Calculus I Midterm 1 Study Guide – Step-by-Step Guidance

Q1. Find the domain of $f(x) = \frac{1}{x^2 - 4}$.

Background

Topic: Domains of Rational Functions

This question tests your understanding of how to determine where a rational function is defined, specifically 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 $\frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$.

Step-by-Step Guidance

1. Set the denominator equal to zero: $x^2 - 4 = 0$.

2. Solve for $x$ to find the values that are not allowed in the domain.

3. State the domain as all real numbers except the values found in step 2, using interval notation.

Try solving on your own before revealing the answer!

Q2. Find the domain of $f(x) = \frac{5x^2}{x^2 - 9}$.

Background

Topic: Domains of Rational Functions

This question asks you to determine where a rational function is defined by finding 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: $\frac{P(x)}{Q(x)}$, undefined where $Q(x) = 0$.

Step-by-Step Guidance

1. Set the denominator equal to zero: $x^2 - 9 = 0$.

2. Solve for $x$ to find the excluded values.

3. Express the domain as all real numbers except those values, using interval notation.

Try solving on your own before revealing the answer!

Q3. Find the domain of $f(x) = \frac{1}{x(x-4)}$.

Background

Topic: Domains of Rational Functions

This problem tests your ability to find where a rational function is undefined due to division by 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: $x(x-4) = 0$.

2. Solve for $x$ to find the values that make the denominator zero.

3. Write the domain as all real numbers except those values, using interval notation.

Try solving on your own before revealing the answer!

Q4. Find the domain of $f(x) = \frac{1}{\sqrt{x^2 - 9}}$.

Background

Topic: Domains with Radicals and Denominators

This question tests your ability to find the domain of a function with a square root in the denominator. The expression under the square root must be positive (not just non-negative) because the denominator cannot be zero.

Key Terms and Formulas

• Square Root Domain: $\sqrt{A}$ is defined for $A \geq 0$.

• Denominator Restriction: Denominator cannot be zero.

Step-by-Step Guidance

1. Set the radicand (expression under the square root) greater than zero: $x^2 - 9 > 0$.

2. Solve the inequality for $x$ to find where the function is defined.

3. Express the domain in interval notation, excluding values where the denominator is zero.

Try solving on your own before revealing the answer!

Q5. Find the domain of $f(x) = \sqrt[3]{16 - x^2}$.

Background

Topic: Domains with Odd Roots

This question tests your understanding of cube roots. Cube roots are defined for all real numbers, so you need to check if there are any restrictions.

Key Terms and Formulas

• Cube Root Domain: $\sqrt[3]{A}$ is defined for all real $A$.

Step-by-Step Guidance

1. Recognize that the cube root function is defined for all real numbers.

2. Check if there are any additional restrictions (e.g., denominators, even roots).

3. State the domain accordingly.

Try solving on your own before revealing the answer!

Q6. Find the domain of $f(x) = \sqrt{2x^2 + 1}$.

Background

Topic: Domains with Even Roots

This question tests your ability to determine where a square root function is defined. The expression under the square root must be non-negative.

Key Terms and Formulas

• Square Root Domain: $\sqrt{A}$ is defined for $A \geq 0$.

Step-by-Step Guidance

1. Set the radicand greater than or equal to zero: $2x^2 + 1 \geq 0$.

2. Solve the inequality for $x$.

3. Express the domain in interval notation.

Try solving on your own before revealing the answer!