Math / Calculus

QuestionSoit $f$ la fonction numérique définie par: $f(x)=\left\{\begin{array}{ll}1+\frac{x-1}{\ln |x|} & \text {, si } x<0 \\ \left(x^{2}+2 x\right) e^{-x}+e^{-x} & , \text { si } x \geq 0\end{array}\right.$ On note $C_{f}$ la courbe représentative de $f$ .
1. Déterminer le domaine de définition de $f$ .
2. Etudier la continuité et la dérivabilité de $f$ en 0 .
3. Etudier les variations de $f$ puis dresser son tableau de variation.
4. Etudier les branches infinies de $f$ .
5. Vérifier que : $f(\alpha)=\alpha+1$ .
6. Construire la courbe $C_{f}$ .

Studdy Solution

STEP 1

1. The function $f(x)$ is piecewise, with different expressions for $x < 0$ and $x \geq 0$ .
2. We need to analyze the domain, continuity, differentiability, variations, asymptotic behavior, and specific function values.
3. The function involves logarithmic, exponential, and polynomial expressions.

STEP 2

1. Determine the domain of $f$ .
2. Study the continuity and differentiability of $f$ at $x = 0$ .
3. Analyze the variations of $f$ and create a variation table.
4. Study the asymptotic behavior (infinite branches) of $f$ .
5. Verify the equation $f(\alpha) = \alpha + 1$ .
6. Construct the graph of $C_f$ .

STEP 3

Determine the domain of $f$ :
- For $x < 0$ , $f(x) = 1 + \frac{x-1}{\ln |x|}$ . The logarithm $\ln |x|$ is defined for $x < 0$ since $|x|$ is positive. - For $x \geq 0$ , $f(x) = (x^2 + 2x)e^{-x} + e^{-x}$ . This expression is defined for all $x \geq 0$ .
Thus, the domain of $f$ is all real numbers except $x = 0$ , where the logarithm is undefined.

STEP 4

Study the continuity and differentiability of $f$ at $x = 0$ :
- Check continuity by evaluating the left-hand limit, right-hand limit, and the function value at $x = 0$ . - For $x \to 0^-$ , evaluate $\lim_{x \to 0^-} f(x)$ . - For $x \to 0^+$ , evaluate $\lim_{x \to 0^+} f(x)$ . - Check if these limits are equal.

STEP 5

Calculate the left-hand limit:
$\lim_{x \to 0^-} \left(1 + \frac{x-1}{\ln |x|}\right)$
This requires careful analysis of the behavior of $\ln |x|$ as $x \to 0^-$ .

STEP 6

Calculate the right-hand limit:
$\lim_{x \to 0^+} \left((x^2 + 2x)e^{-x} + e^{-x}\right)$
This involves evaluating the behavior of the exponential terms as $x \to 0^+$ .

STEP 7

Determine if $f(x)$ is differentiable at $x = 0$ by checking if the left-hand and right-hand derivatives exist and are equal.

STEP 8

Analyze the variations of $f$ :
- Determine the derivative $f'(x)$ for each piece of the function. - Identify critical points where $f'(x) = 0$ or is undefined. - Use the sign of $f'(x)$ to determine intervals of increase or decrease.

STEP 9

Create the variation table based on the analysis of $f'(x)$ .

STEP 10

Study the asymptotic behavior (infinite branches) of $f$ :
- Analyze the behavior of $f(x)$ as $x \to -\infty$ and $x \to \infty$ . - Determine if there are any horizontal, vertical, or oblique asymptotes.

STEP 11

Verify the equation $f(\alpha) = \alpha + 1$ :
- Solve for $\alpha$ such that the equation holds true. - Substitute $\alpha$ into the appropriate piece of $f(x)$ and verify the equality.

STEP 12

Construct the graph of $C_f$ :
- Use the information from the domain, continuity, differentiability, variations, and asymptotic behavior. - Plot key points, asymptotes, and the general shape of the curve.

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Studdy Solution

STEP 1

1. The function $f(x)$ is piecewise, with different expressions for $x < 0$ and $x \geq 0$ .
2. We need to analyze the domain, continuity, differentiability, variations, asymptotic behavior, and specific function values.
3. The function involves logarithmic, exponential, and polynomial expressions.

STEP 2

STEP 3

STEP 4

STEP 5

Calculate the left-hand limit:
$\lim_{x \to 0^-} \left(1 + \frac{x-1}{\ln |x|}\right)$
This requires careful analysis of the behavior of $\ln |x|$ as $x \to 0^-$ .

STEP 6

Calculate the right-hand limit:
$\lim_{x \to 0^+} \left((x^2 + 2x)e^{-x} + e^{-x}\right)$
This involves evaluating the behavior of the exponential terms as $x \to 0^+$ .

STEP 7

Determine if $f(x)$ is differentiable at $x = 0$ by checking if the left-hand and right-hand derivatives exist and are equal.

STEP 8

Analyze the variations of $f$ :
- Determine the derivative $f'(x)$ for each piece of the function. - Identify critical points where $f'(x) = 0$ or is undefined. - Use the sign of $f'(x)$ to determine intervals of increase or decrease.

STEP 9

Create the variation table based on the analysis of $f'(x)$ .

STEP 10

Study the asymptotic behavior (infinite branches) of $f$ :
- Analyze the behavior of $f(x)$ as $x \to -\infty$ and $x \to \infty$ . - Determine if there are any horizontal, vertical, or oblique asymptotes.

STEP 11

Verify the equation $f(\alpha) = \alpha + 1$ :
- Solve for $\alpha$ such that the equation holds true. - Substitute $\alpha$ into the appropriate piece of $f(x)$ and verify the equality.

STEP 12

Construct the graph of $C_f$ :
- Use the information from the domain, continuity, differentiability, variations, and asymptotic behavior. - Plot key points, asymptotes, and the general shape of the curve.

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Studdy Solution

STEP 1

STEP 2

STEP 3

STEP 4

STEP 5

Calculate the left-hand limit:lim⁡x→0−(1+x−1ln⁡∣x∣) \lim_{x \to 0^-} \left(1 + \frac{x-1}{\ln |x|}\right) x→0−lim​(1+ln∣x∣x−1​)This requires careful analysis of the behavior of ln⁡∣x∣\ln |x|ln∣x∣ as x→0− x \to 0^- x→0−.

STEP 6

Calculate the right-hand limit:lim⁡x→0+((x2+2x)e−x+e−x) \lim_{x \to 0^+} \left((x^2 + 2x)e^{-x} + e^{-x}\right) x→0+lim​((x2+2x)e−x+e−x)This involves evaluating the behavior of the exponential terms as x→0+ x \to 0^+ x→0+.

STEP 7

Determine if f(x) f(x) f(x) is differentiable at x=0 x = 0 x=0 by checking if the left-hand and right-hand derivatives exist and are equal.

STEP 8

Analyze the variations of f f f:- Determine the derivative f′(x) f'(x) f′(x) for each piece of the function. - Identify critical points where f′(x)=0 f'(x) = 0 f′(x)=0 or is undefined. - Use the sign of f′(x) f'(x) f′(x) to determine intervals of increase or decrease.

STEP 9

Create the variation table based on the analysis of f′(x) f'(x) f′(x).

STEP 10

Study the asymptotic behavior (infinite branches) of f f f:- Analyze the behavior of f(x) f(x) f(x) as x→−∞ x \to -\infty x→−∞ and x→∞ x \to \infty x→∞. - Determine if there are any horizontal, vertical, or oblique asymptotes.

STEP 11

Verify the equation f(α)=α+1 f(\alpha) = \alpha + 1 f(α)=α+1:- Solve for α\alphaα such that the equation holds true. - Substitute α\alphaα into the appropriate piece of f(x) f(x) f(x) and verify the equality.

STEP 12

Construct the graph of Cf C_f Cf​:- Use the information from the domain, continuity, differentiability, variations, and asymptotic behavior. - Plot key points, asymptotes, and the general shape of the curve.

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Calculate the left-hand limit:
$\lim_{x \to 0^-} \left(1 + \frac{x-1}{\ln |x|}\right)$
This requires careful analysis of the behavior of $\ln |x|$ as $x \to 0^-$ .

Calculate the right-hand limit:
$\lim_{x \to 0^+} \left((x^2 + 2x)e^{-x} + e^{-x}\right)$
This involves evaluating the behavior of the exponential terms as $x \to 0^+$ .

Determine if $f(x)$ is differentiable at $x = 0$ by checking if the left-hand and right-hand derivatives exist and are equal.

Analyze the variations of $f$ :
- Determine the derivative $f'(x)$ for each piece of the function. - Identify critical points where $f'(x) = 0$ or is undefined. - Use the sign of $f'(x)$ to determine intervals of increase or decrease.

Create the variation table based on the analysis of $f'(x)$ .

Study the asymptotic behavior (infinite branches) of $f$ :
- Analyze the behavior of $f(x)$ as $x \to -\infty$ and $x \to \infty$ . - Determine if there are any horizontal, vertical, or oblique asymptotes.

Verify the equation $f(\alpha) = \alpha + 1$ :
- Solve for $\alpha$ such that the equation holds true. - Substitute $\alpha$ into the appropriate piece of $f(x)$ and verify the equality.

Construct the graph of $C_f$ :
- Use the information from the domain, continuity, differentiability, variations, and asymptotic behavior. - Plot key points, asymptotes, and the general shape of the curve.