Topic 1: Functions and Their Properties
Easy Questions 🌱
Find the domain of \( f(x) = \frac{1}{x - 2} \).
Determine if \( f(x) = x^2 \) is even, odd, or neither.
Compute \( f(2) \) for \( f(x) = 3x - 1 \).
Find the range of \( f(x) = x^2 + 1 \) for \( x \geq 0 \).
Is \( f(x) = |x| \) a one-to-one function?
Moderate Questions 🌱
Find the inverse of \( f(x) = 2x + 3 \).
Determine the domain and range of \( f(x) = \sqrt{x - 1} \).
Is \( f(x) = \frac{1}{x^2} \) even, odd, or neither?
Compute the composition \( (f \circ g)(x) \) where \( f(x) = x^2 \) and \( g(x) = x + 1 \).
Hard Questions 🌟
Find the inverse of \( f(x) = \frac{2x + 1}{x - 1} \) and its domain.
Prove that \( f(x) = e^x \) is a one-to-one function.
Topic 2: Limits and Their Intuitive Meaning
Easy Questions 🌱
Guess \( \lim_{x \to 2} x^2 \) using a table of values.
Determine if \( \lim_{x \to 0} \frac{|x|}{x} \) exists.
Estimate \( \lim_{x \to 1} \frac{x^2 - 1}{x - 1} \) intuitively.
What is \( \lim_{x \to 3} (2x + 1) \)?
Moderate Questions 🌱
Guess \( \lim_{x \to 0} \frac{\sin x}{x} \) using a graph.
Determine if \( \lim_{x \to 0} \frac{1 - \cos x}{x^2} \) exists intuitively.
Estimate \( \lim_{x \to \infty} \frac{2x + 1}{x} \) using behavior analysis.
Hard Questions 🌟
Intuitively determine \( \lim_{x \to 0^+} \frac{\ln x}{x} \) and explain.
Guess \( \lim_{x \to 0} \frac{e^x - 1}{x} \) using numerical values.
Topic 3: Limit Laws and Evaluation Techniques
Easy Questions 🌱
Evaluate \( \lim_{x \to 2} (3x^2 - 4x + 1) \) using limit laws.
Find \( \lim_{x \to 0} (2x + 5) \) using direct substitution.
Compute \( \lim_{x \to 1} \frac{x^2 - 1}{x - 1} \).
Evaluate \( \lim_{x \to \infty} \frac{3}{x} \).
Moderate Questions 🌱
Find \( \lim_{x \to 0} \frac{\sin 2x}{x} \) using limit laws.
Evaluate \( \lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4} \) by rationalizing.
Compute \( \lim_{x \to \infty} \frac{2x^2 + 1}{x^2 - x} \).
Hard Questions 🌟
Evaluate \( \lim_{x \to 0} \frac{e^x - 1 - x}{x^2} \) using L'Hôpital's rule.
Find \( \lim_{x \to 0^+} x \ln x \) using a change of variables.
Topic 4: Continuity and Types of Discontinuities
Easy Questions 🌱
Is \( f(x) = x^2 \) continuous at \( x = 1 \)?
Determine if \( f(x) = \frac{1}{x} \) is continuous at \( x = 0 \).
Identify the type of discontinuity at \( x = 2 \) for \( f(x) = \frac{x - 2}{x^2 - 4} \).
Moderate Questions 🌱
Is \( f(x) = \begin{cases} x^2 & x < 1 \\ 2x & x \geq 1 \end{cases} \) continuous at \( x = 1 \)?
Determine the type of discontinuity at \( x = 0 \) for \( f(x) = \sin(1/x) \).
Check continuity of \( f(x) = \frac{x^2 - 4}{x - 2} \) at \( x = 2 \).
Hard Questions 🌟
Prove \( f(x) = \frac{x^2 - 1}{x - 1} \) is continuous everywhere except at \( x = 1 \).
Identify all discontinuities and their types for \( f(x) = \begin{cases} \frac{1}{x} & x \neq 0 \\ 0 & x = 0 \end{cases} \).
Topic 5: Introduction to the Concept of a Derivative
Easy Questions 🌱
Find the derivative of \( f(x) = x^2 \) using the definition.
Compute the slope of \( f(x) = 3x + 2 \) at \( x = 1 \).
Determine the derivative of \( f(x) = x^3 \) at \( x = 2 \).
Moderate Questions 🌱
Find the derivative of \( f(x) = \sqrt{x} \) using the limit definition.
Compute the derivative of \( f(x) = \frac{1}{x} \) at \( x = -1 \).
Determine the instantaneous rate of change of \( f(x) = x^2 + x \) at \( x = 0 \).
Hard Questions 🌟
Find the derivative of \( f(x) = \sin x \) using the definition.
Compute the derivative of \( f(x) = \frac{1}{x^2} \) at \( x = 1 \) using first principles.