Q1. Sketch the graph that possesses the characteristics listed:

It is concave down at (1,4), concave up at (5,-6), and has an inflection point at (3,-1). Choose the correct graph below.

Background

Topic: Concavity and Inflection Points

This question tests your understanding of how to identify and sketch graphs based on concavity and inflection points.

Key Terms:

• Concave up: The graph is shaped like a cup (second derivative > 0).

• Concave down: The graph is shaped like a cap (second derivative < 0).

• Inflection point: Where the graph changes concavity.

Step-by-Step Guidance

1. Identify the regions of concavity: At (1,4) the graph is concave down, at (5,-6) it is concave up.

2. Locate the inflection point at (3,-1): This is where the graph changes from concave down to concave up or vice versa.

3. Sketch or select the graph that matches these features, making sure the inflection point is at (3,-1).

4. Check that the graph transitions from concave down to concave up at the inflection point.

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Q2. Find a) any critical values and b) any relative extrema for

Background

Topic: Critical Points and Relative Extrema

This question tests your ability to find critical points and relative extrema using derivatives.

Key Terms and Formulas:

• Critical value: Where or does not exist.

• Relative extrema: Local maximum or minimum values.

• Product Rule:

Step-by-Step Guidance

1. Find the derivative using the product rule:

2. Simplify the derivative and set to find critical values.

3. Check where does not exist (if any).

4. Use the first or second derivative test to determine if the critical values are relative maxima or minima.

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Q3. For the function , find the absolute maximum and minimum values over the interval .

Background

Topic: Absolute Extrema on a Closed Interval

This question tests your ability to find absolute maximum and minimum values using calculus techniques.

Key Terms and Formulas:

• Absolute maximum/minimum: Highest/lowest value of on the interval.

• Critical points: Where or does not exist.

• Check endpoints: Evaluate at and .

Step-by-Step Guidance

1. Find and solve to find critical points within .

2. Evaluate at each critical point and at the endpoints and .

3. Compare these values to determine which is the absolute maximum and minimum.

4. Be sure to check all points in the interval, including endpoints.

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Q4. For , find any relative extrema.

Background

Topic: Relative Extrema and Critical Points

This question tests your ability to use derivatives to find relative maxima and minima.

Key Terms and Formulas:

• Relative extrema: Local maximum or minimum values.

• Critical points: Where or does not exist.

• Product Rule:

Step-by-Step Guidance

1. Find using the product rule:

2. Simplify and set to find critical points.

3. Use the first or second derivative test to classify the critical points as relative maxima or minima.

4. Check if is undefined anywhere.

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Q5. For , find the coordinates of the relative extrema and intervals of increase/decrease.

Background

Topic: Relative Extrema, Increasing/Decreasing Intervals

This question tests your ability to use derivatives to find where a function is increasing, decreasing, and its relative extrema.

Key Terms and Formulas:

• Relative extrema: Local maximum or minimum values.

• Increasing/Decreasing: Determined by the sign of .

• Critical points: Where or does not exist.

Step-by-Step Guidance

1. Find and solve to find critical points.

2. Use the first derivative test to determine intervals of increase and decrease.

3. Classify the critical points as relative maxima or minima.

4. Write the intervals in proper notation.

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Q6. An employee's monthly productivity , . Find the maximum productivity and the year in which it is achieved.

Background

Topic: Optimization and Applications

This question tests your ability to use calculus to optimize a real-world function.

Key Terms and Formulas:

• Maximum productivity: Highest value of in the interval.

• Critical points: Where or does not exist.

• Check endpoints: Evaluate at and .

Step-by-Step Guidance

1. Find and solve to find critical points within .

2. Evaluate at each critical point and at the endpoints and .

3. Compare these values to determine the maximum productivity and the year it is achieved.

4. Be sure to check all points in the interval, including endpoints.

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