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.3 pls and show full work
![1. **Find the limit.**
Say why LH 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 \( 4x^2 + y^2 = 25 \)**
(a) If \( dy/dt = 1/3 \). Find \( dx/dt \) when \( x = 2 \). (**Note:** \( x \) and \( y \) are length in meters)
(b) If \( dx/dt = 3 \), find \( dy/dt \) when \( x =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F92b3472e-ab59-40f2-b296-ac3727a4d9f3%2F1d204881-6652-4da1-bb59-b443ad6e4cb1%2F03pora_processed.png&w=3840&q=75)
Transcribed Image Text:1. **Find the limit.**
Say why LH 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 \( 4x^2 + y^2 = 25 \)**
(a) If \( dy/dt = 1/3 \). Find \( dx/dt \) when \( x = 2 \). (**Note:** \( x \) and \( y \) are length in meters)
(b) If \( dx/dt = 3 \), find \( dy/dt \) when \( x =
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