Q1. Find the slope of the tangent line to the graph of the function at the given point: at

Background

Topic: Derivatives and Tangent Lines

This question tests your ability to find the derivative of a function and use it to determine the slope of the tangent line at a specific point.

Key Terms and Formulas

• Derivative: The derivative gives the slope of the tangent line to at any point .

• Power Rule:

Step-by-Step Guidance

1. Find the derivative using the power rule for each term in .

2. Substitute into to find the slope at that point.

Try solving on your own before revealing the answer!

Q2. Find the derivative of by the limit process.

Background

Topic: Definition of the Derivative (Limit Process)

This question asks you to use the formal definition of the derivative to find for a linear function.

Key Terms and Formulas

• Limit Definition of Derivative:

Step-by-Step Guidance

1. Write for the given function.

2. Set up the difference quotient .

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

4. Take the limit as .

Try solving on your own before revealing the answer!

Q3. Find the derivative of by the limit process.

Background

Topic: Definition of the Derivative (Limit Process)

This question tests your ability to apply the limit definition of the derivative to a quadratic function.

Key Terms and Formulas

• Limit Definition of Derivative:

Step-by-Step Guidance

1. Compute by substituting into the function.

2. Set up the difference quotient .

3. Expand and simplify the numerator completely.

4. Divide by and then take the limit as .

Try solving on your own before revealing the answer!

Q4. Find an equation of the tangent line to the graph of at the point .

Background

Topic: Tangent Lines and Derivatives

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

Key Terms and Formulas

• Derivative: gives the slope of the tangent line at .

• Point-Slope Form:

Step-by-Step Guidance

1. Find for .

2. Evaluate to get the slope at .

3. Use the point-slope form with and to write the equation of the tangent line.

Try solving on your own before revealing the answer!

Q5. Find an equation of the line tangent to that is parallel to the line .

Background

Topic: Tangent Lines Parallel to a Given Line

This question tests your ability to find where the tangent to a curve is parallel to a given line, using derivatives and algebra.

Key Terms and Formulas

• Derivative: gives the slope of the tangent line.

• Parallel Lines: Parallel lines have the same slope.

• Point-Slope Form:

Step-by-Step Guidance

1. Find the slope of the given line by rewriting it in slope-intercept form ().

2. Set equal to this slope and solve for .

3. Find the corresponding value on for this .

4. Write the equation of the tangent line using the point-slope form.

Try solving on your own before revealing the answer!

Q6. Use the alternate form of the derivative to find for at .

Background

Topic: Alternate Definition of the Derivative

This question asks you to use the alternate (or symmetric) form of the derivative to find the derivative at a specific point.

Key Terms and Formulas

• Alternate Form of Derivative:

Step-by-Step Guidance

1. Compute for the given function.

2. Set up the difference quotient .

3. Simplify the numerator as much as possible.

4. Take the limit as .

Try solving on your own before revealing the answer!