(1) Find the limit. Say why LH rule (l'Hospital's rule) applies to this case and then use that sin x - x rule to find the limit lim x-0 (e²x) - 1
(1) Find the limit. Say why LH rule (l'Hospital's rule) applies to this case and then use that sin x - x rule to find the limit lim x-0 (e²x) - 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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do no.1 pls and show full work.
![### Calculus Practice Problems
#### Problem Set:
1. **Find the limit.** Say why L'Hospital's rule (L'Hospital's rule) applies to this case and then use that rule to find the limit.
\[
\lim_{x \to 0} \frac{\sin x - x}{(e^{2x}) - 1}
\]
2. **Calculate the limit.**
\[
\lim_{x \to \infty} \frac{\ln x}{\sqrt{x}}
\]
3. **Find the intervals on which function \( f \) is concaving up or concaving down and find the inflection point(s).**
\[
f(x) = x^3 - 3x^2 - 9x + 4
\]
4. **Find the critical numbers of the function.**
\[
f(x) = \frac{x^2 + 2}{2x - 1}
\]
5. **Sketch the function. (Must show the details)**
\[
f(x) =
\begin{cases}
x^2 & \text{when } -1 \leq x \leq 0 \\
2 - 2x & \text{when } 0 < x \leq 1
\end{cases}
\]
6. **If \( f(x) = \frac{x^2}{x+1} + \cos x \), find \( f'(1) \).**
7. **Find the equation of the tangent line at \( (\pi, 0) \).** For \( y = \sin (\sin x) \)
8. **Implicit Differentiation problem.** If \( y \cos(x) + x = 5 \), find \( y'' \) where \( x = 0 \).
(MUST use implicit differentiation approach for credit)
9. **Suppose \( 4x^2 + y^2 = 25 \)**:
1. If \( \frac{dy}{dt} = \frac{1}{3} \), find \( \frac{dx}{dt} \) when \( x = 2 \).
**Note:** \( x \) and \( y \) are length in meters.
2. If \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F92b3472e-ab59-40f2-b296-ac3727a4d9f3%2Ff2c2dfab-a921-40a4-b0d5-ee6f008c64a6%2Ftlxtanj_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculus Practice Problems
#### Problem Set:
1. **Find the limit.** Say why L'Hospital's rule (L'Hospital's rule) applies to this case and then use that rule to find the limit.
\[
\lim_{x \to 0} \frac{\sin x - x}{(e^{2x}) - 1}
\]
2. **Calculate the limit.**
\[
\lim_{x \to \infty} \frac{\ln x}{\sqrt{x}}
\]
3. **Find the intervals on which function \( f \) is concaving up or concaving down and find the inflection point(s).**
\[
f(x) = x^3 - 3x^2 - 9x + 4
\]
4. **Find the critical numbers of the function.**
\[
f(x) = \frac{x^2 + 2}{2x - 1}
\]
5. **Sketch the function. (Must show the details)**
\[
f(x) =
\begin{cases}
x^2 & \text{when } -1 \leq x \leq 0 \\
2 - 2x & \text{when } 0 < x \leq 1
\end{cases}
\]
6. **If \( f(x) = \frac{x^2}{x+1} + \cos x \), find \( f'(1) \).**
7. **Find the equation of the tangent line at \( (\pi, 0) \).** For \( y = \sin (\sin x) \)
8. **Implicit Differentiation problem.** If \( y \cos(x) + x = 5 \), find \( y'' \) where \( x = 0 \).
(MUST use implicit differentiation approach for credit)
9. **Suppose \( 4x^2 + y^2 = 25 \)**:
1. If \( \frac{dy}{dt} = \frac{1}{3} \), find \( \frac{dx}{dt} \) when \( x = 2 \).
**Note:** \( x \) and \( y \) are length in meters.
2. If \(
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