Podcast
Questions and Answers
Evaluate the limit as $x$ approaches 3 for the expression $\frac{x^2 - 9}{x^2 - 3x}$. What is the result?
Evaluate the limit as $x$ approaches 3 for the expression $\frac{x^2 - 9}{x^2 - 3x}$. What is the result?
What is the value of the limit as $x$ approaches -3 for $\frac{x^2 - 9}{1 - \cos x}$?
What is the value of the limit as $x$ approaches -3 for $\frac{x^2 - 9}{1 - \cos x}$?
When evaluating $\lim_{x \to -\infty} \frac{x - 9x - 5}{3x^3 + x - 2}$, what is the result?
When evaluating $\lim_{x \to -\infty} \frac{x - 9x - 5}{3x^3 + x - 2}$, what is the result?
What is the result for $\lim_{t \to 0} \frac{t^2}{x^2 - 81}$?
What is the result for $\lim_{t \to 0} \frac{t^2}{x^2 - 81}$?
Signup and view all the answers
Determine if the function $f(x) = \begin{cases} \cos(x - 2) & \text{if } x \geq 2 \ 1 + x & \text{if } x < 2 \end{cases}$ is continuous at $x = 2$. What is the outcome?
Determine if the function $f(x) = \begin{cases} \cos(x - 2) & \text{if } x \geq 2 \ 1 + x & \text{if } x < 2 \end{cases}$ is continuous at $x = 2$. What is the outcome?
Signup and view all the answers
Identify the horizontal asymptote of the function $f(x) = \frac{x^4 - 8x^2 + 7}{1 + x}$. What is the result?
Identify the horizontal asymptote of the function $f(x) = \frac{x^4 - 8x^2 + 7}{1 + x}$. What is the result?
Signup and view all the answers
When is the particle described by the motion equation $s(t) = t^3 - 13t^2 + 35t - 15$ at rest?
When is the particle described by the motion equation $s(t) = t^3 - 13t^2 + 35t - 15$ at rest?
Signup and view all the answers
For the function $f(x) = \frac{2x + 1}{x^2 - 5x + 6}$, what is the limit as $x$ approaches 2?
For the function $f(x) = \frac{2x + 1}{x^2 - 5x + 6}$, what is the limit as $x$ approaches 2?
Signup and view all the answers
Study Notes
Limit Evaluations
- For limit evaluations, key techniques involve direct substitution, factoring, and recognizing indeterminate forms.
- a) Evaluate (\lim_{x \to 3} \frac{x^2 - 9}{x^2 - 3x}) to find a finite value or determine if it diverges.
- b) Evaluate (\lim_{x \to -3^+} \frac{x^2 - 9}{1 - \cos x}) to analyze behavior approaching the limit.
- c) Assess (\lim_{x \to \frac{\pi}{2}} \frac{x \sin(4t)}{\tan(3t)}) to determine the limit's behavior as (x) approaches (\frac{\pi}{2}).
- d) Check (\lim_{t \to 0} \frac{t^2}{x^2 - 81}) for limit value evaluation.
- e) Analyze (\lim_{x \to 9} \frac{\sqrt{x} - 3}{x^2 - 3x^3 + 1}) for finite or infinite limits.
- f) Find (\lim_{x \to -\infty} \frac{x - 9x - 5}{3x^3 + x - 2}) to investigate limit behavior at negative infinity.
- g) Evaluate (\lim_{x \to -\infty} \sqrt{x^3 + 1 + x^6}) for limit trends as (x) approaches negative infinity.
Differentiation
- a) Differentiate (y = \frac{(2x + x^4) \cos x}{\sec x + x}) by applying quotient and product rules.
- b) Differentiate (y = \frac{\tan x - x}{3}) using the quotient rule.
- c) Differentiate (y = (3x^2 - \sin x + 1)(\csc x + \sqrt{x})) using product rule.
- d) Find derivative (y = \frac{e^x \cos x}{1 - \sqrt{x} x}) using the quotient and product rules.
- e) Differentiate (y = \frac{1}{2 + 3\sqrt{x}}) applying the chain rule as necessary.
Higher-Order Derivatives
- a) Find the second derivative (y = 3x^4 - \sqrt{x} + \pi - x).
- b) Differentiate (y = e^x \sin x) to investigate second derivatives.
Definition of the Derivative
- Use the limit definition of the derivative to derive (f'(x)) for (f(x) = 2x + 1).
Finding Horizontal Tangents
- Identify points where the derivative of (y = x^4 - 8x^2 + 7) equals zero to find horizontal tangents.
Tangent Lines
- Establish equation of the tangent line at the point ((-1, 0)) for the curve (f(x) = \frac{2x - 5x + 7}{x < 2}).
Continuity of Functions
- Determine continuity of piecewise function (f(x) = \begin{cases} \cos(x - 2) & \text{if } x \geq 2 \end{cases}) at (x = 2).
Asymptotic Behavior
- Evaluate limits as (x) approaches positive and negative infinities to identify horizontal and vertical asymptotes.
- Assess continuity and differentiability of the function across its domain.
Particle Motion Analysis
- Determine the velocity of the particle from the motion equation (s(t) = t^3 - 13t^2 + 35t - 15).
- Calculate velocity at specific time instances, analyze conditions for rest, and determine directions of movement.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your knowledge on evaluating limits and differentiation techniques with this quiz. Each question presents a different scenario to apply the fundamental concepts of calculus. Challenge yourself to find limits approaching finite and infinite values, along with differentiating complex functions.
