Q1. What is the domain of ? Show algebraically how you determined your result.

Background

Topic: Functions – Domain

This question tests your understanding of how to find the domain of a rational function by identifying values that make the denominator zero.

Key Terms and Formulas

• Domain: The set of all real numbers for which the function is defined.

• Rational Function: A function of the form , where .

Step-by-Step Guidance

1. Identify the denominator: .

2. Set the denominator equal to zero to find values that are not in the domain: .

3. Solve for to find the excluded values.

4. Express the domain as all real numbers except the values found in the previous step.

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Q2. Given , evaluate and simplify the difference quotient: .

Background

Topic: Difference Quotient

This question tests your ability to compute and simplify the difference quotient, which is foundational for understanding derivatives in calculus.

Key Terms and Formulas

• Difference Quotient:

• This expression gives the average rate of change of over the interval .

Step-by-Step Guidance

1. Compute by substituting into the function: .

2. Write the difference quotient: .

3. Combine the two fractions in the numerator over a common denominator.

4. Simplify the numerator as much as possible.

5. Factor and reduce the expression, if possible, to simplify the difference quotient.

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Q3. Use the graph of to determine the following limits and function values:

• a.

• b.

• c.

• d.

• e.

• f.

• g.

• h.

• i.

Background

Topic: Limits and Function Values from Graphs

This question tests your ability to read limits and function values directly from a graph, including one-sided limits and behavior at infinity.

Key Terms and Formulas

• One-sided limits: (from the right), (from the left)

• Limit at a point: exists if both one-sided limits are equal.

• Function value: is the actual value at (may differ from the limit).

• Limit at infinity: describes end behavior as becomes very negative.

Step-by-Step Guidance

1. For each limit, locate the relevant -value on the graph.

2. For one-sided limits, observe the -value as approaches from the left or right.

3. For two-sided limits, check if both one-sided limits agree.

4. For function values, look for filled (closed) or open circles at the given -value.

5. For the limit at , observe the end behavior of the graph as decreases without bound.

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Q4. Use analytical techniques to determine each limit:

• a.

• b.

• c.

• d.

• e.

• f.

• g.

• h.

Background

Topic: Analytical Limits

This question tests your ability to evaluate limits using algebraic simplification, factoring, and knowledge of trigonometric limits.

Key Terms and Formulas

• Factoring: Used to simplify expressions where direct substitution gives .

• Trigonometric Limits:

• Infinity: Consider end behavior for or .

Step-by-Step Guidance

1. For each limit, first try direct substitution to see if you get a determinate value or an indeterminate form like or .

2. If you get , factor numerator and denominator to simplify.

3. For trigonometric limits, use known limit properties or identities.

4. For limits at infinity, analyze the degree of numerator and denominator or the behavior of the function as grows large.

5. Set up the simplified expression for the final calculation.

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Q5. An object is launched into the air so that its position (height in meters) after seconds is . Find the average velocity from to .

Background

Topic: Average Velocity

This question tests your understanding of how to compute average velocity over a time interval using the position function.

Key Terms and Formulas

• Average Velocity:

• is the position function.

Step-by-Step Guidance

1. Identify and .

2. Compute by substituting into the position function.

3. Compute by substituting into the position function.

4. Subtract from to find the change in position.

5. Divide the result by to find the average velocity.

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Q6a. Graph the piecewise function:

Background

Topic: Piecewise Functions and Graphing

This question tests your ability to interpret and graph a piecewise-defined function, showing different expressions for different intervals of .

Key Terms and Formulas

• Piecewise Function: A function defined by different expressions over different intervals.

• Pay attention to open and closed endpoints for each interval.

Step-by-Step Guidance

1. For , graph only for values less than .

2. For , graph for between (inclusive) and $2$ (exclusive).

3. For , graph for values greater than or equal to $2$.

4. Mark open or closed circles at the endpoints to indicate whether the endpoint is included in the interval.

5. Combine all three pieces on the same set of axes for the complete graph.

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Q6b. Determine if the function is continuous at and . Reference the 3-step checklist for continuity at a point.

Background

Topic: Continuity of Piecewise Functions

This question tests your understanding of the definition of continuity at a point and how to apply it to piecewise functions.

Key Terms and Formulas

• Continuity at a point : A function is continuous at if:

  1. is defined.

  2. exists.

  3. .

Step-by-Step Guidance

1. For , check if is defined (which piece applies?).

2. Compute the left-hand and right-hand limits as approaches .

3. Compare the limit to the function value at .

4. Repeat the process for .

5. State whether the function is continuous at each point based on the 3-step checklist.

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Q7. Consider the function on the interval . Explain how the Intermediate Value Theorem guarantees at least one zero on this interval. Then, use your calculator to find the zero.

Background

Topic: Intermediate Value Theorem (IVT)

This question tests your understanding of the IVT and how to apply it to show the existence of a root (zero) for a continuous function on a closed interval.

Key Terms and Formulas

• Intermediate Value Theorem: If is continuous on and is any number between and , then there exists in such that .

• To show a zero exists, check that and have opposite signs.

Step-by-Step Guidance

1. Compute and to determine their signs.

2. Check if and have opposite signs (one positive, one negative).

3. State that is continuous (since it is a polynomial).

4. Apply the IVT to conclude that there is at least one in such that .

5. Use a calculator to approximate the value of where .

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Q8. Consider the function .

• a. What is the domain of ?

• b. Determine the locations of any vertical/horizontal asymptotes and any removable discontinuities. Be thorough in your work!

Background

Topic: Rational Functions – Domain and Asymptotes

This question tests your ability to find the domain, vertical and horizontal asymptotes, and removable discontinuities for a rational function.

Key Terms and Formulas

• Domain: All real numbers except where the denominator is zero.

• Vertical Asymptote: Occurs where the denominator is zero and the numerator is not zero.

• Horizontal Asymptote: Determined by comparing degrees of numerator and denominator.

• Removable Discontinuity: Occurs where a factor cancels in numerator and denominator.

Step-by-Step Guidance

1. Set the denominator and solve for to find domain restrictions.

2. Check if any factors cancel between numerator and denominator (for removable discontinuities).

3. For vertical asymptotes, identify -values where the denominator is zero and the numerator is not zero.

4. For horizontal asymptotes, compare the degrees of numerator and denominator and their leading coefficients.

5. Summarize the domain, asymptotes, and discontinuities.

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