Q1. Find the limit: \( \lim_{x \to 2} (3x^2 - 5x + 4) \)

Background

Topic: Limits and Continuity

This question tests your understanding of how to evaluate limits of polynomial functions as \( x \) approaches a specific value.

Key Terms and Formulas:

• Limit: The value that a function approaches as the input approaches some value.

• For polynomials, \( \lim_{x \to a} f(x) = f(a) \) (direct substitution works if the function is continuous at \( a \)).

Step-by-Step Guidance

1. Recognize that the function \( 3x^2 - 5x + 4 \) is a polynomial, which is continuous everywhere.

2. Apply the direct substitution property for limits of polynomials: substitute \( x = 2 \) into the function.

3. Set up the substitution: \( 3(2)^2 - 5(2) + 4 \).

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Q2. Suppose f(x) has a vertical asymptote at x = 1. Which of the following could be the graph of f(x)?

Background

Topic: Asymptotes and Graphs of Functions

This question tests your ability to identify vertical asymptotes in the graph of a function.

Key Terms and Formulas:

• Vertical Asymptote: A line \( x = a \) where the function grows without bound as \( x \) approaches \( a \).

Step-by-Step Guidance

1. Recall that a vertical asymptote occurs where the function is undefined and the values approach infinity or negative infinity.

2. Examine each graph and look for a vertical line at \( x = 1 \) where the function diverges.

3. Eliminate any graphs that do not have this behavior at \( x = 1 \).

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Q3. Find the derivative of \( f(x) = x^3 - 2x + 1 \).

Background

Topic: Differentiation

This question tests your ability to apply the power rule and sum rule for derivatives.

Key Terms and Formulas:

• Derivative: Measures the rate at which a function changes as its input changes.

• Power Rule: \( \frac{d}{dx} x^n = n x^{n-1} \)

• Sum Rule: The derivative of a sum is the sum of the derivatives.

Step-by-Step Guidance

1. Apply the power rule to each term: \( x^3 \), \( -2x \), and \( 1 \).

2. For \( x^3 \), the derivative is \( 3x^2 \).

3. For \( -2x \), the derivative is \( -2 \).

4. For the constant \( 1 \), the derivative is \( 0 \).

5. Combine the results to write the derivative function, but do not simplify to the final answer yet.

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Q4. Evaluate the definite integral: \( \int_0^2 (4x - 1) dx \)

Background

Topic: Definite Integrals

This question tests your ability to compute the definite integral of a linear function over a given interval.

Key Terms and Formulas:

• Definite Integral: Represents the signed area under the curve from \( a \) to \( b \).

• \( \int_a^b f(x) dx = F(b) - F(a) \), where \( F(x) \) is any antiderivative of \( f(x) \).

Step-by-Step Guidance

1. Find the antiderivative of \( 4x - 1 \). Recall that the antiderivative of \( x \) is \( \frac{1}{2}x^2 \), and the antiderivative of a constant is the constant times \( x \).

2. Write the general antiderivative: \( F(x) = 2x^2 - x \).

3. Apply the Fundamental Theorem of Calculus: \( F(2) - F(0) \).

4. Set up the expressions for \( F(2) \) and \( F(0) \), but do not compute the final value yet.

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Q5. The function \( f(x) = \frac{1}{x^2} \) is continuous for which values of \( x \)?

Background

Topic: Continuity of Functions

This question tests your understanding of where rational functions are continuous.

Key Terms and Formulas:

• Continuous Function: A function is continuous at \( x = a \) if \( \lim_{x \to a} f(x) = f(a) \).

• Rational functions are continuous everywhere they are defined (denominator not zero).

Step-by-Step Guidance

1. Identify the denominator of \( f(x) = \frac{1}{x^2} \), which is \( x^2 \).

2. Set the denominator not equal to zero: \( x^2 \neq 0 \).

3. Solve for the values of \( x \) where the function is defined and thus continuous.

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