Q1(a). Solve for :

Background

Topic: Linear Equations

This question tests your ability to solve a basic linear equation for the variable .

Key Terms and Formulas:

• Linear equation: An equation of the form .

• To solve, isolate by performing inverse operations.

Step-by-Step Guidance

1. Subtract $3x$.

2. Divide both sides by $2x$.

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Q1(b). Solve for :

Background

Topic: Logarithmic Equations

This question tests your ability to use properties of logarithms to solve for .

Key Terms and Formulas:

• Logarithm properties:

• Exponentiation: If , then

Step-by-Step Guidance

1. Combine the logarithms using the quotient rule: .

2. Exponentiate both sides to remove the logarithm: .

3. Multiply both sides by to clear the denominator.

4. Rearrange the equation to isolate on one side.

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Q1(c). Solve for :

Background

Topic: Exponential Equations

This question tests your ability to solve equations involving exponents by expressing both sides with the same base.

Key Terms and Formulas:

• Exponent rules:

• Set exponents equal if bases are the same.

Step-by-Step Guidance

1. Express $49.

2. Rewrite as $7$ raised to a negative exponent.

3. Set the exponents equal to each other since the bases are the same.

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Q2(a). Compute

Background

Topic: Limits and Factoring

This question tests your ability to evaluate limits, especially when the denominator factors and may cause an indeterminate form.

Key Terms and Formulas:

• Limit: is the value approaches as approaches .

• Factoring quadratics:

Step-by-Step Guidance

1. Factor the denominator .

2. Check if causes the denominator to be zero (possible indeterminate form).

3. If so, see if the numerator also becomes zero at (0/0 form).

4. If you have a form, try to simplify the expression by canceling common factors.

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Q2(b). Compute

Background

Topic: Limits at Infinity, Rational Functions

This question tests your understanding of how rational functions behave as approaches infinity.

Key Terms and Formulas:

• For , if degrees are equal, the limit is .

• Divide numerator and denominator by the highest power of in the denominator.

Step-by-Step Guidance

1. Identify the highest power of in both numerator and denominator.

2. Divide every term in the numerator and denominator by .

3. Simplify the expression and analyze the limit as .

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Q2(c). Compute

Background

Topic: Infinite Limits

This question tests your understanding of limits where the denominator approaches zero, possibly leading to infinity or negative infinity.

Key Terms and Formulas:

• If the denominator approaches zero and the numerator is nonzero, the limit may be infinite or does not exist.

• Check the direction from which approaches the value (from left or right).

Step-by-Step Guidance

1. Substitute into the denominator to see if it becomes zero.

2. Analyze the sign of the denominator as approaches $2$ from the left and right.

3. Determine if the limit approaches , , or does not exist.

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Q3. Suppose you invest $5,000 in an account with an annual interest rate (compounded continuously) at 7 percent. How many years pass until there is $15,000 in the account?

Background

Topic: Exponential Growth, Continuous Compounding

This question tests your ability to use the formula for continuous compounding to solve for time.

Key Terms and Formulas:

• Continuous compounding formula:

• = final amount, = initial principal, = annual rate (decimal), = time in years

Step-by-Step Guidance

1. Write the formula: , and plug in , , .

2. Divide both sides by to isolate the exponential term.

3. Take the natural logarithm of both sides to solve for .

4. Rearrange to get by itself.

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Q4. Suppose describes the population (in hundreds) of zombies after years. Compute the instantaneous rate of change of the population after 5 years.

Background

Topic: Derivatives, Rates of Change

This question tests your ability to find the derivative of a rational function and evaluate it at a specific point.

Key Terms and Formulas:

• Derivative: gives the instantaneous rate of change.

• Quotient rule:

Step-by-Step Guidance

1. Identify and .

2. Compute and .

3. Apply the quotient rule to find .

4. Plug in into your derivative expression (but do not compute the final value yet).

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Q5. Use the limit definition of the derivative to find if . What is the equation of the tangent line to the graph of at ? Do the same for .

Background

Topic: Derivative Definition, Tangent Lines

This question tests your ability to use the limit definition of the derivative and to find the equation of a tangent line.

Key Terms and Formulas:

• Limit definition:

• Tangent line at :

Step-by-Step Guidance

1. For , write and set up the difference quotient.

2. Expand and simplify the numerator.

3. Take the limit as to find .

4. Evaluate and to set up the tangent line equation.

5. Repeat the process for .

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Q6(a). Compute the derivative:

Background

Topic: Basic Differentiation Rules

This question tests your ability to differentiate polynomials using the power rule.

Key Terms and Formulas:

• Power rule:

• Sum/difference rule: Differentiate term by term.

Step-by-Step Guidance

1. Differentiate using the power rule.

2. Differentiate using the power rule.

3. Combine the results for the derivative .

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Q6(b). Compute the derivative:

Background

Topic: Quotient Rule, Chain Rule

This question tests your ability to use the quotient rule and differentiate functions involving roots and powers.

Key Terms and Formulas:

• Quotient rule:

• Derivative of :

Step-by-Step Guidance

1. Let , .

2. Find and .

3. Apply the quotient rule formula.

4. Simplify the numerator as much as possible.

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Q6(c). Compute the derivative:

Background

Topic: Quotient Rule, Product Rule

This question tests your ability to use the quotient rule and product rule together.

Key Terms and Formulas:

• Quotient rule:

• Product rule:

Step-by-Step Guidance

1. Let and .

2. Use the product rule to find .

3. Find .

4. Apply the quotient rule formula.

5. Simplify the numerator as much as possible.

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Q7. Let . Compute . For what values of does ?

Background

Topic: Derivatives, Critical Points

This question tests your ability to use the quotient rule to find the derivative and solve for critical points.

Key Terms and Formulas:

• Quotient rule:

• Critical points: Where

Step-by-Step Guidance

1. Let , .

2. Find and .

3. Apply the quotient rule to find .

4. Set the numerator of equal to zero to find where .

5. Solve for .

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