Calculus III Prerequisite Review – Step-by-Step Study Guidance

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.

Try solving on your own before revealing the answer!

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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