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: Curve Sketching, Concavity, and Inflection Points

This question tests your understanding of how to interpret and sketch a graph based on information about concavity and inflection points.

Key Terms and Concepts:

• Concave Up: The graph is shaped like a cup (U), and the second derivative is positive.

• Concave Down: The graph is shaped like a cap (∩), and the second derivative is negative.

• Inflection Point: The point where the graph changes concavity (from up to down or vice versa).

Step-by-Step Guidance

1. Mark the given points on a coordinate plane: (1,4), (5,-6), and (3,-1).

2. At (1,4), the graph should be concave down (∩ shape locally).

3. At (5,-6), the graph should be concave up (U shape locally).

4. At (3,-1), the graph should have an inflection point, meaning the concavity changes here.

5. Sketch a smooth curve passing through these points, ensuring the concavity changes at (3,-1).

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

Background

Topic: Critical Values and Relative Extrema

This question tests your ability to find where the derivative is zero or undefined (critical values) and to determine if these points are relative maxima or minima.

Key Terms and Formulas:

• Critical Value: A value of where or does not exist.

• Relative Extrema: Points where the function has a local maximum or minimum.

• First Derivative Test: Used to classify critical points as maxima, minima, or neither.

Step-by-Step Guidance

1. Find the first derivative: .

2. Set and solve for to find critical values.

3. Test intervals around each critical value using the first derivative to determine if the function is increasing or decreasing.

4. Classify each critical value as a relative maximum, minimum, or neither based on the sign changes of .

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Q3. For the function , identify intervals where the function is increasing or decreasing, and find the coordinates of any relative extrema and inflection points.

Background

Topic: Increasing/Decreasing Intervals, Extrema, and Inflection Points

This question tests your ability to use the first and second derivatives to analyze the behavior of a function.

Key Terms and Formulas:

• Increasing/Decreasing Intervals: Where (increasing) or (decreasing).

• Inflection Point: Where and the concavity changes.

Step-by-Step Guidance

1. Use the first derivative to determine where the function is increasing or decreasing.

2. Find the second derivative and set it to zero to find possible inflection points.

3. Test intervals around these points to confirm changes in concavity.

4. List the coordinates of any relative extrema and inflection points found.

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Q4. Find the absolute maximum and minimum values of on

Background

Topic: Absolute Extrema on a Closed Interval

This question tests your ability to find the highest and lowest values of a function on a given interval using calculus.

Key Terms and Formulas:

• Absolute Maximum/Minimum: The largest/smallest value of on the interval.

• Closed Interval Method: Evaluate at critical points and endpoints.

Step-by-Step Guidance

1. Find the first derivative and solve for critical points in .

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

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

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Q5. For the function , find any relative extrema.

Background

Topic: Relative Extrema of Exponential Functions

This question tests your ability to use derivatives to find and classify relative extrema for a function involving an exponential term.

Key Terms and Formulas:

• Product Rule:

• Critical Value: Where or does not exist.

Step-by-Step Guidance

1. Find the first derivative using the product rule.

2. Set and solve for to find critical values.

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

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

Background

Topic: Optimization of Quadratic Functions

This question tests your ability to find the maximum value of a quadratic function, which models productivity over time.

Key Terms and Formulas:

• Vertex of a Parabola: For , the maximum occurs at if .

Step-by-Step Guidance

1. Identify the coefficients , , and in the quadratic function.

2. Use the vertex formula to find the time when productivity is maximized.

3. Plug this value of back into to find the maximum productivity.

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