This document contains a mathematics exam for high school students in Greece. It is divided into 4 sections with multiple questions in each section. The questions cover topics related to functions, limits, derivatives, and integrals. Some questions ask students to prove statements, find domains of functions, determine if functions are injective or have critical points. The document is 3 pages long and aims to test students' understanding of key concepts in calculus and mathematical analysis.
This document contains a mathematics exam with 4 problems (Themes A, B, C, D) involving functions, derivatives, monotonicity, convexity, extrema, asymptotes and limits.
Theme A involves properties of differentiable functions, the definition of the derivative, and Rolle's theorem. Theme B analyzes the monotonicity, convexity, asymptotes and graph of a given function.
Theme C proves properties of a continuous, monotonically increasing function and finds extrema of related functions. Theme D proves properties of a power function and its relation to a given line, defines a new function, and proves monotonicity and existence of a single real root for a polynomial equation.
Marko Batista Temporary Objects And Hybrid Ambients Batistandumuseika
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Marko Batista Temporary Objects And Hybrid Ambients Batista
Marko Batista Temporary Objects And Hybrid Ambients Batista
Marko Batista Temporary Objects And Hybrid Ambients Batista
2. 痢律 2裏 裏裏
A4. 留 留留虜侶溜竜竜 旅 凌略竜旅 凌 留虜凌了凌慮凌僚, 粒略凌僚留 凌 竜略隆旅
留 凌 粒略亮亮留 凌 留僚旅凌旅竜溜 竜 虜略慮竜 留侶 虜留旅 隆溜了留 凌 粒略亮亮留 侶
了龍両侶 裡, 留僚 侶 留侶 竜溜僚留旅 流, 流 略慮凌, 留僚 侶 留侶 竜溜僚留旅
了留僚慮留亮龍僚侶.
留) 2
x 0
1
lim
x
粒旅留 虜略虜竜
.
硫) 旅 慮龍 僚留旅竜旅
x
Q x
, 亮竜 硫留虜亮 凌 留旅虜亮慮旅
x
亮竜粒留了竜凌
凌了略旅凌僚 虜留略 隆凌 凌 硫留虜亮凌 凌 留凌僚凌亮留旅, 龍凌僚 了略粒旅竜
留亮竜.
粒) 僚
3
f x x 1
鰹居 竜
2
f 3 1
鰹居 逸 .
隆) 僚
f x ln x
虜留旅 x
g x e
, 竜
1
g f x , x
x
.
竜) 僚
0
x x
lim | f x | 0
, 竜
0
x x
lim f x 0
凌僚略隆竜 10
溜僚凌僚留旅 凌旅 僚留旅竜旅 f, g 亮竜 凌
f x ln 1 x
虜留旅 x
g x e 1
留僚溜凌旅留.
1. 留 凌了凌粒溜竜竜 凌
x
lim f x
虜留旅 凌
x
lim g x
.
凌僚略隆竜 5
2. 留 凌隆旅凌溜竜竜 慮 僚略慮慮 g f
.
凌僚略隆竜 5
3. 留 留凌隆竜溜両竜竜 旅 慮 僚略慮慮 f 留僚旅龍竜留旅 虜留旅 僚留 硫竜溜竜 慮僚
留僚溜凌旅 慮.
凌僚略隆竜 5
裡裏 2裏 4 裏裏
3. 痢律 3裏 裏裏
4. 留 硫竜溜竜 慮僚 竜両溜慮 慮 竜留凌亮龍僚慮 慮 粒留旅虜旅 留略留慮 慮 g
慮 凌凌溜留 略粒竜留旅 留 凌 慮亮竜溜凌 (1,1).
凌僚略隆竜 5
5. 留 竜隆旅略竜竜 凌 溜隆旅凌 慮亮留 留両僚僚 旅 粒留旅虜龍 留留略竜旅 僚
僚留旅竜僚 f ,g
.
凌僚略隆竜 5
溜僚竜留旅 慮 僚略慮慮 f 亮竜 凌 x
x
f x , x
1 e
.
1. 留 硫竜溜竜 旅 旅亮龍 僚 留留亮龍僚 ,
, マ竜 慮 竜虜竜溜留 亮竜 竜両溜慮
y 2x 1
僚留 竜溜僚留旅 了略粒旅留 留亮慮 留僚 x . 留 留旅旅凌了凌粒旅竜竜 慮僚
留略僚慮旅 留.
凌僚略隆竜 8
僚 留=2 虜留旅 硫=-1.
2. 留 留凌隆竜溜両竜竜 旅 慮 粒留旅虜旅 留略留慮 慮 f 虜留旅 慮 留亮慮 竜虜竜溜留 亮竜
竜両溜慮 y 2x 1
龍凌僚 亮凌僚留隆旅虜 虜凌旅僚 慮亮竜溜凌.
凌僚略隆竜 4
3. 留 留凌隆竜溜両竜竜 旅 略竜旅 亮凌僚留隆旅虜 醐, 龍凌旅凌 マ竜 慮 僚略慮慮 f 僚留
留凌旅略侶竜旅 凌了旅虜 亮龍粒旅凌 凌 両.
凌僚略隆竜 7
4. 留 了竜竜 慮僚 竜両溜慮
2 2
f x x 2 x f
醐 , 凌 両 慮 旅亮旅 凌
留凌旅略侶竜旅 凌了旅虜 亮龍粒旅凌 慮 f.
凌僚略隆竜 6
裡裏 3裏 4 裏裏
4. 痢律 4裏 裏裏
溜僚竜留旅 留留粒粒溜旅亮慮 僚略慮慮 f :
慮 凌凌溜留 龍竜旅 僚凌了凌 旅亮ホ 凌
隆旅略慮亮留
f 1 f 2
,
2
.
1. 留 留凌隆竜溜両竜竜 旅 略竜旅 龍僚留 凌了略旅凌僚
0
x 1,2
龍凌旅凌 マ竜
0
2f x f 1 f 2
凌僚略隆竜 5
2. 留 留凌隆竜溜両竜竜 旅 慮 竜両溜慮
f x 0
龍竜旅 凌了略旅凌僚 亮溜留 了慮 凌 .
凌僚略隆竜 5
慮 f 竜溜僚留旅 粒僚慮溜 虜溜僚凌留 凌 虜留旅 旅凌僚:
f 1 f 2 2
,
x x
lim f x , lim f x
3. 留 亮竜了竜旅竜竜 慮僚 f 凌 慮 亮凌僚凌凌僚溜留 虜留旅 慮僚 虜慮留.
凌僚略隆竜 5
4. 留 留凌隆竜溜両竜竜 旅 略凌僚 1
3
1,
2
, 2
3
,2
2
龍凌旅留 マ竜,
1 2
f f 0
.
凌僚略隆竜 5
5. 留 硫竜溜竜 凌 了旅虜凌 僚 旅侶ホ 慮 竜両溜慮
f x
e , >0
粒旅留 虜略虜竜
x.
凌僚略隆竜 5
裡裏 4裏 4 裏裏