Derivatives

Topic:Derivatives

Objectives & Outcomes

• Students will be able to define the concept of the derivative as the rate of change of the value of a function depending on the change in its argument.
• Students will be able to calculate the derivative of a given function.
• Students will be able to determine the slope of the tangent line of a given curve at a given point and use that information to analyze the behavior of a function.

Materials

• Graph paper
• Calculators
• Handouts with examples of functions and their graphs

Warm-up

• Begin the lesson by showing the students a graph of a function and asking them to describe the behavior of the function at different points. For example, at the point (1, 2), the function is increasing, while at the point (-1, 2), it is decreasing. Have the students note the different behaviors of the function at different points and discuss the reasons for these behaviors.

Direct Instruction

• Introduce the concept of the derivative as a way to calculate the rate of change of the value of a function depending on the change in its argument. Demonstrate how to calculate the derivative of a function using the formula for the derivative.
• Discuss how the derivative is related to the graph of a function. For example, if the graph of a function is increasing, then its derivative will be positive and if the graph is decreasing, then its derivative will be negative.
• Introduce the concept of the tangent point of a curve and its tangent line. Demonstrate how to find the tangent point of a curve by finding the point of intersection between its tangent line and the x-axis.

Guided Practice

• Have students work in pairs to practice calculating the derivative of a function using the formula for the derivative. The teacher can circulate to provide assistance and support as needed.
• Have students work in pairs to find the tangent points of a few different curves and their corresponding tangent lines. The teacher can circulate to provide assistance and support as needed.

Independent Practice

• For independent practice, have students work on a project-based activity where they are given a function and asked to find the derivative and draw the corresponding graph. The teacher can provide support and guidance as needed.

Closure

• Review the main points of the lesson.
• Ask students to share any questions or challenges they had while learning about derivatives.

Assessment

• Use a quiz or test to assess students' understanding of the concept of derivatives and how to calculate derivatives and use them to identify the rate of change of a function.
• Observe students during the independent practice to assess their ability to understand and apply the concept of the derivative.
• Ask students to write a reflection on the lesson, including any challenges or questions they had, and what they learned.