SCX5000 - Métodos Matemáticos e de Computação I (2017.1)
Lecture 02
1
Squeeze theorem
For an interval I ⊂ R, assume that g(x) ≤ f (x) ≤ h(x) with x ∈ I. Given x0 ∈ I, if
lim g(x) = lim h(x) = L ,
x→x0
2
x→x0
then
lim f (x) = L .
(1)
x→x0
sin x
x→0 x
Limit lim
R
Denote by S△OP Q , S△OP R and SÂOP Q , respectively, the
areas of the triangles OP Q, OP R and the circular section
OP Q, as in figure 1. It is straightforward that
S△OP Q ≤ SÂOP Q ≤ S△OP R .
Q
(2)
1) Show that the inequality (2) implies
x
sin x
cos x ≤
≤ 1.
x
O
(3)
P
2) From the squeeze theorem, show that
lim
x→0
sin x
= 1.
x
(4)
Figure 1: △OP Q, ÂOP Q and △OP R.
3
Derivatives
The derivative of function f at a point x is defined
as
f
d
∆y
f (x + ∆x) − f (x)
f (x) ∶= lim
= lim
, (5)
∆x→0 ∆x
∆x→0
dx
∆x
f (x + ∆x)
where the notation f ′ (x) is also used. A function
is said to be differentiable if the derivative exists at
every point of its domain.
∆y
1) If f and g are two differentiable functions (at x),
show, using (5), that
d
d
d
[f (x) + g(x)] =
f (x) +
g(x) .
dx
dx
dx
f (x)
∆x
(6)
2) If f is a differentiable function (at x), show, using
(5), that
x
d
d
[αf (x)] = α f (x) , for α ∈ R (constant) . (7)
dx
dx
x + ∆x
Figure 2: Graph of f , and ∆x and ∆y.
1
The exercises 1 and 2 show that the differentiation is linear (assuming that the derivatives exist). Note,
from a geometrical standpoint, that the derivative at x corresponds to the tangent of the curve f at the
same point. Therefore,
d
>0
f (x) {
<0
dx
, function f increasing with x
.
, function f decreasing with x
(8)
d
f (x) = 0 is especially important, since it is a necessary (although not sufficient) condition to
The case dx
determine the extremal points (local maxima and minima) of differentiable functions.
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4.1
Derivatives of elementary functions (I)
Power-law functions (I)
3) From the definition (5), show that
d 2
x
dx
= 2x.
4) From the definition (5), show that
d n
x = nxn−1 ,
dx
para n ∈ N .
(9)
The formula (9) can be generalized for n ∈ R, as will be seen later.
4.2
Trigonometric functions
5) From the relation sin(a ± b) = sin a cos b ± sin b cos a, a, b ∈ R, show that
sin x − sin y = 2 sin (
x−y
x+y
) cos (
).
2
2
(10)
Hint: Evaluate sin(a + b) − sin(a − b) and denote x ∶= a + b and y ∶= a − b.
6) From (10) and (4), show, by using the definition (5), that
d
sin x = cos x .
dx
(11)
d
cos x = − sin x .
dx
(12)
d
tan x = sec2 x .
dx
(13)
7) Analogously, show that
8) Analogously, show that
Hint: tan(x + ∆x) =
sin(x+∆x)
,
cos(x+∆x)
et cætera.
2
5
Product and quotient rules
9) Consider two differentiable functions f and g (at point x). By using (5), show that
′
(f ⋅ g) (x) = f ′ (x)g(x) + f (x)g ′ (x)
or
d
d
d
[f (x)g(x)] = [ f (x)] g(x) + f (x) [ g(x)] .
dx
dx
dx
(14)
Hint: Note that adding f (x + ∆x)g(x) − f (x + ∆x)g(x), which is zero zero, does not change the calculation.
10) Consider two differentiable functions f and g (at point x). By using (5), show that
f ′
f ′ (x)g(x) − f (x)g ′ (x)
( ) (x) =
2
g
[g(x)]
ou
d
[ d f (x)] g(x) − f (x) [ dx
g(x)]
d f (x)
.
[
] = dx
2
dx g(x)
[g(x)]
(15)
Here, one should assume that g(x) ≠ 0. Hint: Note that adding f (x + ∆x)g(x) − f (x + ∆x)g(x), which is
zero zero, does not change the calculation.
11) Evaluate the derivative of tan x =
6
sin x
cos x
using the quotient rule.
Exercises
12) Evaluate the derivative of the following functions:
a) sin x + cos x b) sin2 x
c) cos2 x
d) 2x sin x
g)
m)
7
sin x
x
x sin x
3(x2 −cos x)
h)
cos x
x
i)
n)
2
x+3 cos x
o)
x
sin x
π sin2 x
x+cos x
j)
p)
3x2
cos x
e) 3x cos x
k)
sin x(x2 −tan x)
x3 (sin2 x−cos x)
f) 4x tan x
3x+sin x
2 tan x
2(x−sin2 x)
q) (x + 1) sin2 x(x tan x)
l)
sin2 x−5x
tan x
r)
x sin x(cos2 x−tan x)
x+π
Python
13) Write a code that print all integer numbers between 1 and 2017 (increasing order).
14) Write a code that print all even numbers between 1 and 2017 (increasing order).
15) Write a code that print all odd numbers between 1 and 2017 (increasing order).
16) Write a code that print all integer numbers between 1 and 2017 (decreasing order).
17) Write a code that print all even numbers between 1 and 2017 (decreasing order).
18) Write a code that print all odd numbers between 1 and 2017 (decreasing order).
8
Homework
● Assignments: 12q ∼ 12r, 13 (Python code).
● Deadline: 20/03/2017
3