Judging from the college entrance examination questions in recent years, the most valuable questions are mostly a fill-in-the-blank question or multiple-choice question and a solution question; Judging from the score, it accounts for about10% of the total score; From the hot topic of investigation, the maximum problem is more and more included in solid geometry and analytic geometry. From this point of view, although the maximum problem is an old problem, it has been very active, especially the introduction of derivatives, which has injected new vitality into the study of the maximum problem.
Let's choose typical examples and talk about the handling strategy of the maximum value problem in geometry.
The extreme value problem in geometry is related to the properties of geometric figures, and it is often solved by drawing, geometric transformation and using the inequality relationship in geometry. It is also possible to establish a functional relationship and transform the geometric problem into an algebraic problem (that is, algebra) to solve it.
Solid geometry mainly studies the positional relationship between points, lines and surfaces in space, and the maximum problems such as line segments, angles and volumes related to space graphics often appear in test questions. The solution of the maximum problem in solid geometry depends on the analysis of the quantitative relationship between geometric elements in the figure and the selection of an appropriate quantity (angle, line segment, etc.). ) as an independent variable, and establish a functional expression representing the dependent variable (area, volume, etc.). ). Using the method of finding the maximum value of a function in algebra to find the maximum value, we should pay special attention to the selection of independent variables, which has a great influence on the difficulty of solving problems. For the maximum problem in solid geometry, it is often necessary to transform solid graphics into plane, algebra or triangle problems by means of graphic transformation such as translation, rotation and expansion.
The maximum problem in analytic geometry is the main content of studying mathematical problems from a dynamic point of view, so it often appears in the college entrance examination. Find the maximum value with trigonometric function; Find the maximum value by quadratic function value domain; Using the discriminant of the roots of quadratic equation to find the maximum value and using the arithmetic mean value not less than the geometric mean value (mean value theorem) to find the maximum value.
Strategy 1: Turn songs into direct pursuit of maximum value.
Some extreme problems in solid geometry can be solved by graphic transformation, such as translation, rotation and expansion.
Comments: This question is more innovative than in previous years. It reflects the comprehensiveness of a single question, attaches importance to the multiple connections of mathematical knowledge, and designs test questions at the intersection of plane vectors, functions, derivatives, conic curves, curve tangents, inequalities and other knowledge, which reflects the horizontal connection of knowledge, examines the comprehensive ability of candidates from multiple angles and levels, and can distinguish candidates at different levels. This is one of the most successful topics in many years.
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