Q1. Find critical numbers of a function and determine whether each critical number marks a maximum or a minimum using the First or Second Derivative Test.

Background

Topic: Applications of the Derivative

This question tests your ability to analyze a function using its derivatives to find critical points and classify them as maxima or minima.

Key Terms and Formulas:

• Critical Number: A value where or is undefined.

• First Derivative Test: Uses sign changes in around critical points to determine maxima/minima.

• Second Derivative Test: Uses at critical points to classify them.

Step-by-Step Guidance

1. Find the derivative of the given function.

2. Set and solve for to find critical numbers.

3. Check where is undefined for additional critical points.

4. Apply the First Derivative Test: Examine the sign of before and after each critical number.

5. Alternatively, use the Second Derivative Test: Compute at each critical number and interpret the result.

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Q2. Solve applied optimization problems by writing a relevant function and finding its absolute extrema on an appropriate domain.

Background

Topic: Optimization

This question tests your ability to model a real-world scenario with a function, then use calculus to find the minimum or maximum value.

Key Terms and Formulas:

• Objective Function: The function you want to optimize (minimize or maximize).

• Constraints: Equations or inequalities that restrict the domain.

• Critical Points: Found by setting the derivative of the objective function to zero.

Step-by-Step Guidance

1. Identify the quantity to be optimized and write the objective function.

2. Express the objective function in terms of a single variable using the constraints.

3. Find the derivative of the objective function with respect to that variable.

4. Set the derivative equal to zero and solve for the variable to find critical points.

5. Check endpoints and use the First or Second Derivative Test to confirm extrema.

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Q3. Use implicit differentiation to differentiate a function that is not solved for y.

Background

Topic: Implicit Differentiation

This question tests your ability to find derivatives when is not isolated, using the chain rule.

Key Terms and Formulas:

• Implicit Differentiation: Differentiating both sides of an equation with respect to , treating as a function of .

• Chain Rule:

Step-by-Step Guidance

1. Differentiate both sides of the equation with respect to , applying the chain rule to terms involving .

2. Collect all terms involving on one side of the equation.

3. Solve for algebraically.

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Q4. Use logarithmic differentiation when applicable.

Background

Topic: Logarithmic Differentiation

This question tests your ability to use logarithms to simplify differentiation, especially for products, quotients, or variable exponents.

Key Terms and Formulas:

• Logarithmic Differentiation: Take the natural log of both sides, then differentiate.

• Useful for or complicated products/quotients.

Step-by-Step Guidance

1. Take of both sides: .

2. Simplify using log properties: .

3. Differentiate both sides with respect to .

4. Solve for .

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Q5. Solve related rates problems.

Background

Topic: Related Rates

This question tests your ability to relate the rates of change of different quantities using derivatives.

Key Terms and Formulas:

• Related Rates: Problems where two or more variables change with respect to time.

• Chain Rule:

Step-by-Step Guidance

1. Write an equation relating the variables.

2. Differentiate both sides with respect to time .

3. Plug in known values and solve for the unknown rate.

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Q6. Evaluate indefinite integrals relating to the elementary functions (power, exponential, logarithmic, trigonometric, and inverse trigonometric functions).

Background

Topic: Antiderivatives and Integration

This question tests your ability to find antiderivatives of basic functions using integration rules.

Key Terms and Formulas:

• Indefinite Integral:

• Power Rule: (for )

• Exponential Rule:

• Trigonometric Rules: ,

Step-by-Step Guidance

1. Identify the type of function (power, exponential, trig, etc.).

2. Apply the appropriate integration rule.

3. Simplify the integrand if needed before integrating.

4. Add the constant of integration .

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Q7. Find the general and particular solutions to differential equations.

Background

Topic: Differential Equations

This question tests your ability to solve equations involving derivatives and apply initial conditions.

Key Terms and Formulas:

• General Solution: The antiderivative with an arbitrary constant .

• Particular Solution: The solution after applying initial conditions to solve for .

Step-by-Step Guidance

1. Integrate the given derivative to find the general solution.

2. Apply the initial condition to solve for the constant .

3. Write the particular solution.

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