Concept limit and derivative It's an important basis in calculus, which affects the student's ability to understand advanced mathematics. However, many students have difficulty understanding these two concepts. In this article, we're going to talk about some common reasons why students face difficulty in understanding the concept of limit and derivative, as well as the implications faced in calculus learning.

The difficulty of understanding Limit Concepts

1. Limit Concept Abstraction The concept of limit is often considered abstract and undigested by students. Students need to understand the approach of function value to a certain number without ever reaching it in an exact way. For example, understanding that the xxx (x) lif _ {x to a} f (x) limx (x) states that when xxx approaches aaa, the function of f (x) f (x) approaches that particular number, but it doesn't always reach that value. Some studies show that students often misunderstand this concept because of the limitations of their basic mathematical understanding, such as the understanding of real numbers and the concept of infinity. Students tend to see limit static rather than dynamic, which complicates their difficulty in responding to limit related problems.
2. Complex notation usage Limit notation such as lim They often can't connect the notation to the underlying conceptual significance.
3. Error understanding Limit Approach Students often do not understand how the value of a function approaches a particular number when the variable approaches a particular value. They tend to see that number as the final number, when the limit measures what happens. as approaching Those numbers,

It's difficult to understand the concept of decline

1. Linked to the Limit Context The derivative is one of the essential applications of the limit, so students who don't understand the concept of the limit tend to also have difficulty understanding the derivative. For example, the derivative of a function at a certain point f) f '(x) f x + Argx) x) xf' (x) = lim _ {Delta x to 0} frac {f (x + Delta x) - f (x)} {Delta If understanding the limit isn't strong, the derivative concept becomes hazy.
2. Geometric Interpretation Many students have trouble connecting the derivative to the slopes of the tangent line of a function graph. Studies show that students often see the derivative of just as mathematical formula, without understanding the geometric concept or visualization of the slope of lines.
3. Error using Direction Rule Even after understanding the basic definition of the derivative, students often have difficulty applying the rules of differentiation, such as chain rules and product rules. This mistake could be caused by a lack of practice or a superficial concept of how the derivative works in a more complex context.

Strategy Control Adjustment

To address these difficulties, some approaches have been proposed:

• Visualize Approach: Use of graphs and interactive animation can help students understand how the limit works dynamic and how the derivative affects the form of a graph of a function.
• Contextual Approach(Laughter)
• Stage Training: Giving drills, from basic concepts to more complex applications, can strengthen student understanding gradually.

Conclusion

The student's difficulty in understanding the concept of limit and derivative is largely caused by the abstract properties of those two concepts and the lack of basic understanding of limit. The use of complex notations, the lack of visualization, and the lack of understanding about the connection between limit and derivative is worsening this problem. With a more visual, contextual and intermediate learning strategy, these difficulties can be overcome gradually. Teachers need to pay more attention to the conceptual understanding of students before proceeding to more complex applications.

Source: Tall, D. '93. Proceedings of ICME.