Q1. Find for

Background

Topic: Differentiation of Polynomials

This question tests your ability to compute the derivative of a polynomial function using the power rule.

Key Terms and Formulas

• Derivative: The rate at which a function changes with respect to its variable.

• Power Rule:

• Constant Rule: The derivative of a constant is zero.

Step-by-Step Guidance

1. Apply the power rule to each term with a variable: For , , and .

2. Remember that the derivative of a constant (like ) is zero.

3. Write the derivative of each term separately: , , , and .

4. Combine the results to express as a sum of the derivatives you found.

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Q2. Find for

Background

Topic: Differentiation of Polynomials

This question also tests your ability to use the power rule to find the derivative of a polynomial function.

Key Terms and Formulas

• Power Rule:

• Constant Rule: The derivative of a constant is zero.

Step-by-Step Guidance

1. Apply the power rule to each term: , , , and .

2. Find the derivative of each term individually.

3. Sum the derivatives to write .

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Q9. Find for

Background

Topic: Quotient Rule for Derivatives

This question tests your ability to differentiate a rational function using the quotient rule.

Key Terms and Formulas

• Quotient Rule:

• Let and .

Step-by-Step Guidance

1. Identify and in the function: , .

2. Compute and : , .

3. Apply the quotient rule formula: .

4. Substitute , , , and into the formula, but do not simplify to the final answer yet.

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Q17. The total cost function for the production of air conditioners is . Find the marginal cost.

Background

Topic: Marginal Analysis in Business Calculus

This question tests your understanding of marginal cost, which is the derivative of the total cost function with respect to the number of units produced.

Key Terms and Formulas

• Marginal Cost: , the derivative of the cost function .

• Power Rule:

Step-by-Step Guidance

1. Identify the cost function: .

2. Differentiate each term with respect to using the power rule.

3. Combine the derivatives to write , the marginal cost function.

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Q18. Find the revenue function in terms of if the price per unit is $450$.

Background

Topic: Revenue Functions in Business Calculus

This question tests your ability to construct a revenue function, which is the product of the price per unit and the number of units sold.

Key Terms and Formulas

• Revenue Function: , where is the price per unit and is the number of units sold.

Step-by-Step Guidance

1. Identify the price per unit () and the variable for units sold ().

2. Write the revenue function as .

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