Q1. Simplify: \( \frac{x^2 - 9}{x - 3} \)

Background

Topic: Algebraic Simplification

This question tests your ability to factor and simplify rational expressions, a skill often needed when working with limits and derivatives in calculus.

Key Terms and Formulas:

• Factoring: Expressing a polynomial as a product of its factors.

• Simplifying Rational Expressions: Canceling common factors in the numerator and denominator.

Step-by-Step Guidance

1. Factor the numerator \( x^2 - 9 \) as a difference of squares.

2. Write \( x^2 - 9 = (x - 3)(x + 3) \).

3. Rewrite the expression as \( \frac{(x - 3)(x + 3)}{x - 3} \).

4. Identify and cancel any common factors in the numerator and denominator, being careful to note any restrictions on the variable.

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Q2. Solve: \( e^{2x} = 7 \)

Background

Topic: Solving Exponential Equations

This question tests your ability to solve equations involving exponentials, which is important for calculus problems involving growth, decay, and logarithmic differentiation.

Key Terms and Formulas:

• Exponential Equation: An equation where the variable is in the exponent.

• Natural Logarithm: \( \ln(x) \) is the inverse of \( e^x \).

• Key property: \( \ln(e^a) = a \).

Step-by-Step Guidance

1. Start with the equation: \( e^{2x} = 7 \).

2. Take the natural logarithm (\( \ln \)) of both sides to bring the exponent down.

3. Use the property \( \ln(e^{2x}) = 2x \) to simplify the left side.

4. Solve for \( x \) by dividing both sides by 2.

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Q3. Simplify: \( \frac{1}{1 + \tan^2 x} \)

Background

Topic: Trigonometric Identities

This question tests your knowledge of fundamental trigonometric identities, which are essential for calculus, especially in integration and differentiation involving trigonometric functions.

Key Terms and Formulas:

• Pythagorean Identity: \( 1 + \tan^2 x = \sec^2 x \)

• Reciprocal Identity: \( \sec x = \frac{1}{\cos x} \)

Step-by-Step Guidance

1. Recall the Pythagorean identity involving tangent and secant.

2. Substitute \( 1 + \tan^2 x \) with \( \sec^2 x \).

3. Rewrite the expression as \( \frac{1}{\sec^2 x} \).

4. Use the reciprocal identity to further simplify the expression.

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Q4. Verify: \( \sin^2 x + \cos^2 x = 1 \)

Background

Topic: Fundamental Trigonometric Identities

This question asks you to verify the most basic Pythagorean identity, which is foundational for all trigonometric manipulations in calculus.

Key Terms and Formulas:

• Pythagorean Identity: \( \sin^2 x + \cos^2 x = 1 \)

• Unit Circle Definition of Sine and Cosine

Step-by-Step Guidance

1. Recall the definition of sine and cosine on the unit circle: for any angle \( x \), \( \sin x \) and \( \cos x \) are the y- and x-coordinates, respectively, of a point on the unit circle.

2. By the Pythagorean Theorem, the sum of the squares of the coordinates equals 1.

3. Write the equation \( (\cos x)^2 + (\sin x)^2 = 1 \) and recognize this as the identity to verify.

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Q5. Solve: \( 2 \sin x = 1 \) on the interval \( [0, 2\pi] \)

Background

Topic: Solving Trigonometric Equations

This question tests your ability to solve basic trigonometric equations and find all solutions within a given interval, a skill needed for calculus problems involving periodic functions.

Key Terms and Formulas:

• Inverse Sine Function: \( \sin^{-1}(x) \)

• Unit Circle Values

Step-by-Step Guidance

1. Divide both sides by 2 to isolate \( \sin x \).

2. Write \( \sin x = \frac{1}{2} \).

3. Recall the values of \( x \) in \( [0, 2\pi] \) where \( \sin x = \frac{1}{2} \).

4. List all such values within the given interval.

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