(1) Definition:
For the function y = f (x) (x ∈ d), let the real number x with f (x) = 0 be called the zero point of the function y = f (x) (x ∈ d).
(2) the relationship between the zero point of the function and the root of the corresponding equation, and the intersection of the image of the function and the X axis:
Does the equation f (x) = 0 have a real root? Does the image of the function y = f (x) intersect with the x axis? The function y = f (x) has zero.
(3) Determination of zero point of function (zero point existence theorem):
If the image of the function y = f (x) on the interval [a, b] is a continuous curve with f (a) f (b).
2. quadratic function y = ax2+bx+c (a >; 0) The relationship between the image and the zero point.
Step 3 split
For continuous intervals [a, b] and f (a) f (b)
4. The zero point of the function is not a point:
The zero point of the function y = f (x) is the real root of the equation f (x) = 0, that is, the abscissa of the intersection of the image of the function y = f (x) and the x axis, so the zero point of the function is a number, not a point. When writing function zero, it must be a number, not a coordinate.
5. When judging the existence of functional zero, it must be emphasized that:
(1)f(x) is continuous on [a, b];
(2)f(a)f(b )& lt； 0;
(3) Zero exists in (a, b).
This is a sufficient condition for the existence of zero, but it is not a necessary condition.
6. For continuous functions in the definition domain, all function values between two adjacent zeros keep the same sign.
two
Related concepts of 1. geometric series
(1) Definition:
A series is called a geometric series if the ratio of each term to its previous term is equal to the same constant (non-zero) of the second term. This constant is called the common ratio of geometric series, and is usually expressed by the letter Q. The defined expression is an+ 1/an = q (n ∈ n *, q is a non-zero constant).
(2) Proportion:
If a, g and b are in geometric series, then g is called the proportional mean of a and b, that is, g is the proportional mean of a and b? A, g and b do geometric series? G2=ab。
2. Relevant formulas of geometric series.
(1) general formula: an = a 1qn- 1.
3. Common properties of geometric series {an}
(1) in the geometric series {an}, if m+n = p+q = 2r (m, n, p, q, r∈N*), then am an = apaq = a.
Especially a1an = a2an-1= a3an-2 =.
(2) In the geometric series {an} and q, the sequences am, AM+K, AM+2K, AM+3K, … are still geometric series, and the sequences Q: Sm, S2M-SM, S3m-S2m, … are still geometric series (at this time Q ≦-1); an=amqn-m。
4. The characteristics of geometric series
(1) According to the definition of geometric series, any term of geometric series is non-zero and the common ratio q is also non-zero.
(2) If an+ 1 = qan, q≠0 cannot immediately assert that {an} is a geometric series, we need to verify a 1≠0.
5. the first n terms of geometric series and Sn.
The first n terms and Sn of (1) geometric series are obtained by dislocation subtraction. Pay attention to the application of this thinking method in the summation of series.
(2) When using the first n terms and formulas of geometric series, we must pay attention to the classification and discussion of Q = 1 and q≠ 1 to prevent mistakes caused by ignoring the special situation of Q = 1.