I exponential function
(A) the operation of exponent and exponent power
The concept of 1. radical: generally, if, then it is called n-degree radical, where >: 1 and ∈ *.
When it is an odd number, the power root of a positive number is a positive number and the power root of a negative number is a negative number. At this point, the power root of is represented by a symbol. The formula is called radical, here it is called radical index, here it is called radical.
When it is an even number, a positive number has two power roots, and the two numbers are opposite. At this time, the positive power roots of positive numbers are represented by symbols, and the negative power roots are represented by symbols. Positive and negative power roots can be combined into +(>: 0). It can be concluded that negative numbers have no even roots; Any power root of 0 is 0, which is recorded as.
Note: In odd numbers, even numbers,
2. Power of fractional exponent
The meaning of the power of the positive fractional index stipulates:
1, the power of the positive fraction index of 0 is equal to 0,
The negative fractional exponential powers of 2 and 0 are meaningless.
It is pointed out that after defining the meaning of fractional exponent power, the concept of exponent is extended from integer exponent to rational exponent, and the operational nature of integer exponent power can also be extended to rational exponent power.
3. Operational Properties of Exponential Power of Real Numbers
(B) Exponential function and its properties
1, the concept of exponential function: Generally speaking, a function is called an exponential function, where x is the independent variable and the domain of the function is R. 。
Note: The base range of exponential function cannot be negative, zero 1.
2. Images and properties of exponential function
1、a & gt 1
2、0
3. Infinitely extend in the positive and negative directions of X axis and Y axis.
4. the domain of the function is r.
5. The image is asymmetrical about the origin and the y axis.
6. Nonsingular even functions
7. Functional images are all above the X axis.
8. The range of the function is R+
9. Function images all pass through the fixed point (0, 1).
From left to right, the image gradually rises;
Looking from left to right, the image gradually drops.
Add functions; Descending function
The vertical coordinates of the images in the first quadrant are all greater than 1.
The vertical coordinates of the images in the first quadrant are all less than 1.
The vertical coordinates of the images in the second quadrant are all less than 1.
The vertical coordinates of the images in the second quadrant are all greater than 1.
The upward trend of the image is getting steeper and steeper; The upward trend of image is getting slower and slower.
The function value begins to grow slowly, and then grows rapidly after reaching a certain value;
The function value begins to decrease rapidly, and then decreases slowly after reaching a certain value.
Note: Using the monotonicity of the function and combining with the image, we can also see that:
Second, the logarithmic function
(1) logarithm
The concept of 1. Logarithm: Generally speaking, if, then this number is called logarithm with base, written as: (-base,-true number,-logarithmic formula).
Description:
1) Pay attention to the limit of cardinality, and;
2) Pay attention to the writing format of logarithm.
2. Two important logarithms:
Common logarithm of 1: logarithm based on 10;
Natural logarithm: Logarithm based on irrational numbers.
Conversion between Logarithmic and Exponential Expressions
Logarithmic exponential expression
Logarithmic radix → power radix
Logarithm-→ exponent
Real number ←→ power
(B) the operational nature of logarithm
Note: Bottom-changing formula
The following conclusions (1) are derived by using the formula of changing the bottom; (2) .
(2) Logarithmic function
1, the concept of logarithmic function: function, also called logarithmic function, where is the independent variable and the domain of the function is (0, +∞).
note:
1) The definition of logarithmic function is similar to that of exponential function, both of which are formal definitions. Pay attention to discrimination.
For example, none of them are logarithmic functions, only logarithmic functions.
2) The restrictions of logarithmic function on cardinality:, and.
2, the nature of the logarithmic function:
a & gt 1
Functional attribute
1 function images are all on the right side of the y axis.
The domain of 2 function is (0, +∞)
3 The image is asymmetrical about the origin and the Y axis.
4 non-odd non-even function
5 extends infinitely in the positive and negative directions of the Y axis.
The range of 6 functions is r.
All seven function images pass through a fixed point (1, 0).
Looking from left to right, the image gradually rises.
Looking from left to right, the image gradually drops.
increasing function
Descending function
The image ordinate of the first quadrant is greater than 0.
The image ordinate of the first quadrant is greater than 0.
The vertical coordinates of the images in the second quadrant are all less than 0.
The vertical coordinates of the images in the second quadrant are all less than 0.
(3) Power function
1. Definition of power function: Generally speaking, a shape function is called a power function, where is a constant.
2. Summarize the properties of power function.
(1) All power functions are defined at (0, +∞), and the image passes through (1,1);
(2) When the image of the power function crosses the origin, it is an increasing function in the interval. Especially, when the image of power function is convex; When the image of the power function is convex;
(3) The image of power function is a decreasing function in the interval. In the first quadrant, when moving from the right to the origin, the image is infinitely close to the positive semi-axis of the shaft on the right side of the shaft, and infinitely close to the positive semi-axis of the shaft above the shaft when moving to the origin.
Chapter III Functional Application
First, the root of the equation and the zero of the function.
1, the concept of function zero: for a function, the real number that makes it true is called the zero of the function.
2. The meaning of the zero point of the function: the zero point of the function is the real root of the equation, that is, the abscissa of the intersection of the image of the function and the axis. Namely:
The equation has a real root function, the image has an intersection with the axis, and the function has a zero point.
3, the role of zero solution:
Find the zero point of a function:
1 (algebraic method) to find the real root of the equation;
2 (Geometric method) For the equation that can't be solved by the root formula, we can relate it with the image of the function and find the zero point by using the properties of the function.
4. Zero point of quadratic function:
Quadratic function.
1)△& gt； 0, the equation has two unequal real roots, the image of the quadratic function has two intersections with the axis, and the quadratic function has two zeros.
2)△=0, the equation has two equal real roots (multiple roots), the image of the quadratic function intersects with the axis, and the quadratic function has a double zero or a second-order zero.
3)△& lt； 0, the equation has no real root, the image of the quadratic function has no intersection with the axis, and the quadratic function has no zero.
Trigonometric function and inverse trigonometric function
This is the simplest transcendental function originated from geometry. The method of measuring angle in advanced analysis is the so-called radian method, that is, the corresponding central angle is measured by the arc segment on the unit circumference. Trigonometric functions are sinx, cosx and their derivatives, and their definitions are shown in figure 1. The Taylor expansion of sinx and cosx at x=0 is (2) (3), and their convergence radius is. Inverse functions of sinx, cosx, tanx, cotx, secx and cosecx are arcsinx, arccotx, arctanx, arccosecx (or sin- 1x, cos- 1x, tan- 1x, cot-.
Elementary function diagram
It's called inverse trigonometric function. Let α be a positive number of exponential function and logarithmic function, then y=αz represents an exponential function with α as the base (Figure 2). Its inverse function y=logαx is called a logarithmic function with α as the base (Figure 3). Especially when α=e, y=ez (or expx) and y=logαx=lnx (or logx) are exponential and logarithmic functions. Logx can be defined by the following integral formula, which respectively represents the area surrounded by hyperbola, lower T axis, left and right straight lines t= 1 and T = X. So when x changes on the positive real axis, y=logx is on the real axis, and log 1=0. Is the increasing function derivative of x, and logx satisfies the addition theorem, that is, log (x1x2) = logx1+logx2.
Inverse exponential function of logarithmic function
Ex is a increasing function, which defines a real number that is positive on the real axis, e0= 1. The derivative of ex is the same as itself. In addition, ex satisfies the multiplication theorem, that is. When x=0, the Taylor expansion of ex is.
Hyperbolic Function and inverse hyperbolic function
Hyperbolic function can be derived from exponential function through rational operation.
Elementary function
Count. Its properties are very similar to trigonometric functions, and they are represented by sinhx, coshx, tanhx, cothx, sechx and cosechx respectively, and are defined as follows: hyperbolic sine (Figure 4) and hyperbolic cosine (Figure 5) respectively. Like trigonometric functions, the hyperbolic tangent (Figure 6)tanhx=sinhx/coshx and hyperbolic cotangent (Figure 7)cothx=coshx/sinhx derived from them are both called hyperbolic functions. They have the following geometric explanations: hyperbola x2-y2 =1(x >; 0) Take a little m, let O be the origin, and N=( 1, 0). Let the area enclosed by on, OM and the arc on the hyperbola be θ/2, and the coordinate of point M is regarded as a function of θ, which is recorded as coshθ and sinhθ, that is, there is expression (5). Elementary function Elementary function Elementary function Complex variable Elementary function The domain of elementary function is complex number domain.
Rational function, power function and radical function
The ratio of two complex coefficient polynomials is a rational function, which realizes the analytical mapping from the extended complex plane to itself. Fractional linear function is a special rational function, which is of great significance in complex analysis. Another special case is that the power function w=zn, n is a natural number,
Elementary function
It is analytic on the whole plane. Therefore, when n≥2, the * * * shape mapping (conformal mapping) is realized everywhere except z=0. It changes the circumference 丨 z 丨 = r into the circumference |w|=rn, and changes the ray argz=θ into the ray argw=nθ. As long as the difference between the radian angles of any two points in a region is less than 2π/n, any region is a monocotyledonous region with w=zn, and the inverse function of the power function w=zn is the root function of n value (k = 0, 1, …, n- 1), which is called its branch. They are in any region of θ1z.
Exponential function and logarithmic function
In the exponential function formula (4), if X is changed into a complex variable Z, the exponential function w=ez of the complex variable is obtained, and obviously there is (k is an integer). Complex exponential function has similar properties to real exponential function: ez is an integral function and for any complex number z, ez ≠ 0; It satisfies the multiplication theorem:; Ez takes 2kπi as the cycle, that is; And its derivative is the same as itself, that is. The function w=ez realizes the * * * shape mapping in the whole plane. Any region, as long as the difference between imaginary parts of any two points in the region is less than 2π, is a univalent region of ez. For example, the exponential function changes a straight line x=x0 into a circle and a straight line y=y0 into a ray argw=y0, thereby changing the region Sk into a region 0w.