Q1. Find the average rate of change of the following functions between and :

Background

Topic: Average Rate of Change

This question tests your understanding of how to compute the average rate of change of a function over a given interval. This is a foundational concept in calculus, closely related to the slope of the secant line between two points on the graph of the function.

Key Terms and Formulas

• Average Rate of Change: The change in the function's value divided by the change in over an interval .

Formula:

Step-by-Step Guidance

1. Identify the interval: , .

2. For each function, compute and :

   • For , calculate and .

   • For , calculate and .

3. Subtract from for each function.

4. Divide the result by (which is ).

Try solving on your own before revealing the answer!

Q2. Find the instantaneous rate of change for at .

Background

Topic: Instantaneous Rate of Change (Derivative)

This question is about finding the derivative of a function at a specific point, which represents the instantaneous rate of change or the slope of the tangent line at that point.

Key Terms and Formulas

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

Formula for the derivative of :

Step-by-Step Guidance

1. Rewrite as to make differentiation easier.

2. Apply the power rule and chain rule to find .

3. Substitute into your expression for .

4. Simplify the expression as much as possible without calculating the final value.

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Q3a. The distance in feet of an object from a starting point is given by , where is time in seconds. Find the average velocity of the object from 2 seconds to 4 seconds.

Background

Topic: Average Velocity

This question tests your ability to compute the average velocity of an object over a time interval, which is the average rate of change of the position function.

Key Terms and Formulas

• Average Velocity: The change in position divided by the change in time.

Formula:

Step-by-Step Guidance

1. Identify and .

2. Calculate and using the given position function.

3. Subtract from .

4. Divide the result by .

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Q3b. Find the instantaneous velocity at 4 seconds.

Background

Topic: Instantaneous Velocity (Derivative)

This question asks you to find the derivative of the position function at a specific time, which gives the instantaneous velocity.

Key Terms and Formulas

• Instantaneous Velocity: The derivative of the position function with respect to time.

Formula:

Step-by-Step Guidance

1. Find the derivative of .

2. Substitute into .

3. Simplify the expression to get the instantaneous velocity at (but do not compute the final value).

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Q4a. The total cost for manufacturing cases of t-shirts is for . Find the average change, per case, in the total cost if the number of cases manufactured changes from 1 to 5 cases.

Background

Topic: Average Rate of Change (Cost Function)

This question is about finding the average rate of change of the cost function over a specified interval, which tells you the average additional cost per case.

Key Terms and Formulas

• Average Rate of Change:

Step-by-Step Guidance

1. Calculate and using the cost function.

2. Subtract from .

3. Divide the result by .

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Q4b. Find the additional cost when production is increased from 1 to 2 cases.

Background

Topic: Marginal Cost (Discrete Change)

This question asks for the actual increase in total cost when production increases by one case, which is a discrete change in the cost function.

Key Terms and Formulas

• Additional Cost:

Step-by-Step Guidance

1. Calculate and using the cost function.

2. Subtract from to find the additional cost.

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Q4c. Find the instantaneous rate of change of cost with respect to the number of cases produced when just one case is produced.

Background

Topic: Marginal Cost (Derivative)

This question is about finding the derivative of the cost function at a specific value of , which gives the marginal cost when .

Key Terms and Formulas

• Marginal Cost:

Formula:

Step-by-Step Guidance

1. Find the derivative of the cost function.

2. Substitute into .

3. Simplify the expression to find the marginal cost at (but do not compute the final value).

Try solving on your own before revealing the answer!