…..-2,-1,0,1,2,…}
Rational Numbers, R= {1/2,-3/4,0.45,-03.23,0.4444,……,-2.75}
-rational can be expressed as a simple function
Terminating Decimals
Non-terminating, separating decimals
Irrational Numbers, Q1 = { √2,∏,e}
Non-terminating, non-separating decimals
15. Set- wall destined
-defined collection of objects
Elements
-objects that make up set
{} A={1,2,3,4} roster method
A = {x/x is a counting number less than 5}
Seven that set- builder notational
A is a subset of B it all elements at A one element of B
A& B
16. Def: If a,b є IR
1.) a<b if b-a is positive
2.) a>b if a-b is positive
3. ) a>b if a=b or a>b
4.) a<bif a=b or a<b
17. 1. If a > 0 and b> 0, then a + b > 0
2. If a > 0 and b> 0, then ab > 0
3. Transitive Property of Inequality
• If a > b and b < c, then a < c
4. If a < b, then a + c < b + c
5. If a < b and c > 0, then ac < bc
If a < b and c < 0, then ac > bc
18. 1. Open interval
( a, b) = {x/a < x < b }
2. Closed Interval
( a, b) = { x/a ≤ x ≤ b}
3. Half-Open Interval
[ a , b) = { x/a ≤ x < b}
( a, b] = { x/a} < x ≤ b}
19.
20. In mathematics, the concept of a "limit" is used to describe the
value that a function or sequence "approaches" as the input or
index approaches some value. The concept of limit allows one
to, in complete space, define a new point from a Cauchy
sequence of previously defined points. Limits are essential
to calculus and are used to define continuity , derivatives
and integrals.
21. The concept of the limit of a function is further generalized to
the concept of topological net, while the limit of a sequence is
closely related to limit and direct limit in category theory.
In formulas, limit is usually abbreviated as lim as in lim(an) = a or
represented by the right arrow (→) as in an → a.
22. The derivative of f(x) with respect to x is the function
f’(x) and is defined as,
f’(x)=lim f(x+h)-f(x)
h 0 h
23.
24. Consider the following sequence: 1.79, 1.799, 1.7999,... We
could observe that the numbers are "approaching" 1.8, the
limit of the sequence.
Formally, suppose x1, x2, ... is a sequence of real numbers. We
say that the real number L is the limit of this sequence and we
write
to mean
For every real number ε > 0, there exists a natural
number n0 such that for all n > n0, |xn − L| < ε.
Lim xn = L
n ∞
25. Intuitively, this means that eventually all elements of the
sequence get as close as we want to the limit, since
the absolute value |xn − L| is the distance between xn and L.
Not every sequence has a limit; if it does, we call it convergent,
otherwise divergent. One can show that a convergent
sequence has only one limit.
The limit of a sequence and the limit of a function are closely
related. On one hand, the limit of a sequence is simply the
limit at infinity of a function defined on natural numbers. On
the other hand, a limit of a function f at x, if it exists, is the
same as the limit of the sequence xn = f(x + 1/n).
26.
27. In calculus, a one-sided limit is either of the two limits of a
function f(x) of a real variable x as x approaches a specified point
either from below or from above. One should write either:
for the limit as x decreases in value
approaching a (x approaches a "from the right" or "from above"),
and similarly
for the limit as x increases in value
approaching a (x approaches a "from the left" or "from below").
28. The two one-sided limits exist and are equal if and only if the
limit of f(x) as x approaches a exists. In some cases in which the
limit
does not exist, the two one-sided limits nonetheless exist.
Consequently the limit as x approaches a is sometimes called a
"two-sided limit". In some cases one of the two one-sided limits
exists and the other does not, and in some cases neither exists.
Lim f(x) or lim f(x)
x a + x a
Lim f(x) or lim f(x)
x a x a
Lim f(x)
x a
29.
30. A formal definition of convergence can be stated as follows. Suppose pn as n goes
from 0 to ∞ is a sequence that converges to a fixed point p, with λ for all n. If
positive constants λ and α exist with
lim = λ
n ∞
then pn as n goes from 0 to converges to p of order α, with asymptotic error
constant λ
Given a function f(x) = x with a fixed point p, there is a nice checklist for checking
the convergence of p.
1) First check that p is indeed a fixed point:
f(p) = p
2) Check for linear convergence. Start by finding .f’(p) If....
3) If we find that we have something better than linear we should check for
quadratic convergence. Start by finding f’”(p) If....
Pn+1-P
Pn - P a
31.
32. the derivative is a measure of how a function changes as its
input changes. Loosely speaking, a derivative can be thought of
as how much one quantity is changing in response to changes
in some other quantity
33. d
dx
d
dx
d
dx
c = 0 , where c = constant
(Ax) = A, where a = constant
(uv) = uv’ + vu’
d
dx
(xa ) = axa-
1
d
dx
u
x
= xu’ – ux’
x2
38. Factoring a f(x+h) on the first limit and g(x) from the
second limit we get:
(fx(x)g(x))’ =lim
f(x+h)[g(x+h) – g(x)]
+ lim
g(x)[f(x+h)-f(x)
h 0 h h 0 h
39. The key is to subtract and add a term:
f(x+h)g(x)
By doing this, you can get the following:
(fx(x)g(x))’ =lim f(x+h)g(x+h) – f(x+h)g(x)+f(x+h)g(x)-f(x)g(x)
h 0 h
40. Based on the property of limits, we can the limit can
be broken into two because the limit of a sum is the
sum of the limits.
(fx(x)g(x))’ =lim
f(x+h)g(x+h) – f(x+h)g(x)
+ lim
f(x+h)g(x)-f(x)g(x)
h 0 h h 0 h
41. (f(x)g(x))’ =lim f(x+h)lim
g(x+h)-g(x)
+ lim g(x)lim
f(x+h)-f(x)
h 0 h h 0 h 0 h
Another property of limits says that the limit of a product is a
product of the limits. Using this fact, the limit can be written like
this:
44. If the functions f and g are differentiable at x, with g(x) 0,
then the quotient f/g is differentiable at x, and
d f f’ (x)g(x) – f(x)g’(x)
dx g (x) = [g(x)]2
45. lim f(x+h) = f (x)
h 0
lim g(x) = g (x)
h 0
(f(x) g (x))’ = f (x) g’(x) + g (x) f’(x)
Also,
And
Since they do not defend on h
And the last one, what is supposed to be shown:
46. d f f’ (x+h) f(x)
dx g (x) =
lim
g (x+h)
-
g (x)
h
= lim f(x+h)g(x) – f(x)g(x+h)
h 0 g(x+h)g(x)h
= lim f(x+h)g(x) – f(x)g(x) + f(x)g(x) – f(x)g(x+h)
h 0 g(x+h)g(x)h
= lim f(x+h) – f(x)g(x) - f(x)[(g(x+h) – g(x)]
h 0 g(x+h)g(x)h
= lim
f(x+h) -f(x)
g(x) - f(x)
g(x+h) – g(x)
h 0
h
g(x+h)g(x)
h
h 0
47. If we recognize the difference quotients for f and g in this last
expression, we see that taking the limit as h0 replaces them
by the dreivatives f'(x) and g'(x). Further, since g is
differentiable, it is also continuous, and so g(x+h)g(x) as h0.
Putting this all together gives
And that is the quotient rule.
d f f’ (x)g(x) – f(x)g’(x)
dx g (x) = [g(x)]2
48.
49. means that f(t) does not bound a finite area
from a to b
f (t) dt = ∞∫
b
a
50. means that the area under f(t) is infinite.
f (t) dt = ∞∫
∞
-∞
51. means that the area under f(t) equals 1
f (t) dt = 1∫
∞
-∞
