Q1. Find the limit:

Background

Topic: Limits

This question tests your understanding of how to evaluate the limit of a polynomial function as approaches a specific value.

Key Terms and Formulas:

• Limit: The value that a function approaches as the input approaches a certain point.

• For polynomials, limits can be evaluated by direct substitution.

Step-by-Step Guidance

1. Identify the function: .

2. Since this is a polynomial, you can substitute directly into the function.

3. Set up the substitution: .

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Q2. Which graph below matches the limit group?

Given two graphs, select the one that matches the behavior of the function as approaches a certain value.

Background

Topic: Graphical Interpretation of Limits

This question tests your ability to interpret the limit of a function from its graph.

Key Terms:

• Limit from a graph: The value the function approaches as gets close to a specific point.

• Look for continuity and jumps in the graph.

Step-by-Step Guidance

1. Examine the behavior of the function near the point of interest on each graph.

2. Check if the function approaches the same value from both sides (left and right).

3. Identify any discontinuities or jumps that might affect the limit.

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Q3. Find the derivative:

Background

Topic: Derivatives

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

Key Terms and Formulas:

• Derivative: The rate of change of a function with respect to its variable.

• Power Rule:

Step-by-Step Guidance

1. Apply the power rule to each term of the polynomial.

2. For , the derivative is .

3. For , the derivative is .

4. For , the derivative is $1$.

5. The derivative of a constant () is $0$.

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Q4. Evaluate the definite integral:

Background

Topic: Definite Integrals

This question tests your ability to compute the area under a curve using definite integration.

Key Terms and Formulas:

• Definite Integral: gives the net area under from to .

• Power Rule for Integration:

Step-by-Step Guidance

1. Integrate to get .

2. Integrate $1x$.

3. Combine the results: .

4. Evaluate this expression at the upper and lower limits ( and ).

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Q5. Find the critical points of .

Background

Topic: Critical Points

This question tests your ability to find where the derivative of a function is zero or undefined, which are potential maxima, minima, or saddle points.

Key Terms and Formulas:

• Critical Point: Where or is undefined.

• Derivative:

Step-by-Step Guidance

1. Find the derivative: .

2. Set the derivative equal to zero: .

3. Solve for to find the critical point.

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