Preuves

Question 1

lim. cosx-1/x=0

Answer 1

fois (cosx+1)/(cosx+1) en limites.
developer.
separer sinx/x et -sinx/cosx+1
=0

Question 2

lim. sinx/x = 1

Answer 2

aire + gendarmes

Question 3

d. e^x = e^x

Answer 3

utiliser lim. e^x -1/x = 1 et def. d. = f(x+h)-f(x)/h

Question 4

theorem valeurs inter.

Answer 4

f(x) continue sur [a,b]
N, entre f(a) et f(b)
existe c dans [a.b] tel que f(c)=N
restate theorem.

Question 5

theorem gendarmes.

Answer 5

évaluer limite.
condition 1. entre -1<=g(x)<=1 pour x égale pas a …
condition 2. limf(x)=limh(x)=L
restate theorem.

Question 6

def. continuity f(x) continue sur x =a

Answer 6

f(a)=limf(x) x–> a
f(a) existe
limf(x) x–>a existe

Question 7

prouver f(x) dérivable en x = a implique f(x) est continue x=a

Answer 7

prouver que lim.. ( (f(x)-f(a) ) = 0
prouver lim.f(x)=f(a) (utiliser 1er preuve)

Question 8

d. c*f(x)= c * d.f(x)

Answer 8

f(x)=c
d.c=0 (constant)
limh–>0 f(x+h)-f(x)
remplace h = 0 dans limite.
0/h=0 si h nest pas 0

Question 9

d. f(x)g(x) = d.f(x)g(x) + d.g(x)*f(x)

Answer 9

supposer f et g sont derivable.
set limh–>0 f(x+h)-f(x) for f(x) and g(x) respectively
set J(x)= f(x)g(x)
d.J(x)= lim.f(x+h)g(x+h) - f(x)g(x) –> add constant term–. -f(x)g(x+h)+f(x)*g(x+h)
factor out g(x+h) and f(x) –> separate limited (because they exists independ.)
the g(x+h) factored out and f(x) –> take out as constants
use d. definition
restate. cqfd.

Question 10

d. x^n = n*x^(n-1)

Answer 10

f(x)= x^n
use d. def by limit y–>x
lim. y–> x of y^n-x^n/y-x
factor (y-x) –> use a^n-b^n
=limy–>x term after (y-x) cancel
SD of x in place of y
= n*x^n-1
cqfd

Question 11

d.sinx=cosx

Answer 11

lim.h–>0 f(x+h)-f(x)/h replace sin(x)
appliquer sin(a+b) –> sin(x)cos(h)….
factoriser sin(x) et cos(x)
spearer limites selon sin et cos
sortir sin(x) et cos(x) comme constant
limites remarquables
=cosx
cqfd

Question 12

d.a^x = a^x * ln(a)

Answer 12

pour a>0
a^x = (e^ln(a))^x = e^ln(a)
regle de chaine
-ext. f(u)=e^U
-int. xln(a)
d. regle de chain =
e^xln(a) * d.xln(a)
= e^xln(a)(1ln(a) = a^xln(a)
cqfd

Question 13

d. tanx = sec^2x

Answer 13

regle de quotient.
tan=sin/cos
d.sincos-d.cossin/cos^2
cos^2+sin^2=1
1/cos^2=sec^2
cqfd

Question 14

d.secx=secx*tanx

Answer 14

regle de quotient.
secx=1/cosx
d=sinx/cos^2x –> seperate to sinx/cosx * 1/cos x
= tanx * sec x
cqfd