Q1. Estimate the slope of the curve at the designated point.

Background

Topic: Tangent Lines and Derivatives

This question is testing your understanding of how to estimate the slope of a curve at a specific point, which is the geometric meaning of the derivative at that point. The slope of the tangent line to the curve at a point gives the instantaneous rate of change of the function there.

Key Terms and Formulas

• Tangent Line: A straight line that touches a curve at a single point without crossing it locally.

• Derivative: The slope of the tangent line to the curve at a given point, denoted as .

• Slope Formula:

Step-by-Step Guidance

1. Identify the point on the curve where you are asked to estimate the slope (point P in the image).

2. Observe the tangent line drawn at point P. The slope of this line represents the derivative of the function at that point.

3. Choose two points on the tangent line (not necessarily on the curve) that are easy to read from the grid. Record their coordinates as and .

4. Calculate the slope using the formula .

Try solving on your own before revealing the answer!

Q2. Differentiate

Background

Topic: Basic Differentiation Rules

This question tests your ability to apply the power rule for differentiation to a simple polynomial function.

Key Terms and Formulas

• Power Rule: If , then

• Derivative: The function that gives the slope of the original function at any point .

Step-by-Step Guidance

1. Identify the exponent in the function (here, ).

2. Apply the power rule: Multiply the exponent by the coefficient (which is 1 in this case), and subtract 1 from the exponent.

3. Write the resulting expression for , but do not simplify to the final answer yet.

Try solving on your own before revealing the answer!

Q3. Differentiate

Background

Topic: Chain Rule

This question tests your ability to differentiate a composite function using the chain rule. The function inside the square root is itself a quadratic.

Key Terms and Formulas

• Chain Rule: If , then

• Derivative of Square Root:

Step-by-Step Guidance

1. Rewrite the function as to make differentiation easier.

2. Apply the chain rule: Differentiate the outer function (the square root) and multiply by the derivative of the inner function ().

3. Compute the derivative of the inner function.

4. Combine the results, but do not simplify to the final answer yet.

Try solving on your own before revealing the answer!

Q16. For which x values is ? (Use the graph shown)

Background

Topic: Critical Points and Graphical Analysis

This question tests your ability to interpret a graph and identify where the derivative (slope) of the function is zero. These points correspond to local maxima, minima, or points of inflection where the tangent line is horizontal.

Key Terms and Concepts

• Critical Point: A point where or is undefined.

• Horizontal Tangent: The tangent line is flat, indicating a local maximum or minimum.

Step-by-Step Guidance

1. Look for points on the graph where the curve changes direction (peaks and valleys).

2. At these points, the tangent line is horizontal, so the slope is zero.

3. Estimate the x-values where these horizontal tangents occur by reading the graph carefully.

4. List these x-values, but do not write the final list yet.

Try solving on your own before revealing the answer!