Q1. What does it mean for a function to be differentiable?

Background

Topic: Differentiability and Continuity

This question explores the concept of differentiability, which is central to calculus. A function is differentiable at a point if its derivative exists at that point, meaning the function is both continuous and smooth (no sharp corners or breaks).

Key Terms:

• Differentiable: A function is differentiable at a point if the derivative exists there.

• Continuous: No breaks or jumps in the graph.

• Derivative: The instantaneous rate of change (IROC) or slope of the tangent line.

Step-by-Step Guidance

1. Recall that differentiability requires the function to be continuous at the point in question.

2. Check for smoothness: The function should not have sharp corners (like ) or vertical tangents (like at ).

3. Understand that if the derivative does not exist at a point (e.g., due to a discontinuity or sharp corner), the function is not differentiable there.

4. Review graphical examples to identify points of non-differentiability.

Try solving on your own before revealing the answer!

Q2. Estimate the slope of the tangent line to the curve at a given point.

Background

Topic: Tangent Lines and Instantaneous Rate of Change

This question tests your ability to visually estimate the slope of a tangent line to a curve, which represents the derivative at that point.

Key Terms:

• Slope: The ratio of rise over run ().

• Tangent Line: A line that touches the curve at one point and has the same slope as the curve at that point.

Step-by-Step Guidance

1. Identify the point on the curve where you need to estimate the slope.

2. Draw or visualize the tangent line at that point.

3. Choose two points on the tangent line and calculate the rise (change in ) and run (change in ).

4. Compute the slope using .

Try solving on your own before revealing the answer!

Q3. Sort points A–E in increasing order of the slope of the tangent line for the function at that point.

Background

Topic: Comparing Slopes of Tangent Lines

This question asks you to analyze a graph and determine the relative slopes of tangent lines at different points, which is a visual application of derivatives.

Key Terms:

• Tangent Line Slope: The derivative at a specific point.

• Increasing Order: Arrange from smallest to largest slope.

Step-by-Step Guidance

1. Examine the graph and identify the tangent line at each labeled point (A–E).

2. Estimate whether the slope is positive, negative, or zero at each point.

3. Rank the points based on the steepness and direction of the tangent lines.

4. Arrange the points from the most negative slope to the most positive slope.

Try solving on your own before revealing the answer!

Q4. Find the derivative of the function .

Background

Topic: Differentiation Using Power Rule

This question tests your ability to apply the power rule to functions involving roots and fractional exponents.

Key Terms and Formulas:

• Power Rule:

• Root as Exponent:

• Fractional Exponent:

Step-by-Step Guidance

1. Rewrite as and as is.

2. Apply the power rule to each term: and .

3. Multiply the coefficient by the exponent and reduce the exponent by 1 for each term.

4. Combine the results to form the derivative.

Try solving on your own before revealing the answer!

Q5. Find the derivative of the function .

Background

Topic: Differentiation of Rational and Constant Functions

This question tests your ability to differentiate terms involving in the denominator and constants.

Key Terms and Formulas:

• Power Rule:

• Constant Rule: The derivative of a constant is 0.

• Rewrite as for differentiation.

Step-by-Step Guidance

1. Rewrite each term with in the denominator as .

2. Apply the power rule to each term: , , $\frac{d}{dx}6x^{-1}$.

3. Remember that the derivative of (a constant) is 0.

4. Combine the derivatives of each term.

Try solving on your own before revealing the answer!

Q6. Find the slope of the tangent line to the graph of at . Find the equation of the tangent line.

Background

Topic: Tangent Line Equation Using Derivatives

This question tests your ability to find the slope of a tangent line at a specific point and write the equation of the tangent line.

Key Terms and Formulas:

• Derivative:

• Tangent Line Equation:

• Plug into the derivative to find the slope.

Step-by-Step Guidance

1. Find the derivative of using the power rule.

2. Evaluate the derivative at to get the slope of the tangent line.

3. Find the -value at by plugging into the original function.

4. Use the point-slope form to write the equation of the tangent line.

Try solving on your own before revealing the answer!

Q7. Find all values of where the tangent line to is horizontal.

Background

Topic: Horizontal Tangent Lines and Critical Points

This question tests your ability to find where the derivative is zero, which corresponds to horizontal tangent lines.

Key Terms and Formulas:

• Horizontal Tangent: Occurs where .

• Derivative:

Step-by-Step Guidance

1. Find the derivative of .

2. Set the derivative equal to zero: .

3. Solve the resulting quadratic equation for .

4. Express the solutions in exact form, using radicals if necessary.

Try solving on your own before revealing the answer!

Q8. At what point(s) on the graph of is the slope of the tangent line 6?

Background

Topic: Finding Points with a Given Tangent Slope

This question tests your ability to set the derivative equal to a specific value and solve for the corresponding .

Key Terms and Formulas:

• Derivative:

• Set and solve for .

Step-by-Step Guidance

1. Find the derivative of .

2. Set the derivative equal to 6: .

3. Solve the resulting equation for .

4. Plug the value(s) back into the original function to find the corresponding value(s).

Try solving on your own before revealing the answer!

Q9. Suppose the demand for a certain item is given by , where represents the price of the item in dollars. Find the rate of change of demand with respect to price. Find and interpret the rate of change of demand when the price is $14$.

Background

Topic: Marginal Analysis and Business Applications of Derivatives

This question tests your ability to find the derivative of a demand function and interpret its meaning in a business context.

Key Terms and Formulas:

• Demand Function:

• Marginal Demand: , the rate of change of demand with respect to price.

• Interpretation: How demand changes as price changes.

Step-by-Step Guidance

1. Find the derivative of with respect to .

2. Plug into the derivative to find the rate of change at that price.

3. Interpret the sign and value of the derivative in the context of demand and price.

4. Relate the result to business decisions (e.g., increasing price decreases demand).

Try solving on your own before revealing the answer!