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, which is a foundational concept in Calculus. The domain is the set of all real numbers for which the function is defined.

Key Terms and Formulas:

• Domain: The set of input values () for which the function produces a real output.

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

Step-by-Step Guidance

1. Identify the denominator: . The function is undefined when the denominator equals zero.

2. Set the denominator equal to zero and solve for : .

3. Find the values of that make the denominator zero. These values are excluded from the domain.

4. Express the domain in interval notation, excluding 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 a fundamental concept for understanding derivatives in Calculus.

Key Terms and Formulas:

• Difference Quotient:

• Derivative: The difference quotient is the basis for the definition of the derivative.

Step-by-Step Guidance

1. Compute by substituting into the function: .

2. Write as given: .

3. Set up the difference quotient: .

4. Simplify the numerator by combining like terms and finding a common denominator if needed.

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

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Q3. Use the graph of to determine 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 .

• Limit at infinity: describes the behavior as goes to negative infinity.

Step-by-Step Guidance

1. For each limit, examine the graph near the specified -value. Observe how behaves as approaches the point from the left and right.

2. For one-sided limits, focus only on the direction indicated (right or left).

3. For the two-sided limit, check if the left and right limits are equal. If not, the limit does not exist.

4. For , look for the value of the function at (may be a filled or open dot).

5. For the limit 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, substitution, and knowledge of trigonometric limits.

Key Terms and Formulas:

• Limit Laws: Direct substitution, factoring, and simplification.

• Trigonometric Limits:

Step-by-Step Guidance

1. For each limit, check if direct substitution is possible. If so, substitute the value for .

2. If substitution leads to an indeterminate form (like ), try factoring or using trigonometric identities.

3. For limits at infinity, consider the degree of the numerator and denominator.

4. For trigonometric limits, recall special limits and identities.

5. Stop before the final calculation; set up the expressions for the student to finish.

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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 understanding of how to compute the average velocity of an object over a time interval using its position function.

Key Terms and Formulas:

• Average Velocity:

• Position Function: gives the height at time .

Step-by-Step Guidance

1. Identify the interval: , .

2. Compute and by plugging these values into the position function.

3. Set up the average velocity formula: .

4. Calculate the numerator and denominator separately.

5. Stop before the final division; set up the expression for the student to finish.

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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, which is a function defined by different expressions over different intervals.

Key Terms and Formulas:

• Piecewise Function: A function defined by multiple sub-functions, each with its own domain.

Step-by-Step Guidance

1. Identify the intervals for each piece: , , .

2. For , graph (a parabola shifted up by 3).

3. For , graph (a straight line).

4. For , graph (a horizontal line).

5. Pay attention to endpoints: use open or closed circles as appropriate for each interval.

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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, especially for piecewise functions. The 3-step checklist is: (1) is defined, (2) exists, (3) .

Key Terms and Formulas:

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

  • is defined

  • exists

Step-by-Step Guidance

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

2. Compute the left and right limits as using the appropriate pieces.

3. Compare the left and right limits to see if the overall limit exists.

4. Check if the limit equals .

5. Repeat the process for .

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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, which states that if a function is continuous on and and have opposite signs, then there is at least one in such that .

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Check that is continuous on (it is, since it's a polynomial).

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

3. If and are positive and negative (or vice versa), IVT guarantees a zero.

4. Set up the equation and use a calculator to approximate the zero.

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Q8a. What is the domain of ?

Background

Topic: Domain of Rational Functions

This question tests your ability to find the domain of a rational function 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 excluded values.

3. Express the domain in interval notation, excluding these values.

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Q8b. Determine the locations of any vertical/horizontal asymptotes for , 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 to identify removable discontinuities.

Key Terms and Formulas:

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

• Horizontal Asymptote: Determined by the degrees of the 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 equal to zero and checking if the numerator is also zero at those points.

2. Check for removable discontinuities by seeing if the numerator and denominator share a common factor.

3. Determine horizontal asymptotes by comparing the degrees of the numerator and denominator.

4. Set up the expressions for the student to finish the analysis.

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