Q1. Solve the equation:

Background

Topic: Exponential and Logarithmic Equations

This question tests your understanding of logarithmic properties and how to solve equations involving natural logarithms.

Key Terms and Formulas

• Natural logarithm: is the logarithm base .

• Logarithm property:

• To solve , recall that .

Step-by-Step Guidance

1. Combine the two logarithms using the property .

2. Set the resulting logarithmic expression equal to $0$ and rewrite it in exponential form.

3. Solve the resulting equation for .

4. Check for any extraneous solutions by ensuring the arguments of all logarithms are positive.

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Q2. Find the exact value:

Background

Topic: Properties of Logarithms

This question tests your ability to use logarithm addition properties and evaluate logarithms.

Key Terms and Formulas

• Logarithm addition:

• Change of base and evaluating logarithms.

Step-by-Step Guidance

1. Combine the two logarithms using the addition property.

2. Multiply the arguments: .

3. Express the product as a power of $15$ if possible.

4. Evaluate the logarithm using the definition: .

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Q3. Compute the limit:

Background

Topic: Limits and Trigonometric Functions

This question tests your understanding of limits involving trigonometric functions, especially the standard limit .

Key Terms and Formulas

• Standard limit:

• Algebraic manipulation to match the standard form.

Step-by-Step Guidance

1. Rewrite the limit to resemble the standard form .

2. Factor and multiply numerator and denominator as needed to introduce in the denominator.

3. Express the limit in terms of and a constant multiple.

4. Recall the value of the standard limit and use it to simplify the expression.

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Q4. Using the Intermediate Value Theorem, determine on which interval the equation has a root:

Background

Topic: Intermediate Value Theorem (IVT)

This question tests your understanding of the IVT, which states that if a function is continuous on and takes values of opposite sign at and , then it must cross zero somewhere in .

Key Terms and Formulas

• Intermediate Value Theorem: If is continuous on and and have opposite signs, then $f$ has a root in .

• Evaluate at the endpoints of each interval.

Step-by-Step Guidance

1. Define .

2. Compute at the endpoints of each interval: .

3. Check the sign of at each endpoint.

4. Identify the interval(s) where changes sign between endpoints.

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Q5. Compute the derivative of at and find where the tangent line intersects the y-axis.

Background

Topic: Derivatives and Tangent Lines

This question tests your ability to compute derivatives using the product rule and to find the equation of a tangent line, then determine its y-intercept.

Key Terms and Formulas

• Product rule:

• Equation of tangent line:

• To find the y-intercept, set in the tangent line equation.

Step-by-Step Guidance

1. Compute using the product rule for .

2. Evaluate at .

3. Find .

4. Write the equation of the tangent line at .

5. Set in the tangent line equation to find the y-intercept.

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