Math  /  Calculus

Questiong(x)=ex2x+2g(x)=e^{x}-2 x+2 الدالة العددية g معرفة عالى (I  - 1 ( limxg(x)،limx+g(x) احسب \text { - } 1 \text { ( } \lim _{x \rightarrow-\infty} g(x) ، \lim _{x \rightarrow+\infty} g(x) \text { احسب } 2) أدرس إتجاه تغير الدالة g ثم شكل جدول تغيراتها. 3) بين أنه من أجل كل عدد حقيقي x: 0 > f(x)=xex+x2+1f(x)=x e^{-x}+\frac{x}{2}+1 الدالة العددية f معرفة على و ليكن (C) منحناها البياني في المستوي المنسوب إلى معلم (C, (O,i)(O, \vec{i}).  احسب limxf(x)،limx+f(x)\text { احسب } \lim _{x \rightarrow-\infty} f(x) ، \lim _{x \rightarrow+\infty} f(x)
أ. بين أنه من أجل كل عدد حقيقي x ب. استنتج إتجاه تغير الدالة f ثم شكل جدول تغيراتها. احسب أدرس وضعية المستقيم (د) بالنسبة لـ (C).

Studdy Solution

STEP 1

1. We are analyzing two functions: g(x)=ex2x+2 g(x) = e^x - 2x + 2 and f(x)=xex+x2+1 f(x) = xe^{-x} + \frac{x}{2} + 1 .
2. We need to calculate limits as x x approaches + +\infty and -\infty .
3. We will study the direction of change for both functions and create their variation tables.
4. We will prove certain inequalities for the functions.
5. We will examine the relationship between a line and a curve.

STEP 2

1. Calculate the limits of g(x) g(x) as x x \rightarrow -\infty and x+ x \rightarrow +\infty .
2. Determine the direction of change for g(x) g(x) and create its variation table.
3. Prove the inequality for f(x) f(x) and calculate its limits as x x \rightarrow -\infty and x+ x \rightarrow +\infty .
4. Determine the direction of change for f(x) f(x) and create its variation table.
5. Examine the position of the line with respect to the curve.

STEP 3

Calculate limxg(x) \lim_{x \rightarrow -\infty} g(x) .
As x x \rightarrow -\infty , ex0 e^x \rightarrow 0 and 2x+ -2x \rightarrow +\infty , so g(x)=ex2x+2+ g(x) = e^x - 2x + 2 \rightarrow +\infty .

STEP 4

Calculate limx+g(x) \lim_{x \rightarrow +\infty} g(x) .
As x+ x \rightarrow +\infty , ex+ e^x \rightarrow +\infty and 2x -2x \rightarrow -\infty , but ex e^x grows faster than 2x -2x , so g(x)=ex2x+2+ g(x) = e^x - 2x + 2 \rightarrow +\infty .

STEP 5

Find the derivative g(x) g'(x) to determine the direction of change.
g(x)=ex2 g'(x) = e^x - 2
Set g(x)=0 g'(x) = 0 to find critical points: ex2=0ex=2x=ln(2) e^x - 2 = 0 \Rightarrow e^x = 2 \Rightarrow x = \ln(2) .

STEP 6

Analyze the sign of g(x) g'(x) to determine intervals of increase or decrease.
- For x<ln(2) x < \ln(2) , g(x)<0 g'(x) < 0 (decreasing). - For x>ln(2) x > \ln(2) , g(x)>0 g'(x) > 0 (increasing).

STEP 7

Calculate limxf(x) \lim_{x \rightarrow -\infty} f(x) .
As x x \rightarrow -\infty , xex0 xe^{-x} \rightarrow 0 , x2 \frac{x}{2} \rightarrow -\infty , so f(x)=xex+x2+1 f(x) = xe^{-x} + \frac{x}{2} + 1 \rightarrow -\infty .

STEP 8

Calculate limx+f(x) \lim_{x \rightarrow +\infty} f(x) .
As x+ x \rightarrow +\infty , xex0 xe^{-x} \rightarrow 0 , x2+ \frac{x}{2} \rightarrow +\infty , so f(x)=xex+x2+1+ f(x) = xe^{-x} + \frac{x}{2} + 1 \rightarrow +\infty .

STEP 9

Find the derivative f(x) f'(x) to determine the direction of change.
f(x)=exxex+12 f'(x) = e^{-x} - xe^{-x} + \frac{1}{2}
Analyze the sign of f(x) f'(x) to determine intervals of increase or decrease.

STEP 10

Set f(x)=0 f'(x) = 0 to find critical points and analyze intervals.

STEP 11

Examine the position of the line with respect to the curve C C .
The solution involves detailed analysis of derivatives and critical points, which may require additional steps to fully complete the variation tables and position analysis.

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