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

Background

Topic: Functions and Domain

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

Key Terms and Formulas:

• Domain: The set of all possible input values () for which the function is defined.

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

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 values that make the denominator zero.

4. Exclude these values from the domain.

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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:

• Function:

Step-by-Step Guidance

1. Compute by substituting into the function: .

2. Write the difference quotient: .

3. Expand and simplify the numerator: .

4. Factor and simplify the expression as much as possible, but stop before the final simplification.

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Q3. Consider the graph of shown below. Use the graph to determine each of the following limits:

• a.

• b.

• c.

• d.

• e.

• f.

• g.

• h.

• i.

Background

Topic: Limits from Graphs

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

Key Terms and Formulas:

• Limit: is the value approaches as gets close to .

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

• Function value: is the actual value at .

Step-by-Step Guidance

1. For each limit, examine the graph near the specified -value.

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

3. For , check the value of the function at (may be a filled or open dot).

4. For limits at infinity, observe the end behavior of the graph as goes to .

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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 manipulation, factoring, and knowledge of trigonometric limits.

Key Terms and Formulas:

• Limit Laws: Use algebraic simplification, factoring, and substitution.

• Trigonometric Limits:

Step-by-Step Guidance

1. For each limit, check if direct substitution gives an indeterminate form (like or ).

2. If indeterminate, factor or simplify the expression to resolve the limit.

3. For trigonometric limits, use known limit results or apply L'Hospital's Rule if appropriate.

4. Stop before plugging in the value or completing the final calculation.

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Q5. An object is launched into the air. Its position (height in meters) above the ground seconds after launch is given by . Find the average velocity of the object from to .

Background

Topic: Average Velocity

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

Key Terms and Formulas:

• Average Velocity:

• Position Function: gives the height at time .

Step-by-Step Guidance

1. Identify and .

2. Compute and using the position function.

3. Set up the average velocity formula: .

4. Stop before calculating the final value.

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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 function, showing different expressions for different intervals of .

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Identify the intervals and the corresponding expressions for .

2. For , plot .

3. For , plot .

4. For , plot .

5. Mark endpoints carefully, using open or closed circles as appropriate.

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

Background

Topic: Continuity of Piecewise Functions

This question tests your understanding of continuity at a point, using the formal 3-step checklist.

Key Terms and Formulas:

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

  1. is defined

  2. exists

Step-by-Step Guidance

1. Check if and are defined using the piecewise function.

2. Compute the left and right limits at and .

3. Compare the limits to the function value at each point.

4. Stop before stating whether the function is continuous at each point.

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Q7. Consider the function on the interval . Explain clearly how the Intermediate Value Theorem guarantees that this function must have 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 it applies to finding zeros of a function on a closed interval.

Key Terms and Formulas:

• Intermediate Value Theorem: If is continuous on and and have opposite signs, then has at least one zero in .

Step-by-Step Guidance

1. Check that is continuous on .

2. Compute and to see if they have opposite signs.

3. State that by IVT, there must be a zero between and .

4. Use a calculator to approximate the zero, but stop before stating the value.

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Q8a. Consider the function . What is the domain of ?

Background

Topic: Domain of Rational Functions

This question tests your ability to find the domain by identifying values that make the denominator zero.

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Set the denominator equal to zero: .

2. Solve for to find values to exclude from the domain.

3. State the domain as all real numbers except those values.

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Q8b. Determine the locations of any vertical/horizontal asymptotes for the function, and any locations of removable discontinuities. Be thorough in your work!

Background

Topic: Asymptotes and Discontinuities

This question tests your ability to analyze rational functions for vertical and horizontal asymptotes, and removable discontinuities.

Key Terms and Formulas:

• 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 both numerator and denominator are zero at the same -value.

Step-by-Step Guidance

1. Find vertical asymptotes by setting the denominator to zero and checking if the numerator is also zero at those points.

2. Find horizontal asymptotes by comparing the degrees of numerator and denominator.

3. Check for removable discontinuities by factoring numerator and denominator and seeing if they share a common factor.

4. Stop before stating the exact locations.

Try solving on your own before revealing the answer!