Teaching knowledge points of compulsory five in senior high school mathematics 1
Monotonicity, parity and periodicity of functions
Monotonicity: Definition: Note that the definition is relative to a specific interval.
The judgment methods are: definition method (difference comparison method and quotient comparison method)
Derivative method (for polynomial function)
Composite function method and mirror image method.
Application: compare sizes, prove inequalities and solve inequalities.
Parity check:
Definition: Pay attention to whether the interval is symmetrical about the origin, and compare the relationship between f(_) and f(-_). F(_)-f(-_)=0f(_)=f(-_)f(_) is an even function;
F(_)+f(-_)=0f(_)=-f(-_)f(_) is odd function.
Discrimination methods: definition method, image method and compound function method.
Application: function value transformation solution.
Periodicity: Definition: If the function f(_) satisfies: f(_+T)=f(_), then t is the period of the function f(_).
Others: If the function f(_) satisfies any _ in the domain: f(_+a)=f(_-a), then 2a is the period of the function f(_).
Application: Find the function value and resolution function in a certain interval.
Fourth, graphic transformation: function image transformation: (key) It is required to master the images of common basic functions and master the general rules of function image transformation.
Regularity of common image changes: (Note that translation changes can be explained by vector language, which is related to vector translation)
Translation transformation y = f (_) → y = f (_+a), y = f (_)+b.
Note: (1) If there is a coefficient, first extract the coefficient. For example, translate the function y=f(2_) to get the image of the function y=f(2_+4).
(2) Combining with the translation of vector, understand the meaning of translation according to vector (m, n).
Symmetric transformation y=f(_)→y=f(-_), which is symmetric about y.
Y=f(_)→y=-f(_), about _ axis symmetry.
Y=f(_)→y=f|_|, keep the images above _ axis, and the images below _ axis are symmetrical about _.
Y=f(_)→y=|f(_)| Keep the image on the right side of the Y axis, and then make the right part of the Y axis symmetrical about the Y axis. (Note: it is an even function)
Telescopic transformation: y=f(_)→y=f(ω_),
Y=f(_)→y=Af(ω_+φ) Image transformation of specific reference trigonometric function.
An important conclusion: If f(a-_)=f(a+_), then the image of function y=f(_) is symmetrical about the straight line _=a;
Teaching of compulsory five knowledge points in senior two mathematics II
A, set, simple logic (14 class, 8)
1. setting; 2. subset; 3. supplement; 4. Intersection; 5. Trade unions; 6. Logical connector; 7. Four propositions; 8. Necessary and sufficient conditions.
Second, the function (30 class hours, 12)
1. mapping; 2. Function; 3. Monotonicity of the function; 4. Inverse function; 5. The relationship between function images of reciprocal function; 6. Extension of the concept of index; 7. Operation of rational exponential power; 8. Exponential function; 9. Logarithm; 10. Operational properties of logarithm; 1 1. logarithmic function. 12. Application example of function.
III. Series (12 class hours, 5)
1. sequence; 2. arithmetic progression and its general formula; 3. arithmetic progression's first N terms and formulas; 4. Geometric series and its topping formula; 5. The first n terms and formulas of geometric series.
Fourth, trigonometric function (46 class hours, 17)
The generalization of the concept of 1. angle; 2. Curvature system; 3. Trigonometric function at any angle; 4. The trigonometric function line in the unit circle; 5. Basic relations of trigonometric functions with the same angle; 6. Inductive formulas of sine and cosine; 7. Sine, cosine and tangent of sum and difference of two angles; 8. Sine, cosine and tangent of double angles; 9. Images and properties of sine function and cosine function; 10. Periodic function; The parity of 1 1. function; 12. Image of the function; 13. Images and properties of tangent function; 14. Find the angle with the known trigonometric function value; 15. Sine theorem; 16. Cosine theorem; 17. Examples of oblique triangle solution.
V. Plane Vector (12 8 class hours)
1. vector; 2. Addition and subtraction of vectors; 3. Product of real number and vector; 4. Coordinate representation of plane vector; 5. The demarcation point of the line segment; 6. The product of plane vectors; 7. The distance between two points on the plane; 8. Translation.
Inequalities of intransitive verbs (22 class hours, 5)
1. Inequality; 2. Basic properties of inequality; 3. Proof of inequality; 4. Solving inequality; 5. Inequalities with absolute values.
VII. Equation of Line and Circle (22 class hours, 12)
1. Angle and slope of straight line; 2. Point-oblique and two-point linear equations; 3. General formula of linear equation; 4. Conditions for two straight lines to be parallel and vertical; 5. Angle of intersection of two straight lines; 6. Distance from point to straight line; 7. The plane area is expressed by binary linear inequality; 8. Simple linear programming problem; 9. Concepts of curves and equations; 10. The curve equation is listed by known conditions; Standard equation and general equation of 1 1. circle; Parametric equation of a circle.
VIII. Conic Curve (18 7 class hours)
1. ellipse and its standard equation; 2. Simple geometric properties of ellipse; 3. Parametric equation of ellipse; 4. Hyperbola and its standard equation; 5. Simple geometric properties of hyperbola; 6. Parabola and its standard equation; 7. Simple geometric properties of parabola.
Nine, straight line, plane, simple (36 class hours, 28 points)
1. plane and its basic properties; 2. Intuitive drawing of plane graphics; 3. Plane straight line; 4. Determination and properties of parallelism between a straight line and a plane: 5. Determination and properties of verticality between a straight line and a plane: 6. Three perpendicularity theorem and its inverse theorem; 7. The positional relationship between two planes; 8. Space vector and its addition, subtraction, multiplication and division; 9. Coordinate representation of space vector; 10. the product of space vectors; 1 1. The direction vector of the straight line; 12. angles formed by straight lines on different planes; 13. Common perpendicular of straight lines on different planes; 14. Distance of straight lines in different planes; 15. Verticality of straight line and plane; 16. The normal vector of the plane; 17. Distance from point to plane; 18. The angle formed by a straight line and a plane; 19. The projection of the vector on the plane; 20. The nature that the plane is parallel to the plane; 2 1. Distance between parallel planes; 22. dihedral angle and its plane angle; 23. Determination and nature of verticality of two planes; 24. Polyhedron; 25. Prism; 26. pyramids; 27. Regular polyhedron; 28. Ball.
Ten, permutation, combination, binomial theorem (18 class, 8)
1. Classification counting principle and step-by-step counting principle; 2. Arrangement; 3. Formula of permutation number; 4. combination; 5. Combination number formula; 6. Two properties of combinatorial numbers: 7. Binomial theorem; 8. The nature of binomial expansion.
XI。 Probability (12 class hours, 5)
1. Probability of random events; 2. The probability of this possible event; 3. mutually exclusive events has the probability of occurrence; 4. The probability of mutually independent events occurring simultaneously; 5. Repeat the test independently.
Elective 2 (24)
XII. Probability and Statistics (14 class hours, 6)
1. Distribution table of discrete random variables; 2. Expected value and variance of discrete random variables; 3. Sampling method; 4. Estimation of the overall distribution; 5. Normal distribution; 6. Linear regression.
Thirteen. Restrictions (12 class hours, 6)
1. Mathematical induction; 2. Examples of application of mathematical induction; 3. Limit of sequence; 4. Limit of function; 5. Four operations of limit; 6. Functional continuity.
Fourteen Derivative (18 class hour, 8)
The concept of 1. derivative; 2. Geometric meaning of derivative; 3. Derivatives of several common functions; 4. Derivative of sum, difference, product and quotient of two functions; 5. Derivative of composite function; 6. Basic derivative formula; 7. Using derivative to study monotonicity and extremum of function: 8. Value and minimum value of function.
Fifteen, plural (4 class hours, 4)
The concept of 1. complex number; 2. Addition and subtraction of complex numbers; 3. Multiplication and division of complex numbers; 4. The solution of the univariate quadratic equation and binomial equation of complex numbers.
Teaching of five knowledge points in senior high school mathematics 3
Test center 1: Derive the formula.
Example 1.f(_) is the derivative function of f(_) 13_2_ 1, so the value of f( 1) is 3.
Test site 2: the geometric meaning of derivative.
Example 2. It is known that the tangent equation of the image of function yf(_) at point m (1, f( 1)) is y.
1_2, then f( 1)f( 1)2.
The tangent equation at 3) is Example 3. Curve y_32_24_2 is located at point (1
Comments: The above two small questions are all about the geometric meaning of derivatives.
Test site 3: the application of derivative geometric meaning.
Example 4. Given the curve C: Y _ 33 _ 22 _ and the straight line l:yk_, and the straight line L and the curve C are tangent to the points _0 and y0_00, the equation and tangent coordinates of the straight line L are found. ..
Comments: This little question examines the application of derivative geometric meaning. When solving this kind of problems, we should pay attention to the application of the condition that the tangent point is both on the curve and on the tangent. The differentiability of a function at a certain point is a sufficient condition for the existence of a tangent at that point on the corresponding curve, but it is not a necessary condition.
Test site 4: monotonicity of function.
Example 5. Given that F _ A _ 3 _ 1 is a decreasing function on r, find the range of a .. 32.
Comments: This topic examines the application of derivatives in monotonicity of functions. For the monotonicity problem of higher-order functions, we should have the consciousness of derivative.
Test site 5: extreme value of function.
Example 6. Let the function f(_)2_33a_23b_8c take the extreme value at _ 1 and _2.
(1) Find the values of a and b;
(2) For any value and minimum value on _.
Comments: This topic examines the basic knowledge of parity, monotonicity, the maximum value of quadratic function, the application of derivative, and the ability of reasoning and operation.
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