Q1. Find the domain of the following functions. Decide if the graph is even or odd.

Background

Topic: Functions – Domain, Even/Odd Properties

This question tests your understanding of how to determine the set of all possible input values (domain) for a function, and how to classify a function as even, odd, or neither based on its symmetry.

Key Terms and Concepts:

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

• Even Function: for all in the domain.

• Odd Function: for all in the domain.

Step-by-Step Guidance

1. For each function, identify any restrictions on the input (e.g., denominators cannot be zero, expressions under square roots must be non-negative, etc.).

2. Write the domain in interval notation, considering all restrictions.

3. To determine if the function is even, substitute for and simplify. Compare to .

4. To determine if the function is odd, substitute for and check if .

Try solving on your own before revealing the answer!

Q2. Sketch the graph of for various forms.

Background

Topic: Graphing Functions

This question asks you to visualize and sketch the graph of a function for different algebraic forms, which is a key skill for understanding function behavior.

Key Terms and Concepts:

• Parent Function: The simplest form of a function (e.g., , ).

• Transformations: Shifts, stretches, and reflections applied to the parent function.

Step-by-Step Guidance

1. Identify the parent function and note its basic shape.

2. Analyze any transformations (shifts, stretches, reflections) applied to the parent function.

3. Plot key points (intercepts, vertex, etc.) and sketch the overall shape.

4. Label axes and important features (e.g., asymptotes, maxima/minima).

Try sketching before checking the answer!

Q3. Determine if the following function has an inverse. If so, find it.

Background

Topic: Inverse Functions

This question tests your ability to determine if a function is one-to-one (and thus invertible), and to find the inverse function algebraically if it exists.

Key Terms and Formulas:

• One-to-One Function: Each output is produced by exactly one input.

• Horizontal Line Test: If any horizontal line crosses the graph more than once, the function is not one-to-one.

• Finding the Inverse: Swap and in , then solve for .

Step-by-Step Guidance

1. Check if the function passes the horizontal line test (or algebraically, if implies ).

2. If invertible, write , then swap and to get .

3. Solve for in terms of to find .

4. State the domain and range of the inverse function.

Try working through the steps before checking the answer!

Q4. Simplify the given expressions.

Background

Topic: Algebraic Manipulation

This question tests your ability to simplify algebraic expressions, including rational, exponential, and logarithmic forms.

Key Terms and Formulas:

• Exponent Rules: ,

• Logarithm Rules: ,

Step-by-Step Guidance

1. Apply exponent and logarithm rules as appropriate to combine or simplify terms.

2. Factor or expand expressions where possible to further simplify.

3. Check for restrictions on the variable (e.g., denominators, logarithms).

Try simplifying before checking the answer!

Q5. For each of the following limits, decide if the limit exists or does not exist. If it does exist, find it. If it does not exist, explain or show why. (Do not use L'Hôpital's Rule.)

Background

Topic: Limits

This question tests your understanding of how to evaluate limits using algebraic manipulation, factoring, rationalization, and knowledge of limit laws.

Key Terms and Formulas:

• Limit Laws:

• Factoring: Useful for canceling terms causing indeterminate forms.

• Rationalization: Multiply numerator and denominator by a conjugate if needed.

Step-by-Step Guidance

1. Substitute the value into the function to check if you get a determinate value or an indeterminate form (like ).

2. If indeterminate, factor or rationalize to simplify the expression.

3. Apply limit laws to evaluate the simplified expression.

4. If the limit does not exist, explain why (e.g., unbounded behavior, oscillation).

Try evaluating the limits before checking the answer!

Q6. Find all values of (and if it exists) such that is continuous at .

Background

Topic: Continuity

This question tests your understanding of the definition of continuity and how to ensure a piecewise function is continuous at a given point.

Key Terms and Formulas:

• Continuity at :

Step-by-Step Guidance

1. Write the left-hand and right-hand limits of as approaches .

2. Set the left-hand limit equal to the right-hand limit and to .

3. Solve for (and if needed) to ensure continuity at .

Try setting up the equations before checking the answer!

Q7. Decide whether the function has any vertical or horizontal asymptotes. Explain your answer.

Background

Topic: Asymptotes of Rational Functions

This question tests your ability to identify vertical and horizontal asymptotes by analyzing the behavior of rational functions as approaches certain values or infinity.

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.

Step-by-Step Guidance

1. Set the denominator equal to zero to find possible vertical asymptotes.

2. Compare the degrees of the numerator and denominator to determine horizontal asymptotes.

3. Check limits as and for horizontal asymptotes.

Try identifying the asymptotes before checking the answer!

Q8. Use the limit definition of derivative to find , if it exists.

Background

Topic: Derivative – Limit Definition

This question tests your ability to use the formal (first principles) definition of the derivative to find .

Key Formula:

Step-by-Step Guidance

1. Write out and for the given function.

2. Substitute into the limit definition formula.

3. Simplify the numerator and denominator as much as possible.

4. Take the limit as to find .

Try applying the definition before checking the answer!

Q9. Use the limit definition of derivative to find at a specific value.

Background

Topic: Derivative at a Point – Limit Definition

This question tests your ability to compute the derivative at a specific value using the limit definition.

Key Formula:

Step-by-Step Guidance

1. Compute and for the given function and value of .

2. Substitute into the limit definition formula.

3. Simplify the expression and take the limit as .

Try working through the steps before checking the answer!

Q10. Find the second derivative of the function.

Background

Topic: Second Derivative

This question tests your ability to find the second derivative, which is the derivative of the derivative, and is important for analyzing concavity and inflection points.

Key Formula:

Step-by-Step Guidance

1. Find the first derivative using differentiation rules.

2. Differentiate to obtain .

3. Simplify your result as much as possible.

Try differentiating before checking the answer!

Q11. Find the second derivative at the indicated point.

Background

Topic: Second Derivative at a Point

This question tests your ability to evaluate the second derivative at a specific value of .

Key Formula:

• First, find as in Q10, then substitute the given value for .

Step-by-Step Guidance

1. Find and then .

2. Substitute the given value of into .

Try evaluating before checking the answer!

Q12. Find the equation of the tangent line to the graph of at a given point.

Background

Topic: Tangent Line to a Curve

This question tests your ability to find the equation of the tangent line to a function at a specific point using derivatives.

Key Formula:

• The tangent line at :

Step-by-Step Guidance

1. Find , the derivative of the function.

2. Evaluate at the given -value to find the slope .

3. Use the point-slope form with the given point and calculated slope.

Try setting up the tangent line before checking the answer!

Q13. Find the first derivatives of each of the following functions. Simplify your final answer.

Background

Topic: Differentiation Rules

This question tests your ability to apply various differentiation rules (power, product, quotient, chain) and to simplify the result.

Key Formulas:

• Power Rule:

• Product Rule:

• Quotient Rule:

• Chain Rule:

Step-by-Step Guidance

1. Identify which differentiation rule(s) apply to each function.

2. Apply the rule(s) step by step, showing all intermediate steps.

3. Simplify the derivative, avoiding negative exponents or complex fractions.

Try differentiating and simplifying before checking the answer!