Q1. How can the graph of be obtained from the graph of ?

Background

Topic: Transformations of Trigonometric Functions

This question tests your understanding of how to apply horizontal and vertical shifts, stretches, and compressions to the sine function.

Key Terms and Formulas:

• Amplitude: The coefficient in front of the sine function ( in ) affects vertical stretch/compression.

• Period: Determined by ; period is .

• Phase Shift: Horizontal shift is .

• Vertical Shift: moves the graph up or down.

Step-by-Step Guidance

1. Identify the amplitude, period, phase shift, and vertical shift in .

2. Rewrite as to clarify the phase shift.

3. Determine the horizontal shift: The graph is shifted by units to the right.

4. Determine the horizontal compression/stretch: The coefficient $2.

5. Identify the vertical stretch: The coefficient $3.

6. Identify the vertical shift: The shifts the graph up by $5$ units.

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Q2. Evaluate the following limit:

Background

Topic: Limits and Rational Functions

This question tests your ability to evaluate limits of rational functions, especially when the numerator and denominator both approach zero (indeterminate form).

Key Terms and Formulas:

• Indeterminate Form:

• Factoring: Factor numerator and denominator to simplify.

• Limit Laws: Substitute after simplification.

Step-by-Step Guidance

1. Factor the numerator: .

2. Factor the denominator: .

3. Notice that both numerator and denominator have as a factor.

4. Cancel the common factor from numerator and denominator.

5. Substitute into the simplified expression.

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Q3. An object has position function . Find the average velocity of the object on the interval .

Background

Topic: Average Rate of Change

This question tests your understanding of how to compute the average velocity (average rate of change) of a function over a given interval.

Key Terms and Formulas:

• Average velocity: for interval .

Step-by-Step Guidance

1. Identify and .

2. Compute and using .

3. Calculate .

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

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Q4. Where is the function not continuous?

Background

Topic: Continuity of Rational Functions

This question tests your understanding of where rational functions are discontinuous, typically where the denominator is zero.

Key Terms and Formulas:

• Discontinuity: Occurs where denominator is zero.

• Find zeros of denominator: .

Step-by-Step Guidance

1. Set denominator equal to zero: .

2. Solve for to find points of discontinuity.

3. Check if these points are removable or non-removable discontinuities.

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Q5. Determine the limit:

Background

Topic: Limits at Infinity

This question tests your understanding of how exponential and trigonometric functions behave as approaches negative infinity.

Key Terms and Formulas:

• approaches $0x \to -\infty$.

• oscillates between and $1$.

• Sum of functions: Consider the limit of each part separately.

Step-by-Step Guidance

1. Analyze as ; it approaches $0$.

2. Multiply by ; since $e^x$ is approaching $0.

3. Add $3x \to -\infty$.

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Q6. For the function shown below, determine which of the following statements is FALSE.

Background

Topic: Limits and Discontinuities

This question tests your ability to interpret statements about limits, asymptotes, and discontinuities for a given function.

Key Terms and Formulas:

• Vertical asymptote: Where function approaches infinity.

• Discontinuity: Where function is not continuous.

• One-sided limits: and .

Step-by-Step Guidance

1. Review each statement and consider what it means for the function.

2. Check if the limit exists at .

3. Check the behavior as ; does the function approach infinity?

4. Check if the left and right limits at are equal.

5. Check for vertical asymptotes and discontinuities at .

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Q7. Suppose the domain of is . If , then what is the domain of ?

Background

Topic: Domain of Transformed Functions

This question tests your ability to find the domain of a function after applying linear transformations to the input variable.

Key Terms and Formulas:

• Domain: Set of input values for which the function is defined.

• Transformation: .

Step-by-Step Guidance

1. Set equal to the original domain endpoints: and $8$.

2. Solve for in both cases to find the new domain.

3. Write the new domain as an interval.

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Q8. Find all the asymptotes of .

Background

Topic: Asymptotes of Rational Functions

This question tests your ability to find vertical and horizontal asymptotes for a rational function involving a square root.

Key Terms and Formulas:

• Vertical asymptotes: Where denominator is zero.

• Horizontal asymptotes: Compare degrees of numerator and denominator.

Step-by-Step Guidance

1. Set denominator and solve for to find vertical asymptotes.

2. Analyze the degrees: Numerator is degree 4 (inside the square root), denominator is degree 2.

3. For large , approximate .

4. Find the horizontal asymptote by dividing leading terms: .

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Q9. Suppose and , then?

Background

Topic: Logarithmic Properties

This question tests your ability to use properties of logarithms to simplify expressions.

Key Terms and Formulas:

Step-by-Step Guidance

1. Simplify using logarithm subtraction: .

2. Simplify : .

3. Combine the results to match the answer choices.

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Q10.

Background

Topic: Inverse Trigonometric Functions

This question tests your ability to evaluate inverse trigonometric functions and understand their principal values.

Key Terms and Formulas:

• returns the angle in whose sine is .

• .

Step-by-Step Guidance

1. Compute .

2. Find the angle in such that .

3. Match the value to the answer choices.

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Q11. Evaluate the following limit:

Background

Topic: Limits and Difference Quotients

This question tests your ability to evaluate a limit that resembles the definition of the derivative.

Key Terms and Formulas:

• Difference quotient: as .

• Algebraic simplification: Combine fractions and simplify.

Step-by-Step Guidance

1. Combine the fractions in the numerator: .

2. Find a common denominator and simplify the numerator.

3. Divide by and simplify the expression.

4. Take the limit as .

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Q12. Determine the limit:

Background

Topic: One-Sided Limits and Square Roots

This question tests your ability to evaluate one-sided limits involving square roots and rational expressions.

Key Terms and Formulas:

• One-sided limit: Approach from the left ().

• Square root: Consider domain and sign as approaches 2 from below.

Step-by-Step Guidance

1. Factor the expression under the square root: .

2. As , approaches $0$ from the negative side.

3. Analyze the sign of the denominator as approaches 2 from the left.

4. Consider the behavior of the numerator and denominator to determine the limit.

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