9. Graph each of the following functions, using the steps given at the end of this assign- ment sheet. Note: In parts (a) and (d), you will need to get decimal approximations for the coordinates of the local extrema and the inflection points. You will also need to know that (a) f(x) = xe* (b) f(x) = lim xe* = 0, x118 We will verify these limits using L'Hospital's rule, which is covered in the next home- work. x+1 x - 3 lim x ln(x) = 0. +0+x (c) f(x) 2 x² = 2 X 4 (d) f(x) = x ln x Hint for (9c)¹
9. Graph each of the following functions, using the steps given at the end of this assign- ment sheet. Note: In parts (a) and (d), you will need to get decimal approximations for the coordinates of the local extrema and the inflection points. You will also need to know that (a) f(x) = xe* (b) f(x) = lim xe* = 0, x118 We will verify these limits using L'Hospital's rule, which is covered in the next home- work. x+1 x - 3 lim x ln(x) = 0. +0+x (c) f(x) 2 x² = 2 X 4 (d) f(x) = x ln x Hint for (9c)¹
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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This is just a practice problem, not graded
![**Graphing Functions Assignment**
**Problem 9:**
Graph each of the following functions using the steps at the end of this assignment sheet. **Note:** In parts (a) and (d), you'll need to get decimal approximations for the coordinates of the local extrema and inflection points. You will also need to know:
\[
\lim_{x \to \infty} xe^x = 0, \quad \lim_{x \to 0^+} x \ln(x) = 0.
\]
We will verify these limits using L'Hospital's rule, which is covered in the next homework.
(a) \( f(x) = xe^x \)
(b) \( f(x) = \frac{x + 1}{x - 3} \)
(c) \( f(x) = \frac{x^2}{x^2 - 4} \) \quad *Hint for (9c): To simplify \( f''(x) \), first pull a factor of \( x^2 - 4 \) out of the numerator.*
(d) \( f(x) = x \ln x \)
*OVER for Steps for Graphing a Function* →
---
**Steps for Graphing a Function:**
1. Find the domain of \( f(x) \).
2. Find the vertical asymptotes of \( f(x) \) (if any).
3. Find the horizontal asymptotes of \( f(x) \) (if any).
4. Find the intervals on which \( f(x) \) is increasing/decreasing, and find the x- and y-coordinates of all the local extrema.
5. Find the intervals on which \( f(x) \) is concave up/concave down, and find the x- and y-coordinates of all the inflection points.
6. Find the x-intercept(s) (if algebraically possible) and the y-intercept of \( f(x) \).
7. Plot the specific points that you have found, and then sketch the graph of \( f(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e5ea172-2838-4041-9e7f-74e8cb5bb4e3%2Fb35b81ae-0266-4430-8a1b-d2903ac92f0f%2F84kqwta_processed.png&w=3840&q=75)
Transcribed Image Text:**Graphing Functions Assignment**
**Problem 9:**
Graph each of the following functions using the steps at the end of this assignment sheet. **Note:** In parts (a) and (d), you'll need to get decimal approximations for the coordinates of the local extrema and inflection points. You will also need to know:
\[
\lim_{x \to \infty} xe^x = 0, \quad \lim_{x \to 0^+} x \ln(x) = 0.
\]
We will verify these limits using L'Hospital's rule, which is covered in the next homework.
(a) \( f(x) = xe^x \)
(b) \( f(x) = \frac{x + 1}{x - 3} \)
(c) \( f(x) = \frac{x^2}{x^2 - 4} \) \quad *Hint for (9c): To simplify \( f''(x) \), first pull a factor of \( x^2 - 4 \) out of the numerator.*
(d) \( f(x) = x \ln x \)
*OVER for Steps for Graphing a Function* →
---
**Steps for Graphing a Function:**
1. Find the domain of \( f(x) \).
2. Find the vertical asymptotes of \( f(x) \) (if any).
3. Find the horizontal asymptotes of \( f(x) \) (if any).
4. Find the intervals on which \( f(x) \) is increasing/decreasing, and find the x- and y-coordinates of all the local extrema.
5. Find the intervals on which \( f(x) \) is concave up/concave down, and find the x- and y-coordinates of all the inflection points.
6. Find the x-intercept(s) (if algebraically possible) and the y-intercept of \( f(x) \).
7. Plot the specific points that you have found, and then sketch the graph of \( f(x) \).
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