Knowledge points in the second volume of senior high school mathematics
1. Find the monotonicity of a function:
The basic method of finding monotonicity of function by using derivative: Let function yf(x) be derivable in the interval (a, b), (1) If f(x) is constant, then function yf(x) is increasing function in the interval (a, b); (2) If F (x) is a constant, the function yf(x) is a decreasing function in the interval (a, b); (3) If f (x) is constant, then the function yf(x) is a constant function in the interval (a, b).
The basic steps of finding monotonicity of function by derivative are as follows: ① finding the domain of function yf(x); ② Find the derivative of f(x); ③ Solve the inequality f(x)0, and define the uninterrupted interval of the solution set on the domain as the increasing interval; ④ If the inequality f(x)0 is solved, the uninterrupted interval of the solution set on the domain is a decreasing interval.
Conversely, we can also use derivatives to solve related problems (such as determining the range of parameters) through the monotonicity of functions: let function yf(x) be derivable in the interval (a, b),
(1) If the function yf(x) is increasing function in the interval (a, b), then f(x)0 (where the x value of f(x)0 does not constitute the interval);
(2) If the function yf(x) is a subtraction function in the interval (a, b), then f(x)0 (where the x value of f(x)0 does not constitute the interval);
(3) If the function yf(x) is a constant function in the interval (a, b), then f(x)0 holds.
2. Find the extreme value of the function:
Let the function yf(x) be defined in x0 and its vicinity. If all points near x0 have f(x)f(x0) (or f(x)f(x0)), it is said that f(x0) is the minimum (or maximum) of the function f(x).
The extreme value of differentiable function can be obtained by studying the monotonicity of function. The basic steps are as follows:
(1) Determine the domain of the function f(x); (2) find the derivative f (x); (3) Find all the real roots of the equation f(x)0, x 1x2xn, divide the domain into several cells in sequence, and list the changes of the values of f(x) and f(x) when x changes:
(4) Look up the sign of f(x) and judge the extreme value from the table.
3. Find the value and minimum value of the function:
If the function f(x) has x0 in the domain I, so that there is always f(x)f(x0) for any xI, it is said that f(x0) is the value of the function in the domain. The extreme value of the function in the definition domain is not necessarily, but the maximum value in the definition domain is.
Step of finding the value and minimum value of the function f(x) in the interval [a, b]: (1) Find the extreme value of f(x) in the interval (a, b);
(2) Compare the extreme value obtained in the first step with f (a) and f (b) to obtain the value and minimum value of f(x) in the interval [a, b].
4. Solve the related problems of inequality:
The scope of (1) inequality problem (absolute inequality problem) can be considered.
When the range of f(x)(xA) is [a, b],
The necessary and sufficient condition for the inequality f(x)0 is f(x)max0, i.e B0;
The necessary and sufficient condition for the inequality f(x)0 is f(x)min0, which is a0.
When the range of f(x)(xA) is (a, b),
The necessary and sufficient condition for inequality f(x)0 to be constant is B0; The necessary and sufficient condition for inequality f(x)0 to be constant is a0.
(2) Proving the inequality f(x)0 can be transformed into proving f(x)max0, or using the monotonicity of function f(x) to prove f(x)f(x0)0.
5. The application of derivative in real life:
Solving the (small) value problem in real life can usually be transformed into the maximum value of a function. When calculating the maximum value of a function by derivative, we must pay attention to the unimodal function at the extreme point, which is the maximum value point, and we should explain it when solving the problem.
Knowledge points in the second volume of senior high school mathematics
Compound functional domain
If the definition domain of the function y=f(u) is b and the definition domain of u = g (x) is a, then the definition domain of the compound function y=f[g(x)] is D={x|x∈A, g(x)∈B}, and the range of x in each part should be considered comprehensively.
To find the domain of a function, you should consider the following points:
(1) When it is an algebraic expression or an odd root, the range of r;
(2) When there are even roots, the number of roots shall not be less than 0 (i.e. ≥ 0);
(3) When it is a fraction, the denominator is not 0; When the denominator is an even root, the number of roots is greater than 0;
(4) When it is exponential, the radix is not 0 for zero exponential power or negative integer exponential power.
5] Dang is a combination of some basic functions through four operations, and its domain should be a set of independent variable values that make all parts meaningful, that is, to find the intersection of the domain sets of all parts.
The domain of piecewise function is the union of the value sets of independent variables on each segment.
(7) For the function established in practical problems, we should not only consider making the analytical expression meaningful, but also consider the requirements of practical significance for the independent variables.
⑻ For the function with parameter letters, the values of letters should be classified and discussed when finding the domain. It should be noted that the domain of the function is a non-empty set.
⑼ The truth value of logarithmic function must be greater than zero, and the radix must be greater than zero and not equal to 1.
⑽ The cross-section function in trigonometric function should pay attention to the restriction of diagonal variables.
Frequently asked questions of compound function
(1) Knowing that the domain of f(x) is A, finding the domain of f[g(x)]: In essence, knowing the domain of G (x) is A, so finding the domain of X. ..
(2) Knowing that the domain of f[g(x)] is B, finding the domain of f(x): In essence, knowing the domain of X is B, finding the domain of G (x).
(iii) The domain of f [g(x)] is called C, and the domain of f[h(x)] is found: in essence, the domain of g (x) (that is, the domain of f(x)) is called C; Then take it as the range of h(x), and then find the range of x.
Knowledge points in the second volume of senior high school mathematics
Line, plane, simple geometry:
1, learning three views analysis:
2, oblique mapping method should pay attention to the place:
(1) Take the mutually perpendicular axes Ox and Oy in the known graph. When drawing a vertical view, draw it as the corresponding axes o'x' and o'y' so that ∠ x' o' y' = 45 (or135);
(2) The length of the line segment parallel to the X axis is unchanged, and the length of the line segment parallel to the Y axis is halved.
(3) A 45-degree manuscript under direct vision is 90 degrees, and a 90-degree manuscript under direct vision shall not be 90 degrees.
3, table (edge) area and volume formula:
(1) column: (1) surface area: S=S side +2S bottom; ② Lateral area: S side =; ③ volume: V=S bottom h
⑵ Cone: ① Surface area: S=S side +S bottom; ② Lateral area: S side =; ③ volume: V=S bottom h:
(3) Platform surface area ①: S=S side +S upper bottom S lower bottom ② side area: S side =
⑷ Sphere: ① Surface area: S =；; ② Volume: V=
4. Proof of position relationship (main method): Pay attention to the writing of solid geometry proof.
(1) Straight lines are parallel to the plane: ① Straight lines are parallel to each other; (2) Parallel lines are parallel to each other.
(2) Plane is parallel to plane: ① Line is parallel to plane, and surface is parallel to surface.
(3) Vertical problem: The vertical plane of the line is vertical. The core is line-plane verticality: two intersecting straight lines in a vertical plane.
5. turning: (step-I. find or make an angle; Two. Cornering)
(1) Solution of included angle formed by straight lines on different planes: translation method: translating straight lines to construct triangles;
⑵ Angle between straight line and plane: Angle between straight line and projection.
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