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...
Related questions
Question
do no.2 pls and show full work
![### Calculus Problem Set
#### (1) Find the limit.
Say why LH rule (L’Hôpital’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 the 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 \( y = \sin(x) \).
#### (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 lengths in meters.
- (b) If \( dx/dt = 3 \), find \( dy/dt \) when \( x = -1 \).
- Note: Both \( x \) and \( y \) have negative values](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F92b3472e-ab59-40f2-b296-ac3727a4d9f3%2Fc8f48d61-4379-40e5-ae8a-4746b1fd8f00%2Fwvvh7qf_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculus Problem Set
#### (1) Find the limit.
Say why LH rule (L’Hôpital’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 the 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 \( y = \sin(x) \).
#### (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 lengths in meters.
- (b) If \( dx/dt = 3 \), find \( dy/dt \) when \( x = -1 \).
- Note: Both \( x \) and \( y \) have negative values
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning