The general form of I exponential function is y = a x(a >；; 0 and ≠ 1) (x∈R). From the above discussion of power function, we can know that if x can take the whole set of real numbers as the domain, then we only need to make.
As shown in the figure, different sizes of a will affect the function diagram.
The domain of (1) exponential function is the set of all real numbers, provided that a is greater than 0 and not equal to 1. For the case that a is not greater than 0, there must be no continuous interval in the definition domain of the function, so we will not consider it.
At the same time, the role of a equal to 0 is meaningless and is generally not considered.
(2) The range of exponential function is a set of real numbers greater than 0.
(3) The function graph is concave.
(4) If a is greater than 1, the exponential function increases monotonically; If a is less than 1 and greater than 0, it is monotonically decreasing.
(5) We can see an obvious law, that is, when a tends to infinity from 0 (of course, it can't be equal to 0), the curves of the functions tend to approach the positions of monotonic decreasing functions of the positive semi-axis of Y axis and the negative semi-axis of X axis respectively. The horizontal straight line y= 1 is the transition position from decreasing to increasing.
(6) Functions always infinitely tend to a certain direction on the X axis and never intersect.
(7) The function always passes (0, 1), (if y = a x+b, then the function passes (0, 1+b).
Obviously the exponential function is unbounded.
(9) Exponential function is neither odd function nor even function.
(10) When a in two exponential functions is reciprocal to each other, the two functions are symmetric about y, but neither of them has parity.
Conversion of cardinality:
For any meaningful exponential function:
Add a number to the index and the image will move to the left; Subtract a number and the image will move to the right.
Add a number after f(X), and the image will shift upwards; Subtract a number and the image will pan down.
That is, "addition, subtraction, multiplication and division, left plus right subtraction"
Images of basis functions and exponential functions:
(1) From the point where the exponential function y = a x intersects the straight line x= 1 (1), it can be known that on the right side of the y axis, the base corresponding to the image changes from bottom to top.
(2) From the point (-1, 1/a) where the exponential function y = a x intersects with the straight line x=- 1, it can be seen that on the left side of the Y axis, the corresponding base of the image changes from large to small.
(3) The relationship between the base of the exponential function and the image can be summarized as follows: on the right side of the Y-axis, the base is large and the graph is high; On the left side of the Y axis, "the bottom is big and the picture is low". (as shown on the right)
Power contrast:
Common methods for comparing sizes: (1) ratio difference (quotient) method: (2) function monotonicity method; (3) Intermediate value method: compare the sizes of A and B, first find an intermediate value C, then compare the sizes of A and C and B, and get the size between A and B from the transitivity of inequality.
When comparing the size of the two forces, in addition to the above general methods, we should also pay attention to:
The comparison of two powers of (1) different exponents with the same base can be judged by the monotonicity of the exponential function.
For example: y 1 = 3 4, y2 = 3 5, because 3 is greater than 1, so the function monotonically increases (that is, the greater the value of x, the greater the corresponding value of y), because 5 is greater than 4 and y2 is greater than y 1.
(2) The comparison of the powers of the same exponent with two different bases can be judged by the changing law of the exponential function image.
For example: y 1 = 1/2 4, y2 = 3 4, because1/2 is less than1,the function image monotonously decreases in the definition domain; 3 is greater than 1, so the function image monotonically increases in the definition domain. When x=0, both function images pass (0, 1). Then with the increase of x, the image of y 1 drops, while y2 rises. When x is equal to 4, y2 is greater than y 1.
(3) For the comparison of the power of different cardinality and different exponents, the intermediate value can be used for comparison. For example:
& lt 1 & gt； For the size comparison of three (or more) numbers, we should first group them according to the size of the numbers (especially the size of 0, 1), and then compare the sizes of each group.
& lt2> When comparing the sizes of two powers, if we can make full use of "1" to build a "bridge" (that is, compare it with "1"), we can get the answer quickly. How to judge the size of a power sum "1"? From the images and properties of exponential function, we can know that "if the cardinality a and 1 are inequalities with exponents x and 0 in the same direction (for example, A > 1 and X > 0, or 0 < A < 1 and X < 0), A X is greater than 1, a x.
< 3 > Example: Is the following function a increasing function or a subtraction function on R? Explain why.
⑴y=4^x
Because 4> 1, y = 4 x is the increasing function on R;
⑵y=( 1/4)^x
Because 0
Ⅱ (see: functional diagram curve)
In the plane rectangular coordinate system xOy, draw a ray OP from point O, let the rotation angle be θ, let OP=r, and the coordinate of point P be (x, y).
Sinθ=y/r
Cosine function cosθ=x/r
Tangent function tanθ=y/x
Cotangent function cotθ=x/y
Secθ secθ=r/x
Cotangent function csθ= r/y
(The hypotenuse is R, the opposite side is Y, and the adjacent side is X ...)
And two functions that are not commonly used and easily eliminated:
Positive vector function version θ = 1-cosθ
Covector function coversθ = 1-sinθ.
Sine: The opposite side of angle α is higher than the hypotenuse.
Cosine (cos): The adjacent side of angle α is the upper hypotenuse.
Tangent (tan): The opposite side of angle α is greater than the adjacent side.
Cotangent: The adjacent side of angle α is higher than the opposite side.
Secant: the hypotenuse of angle α is larger than the adjacent side.
Cotangent: The hypotenuse of angle α is higher than the edge.
Square relation:
Sin? α+cos？ α= 1
1+ Tan? α = seconds? α
1+cot？ α=csc？ α
Relationship between products:
sinα=tanα×cosα
cosα=cotα×sinα
tanα=sinα×secα
cotα= cosα×csα
secα=tanα×cscα
cscα=secα×cotα
Reciprocal relationship:
tanα cotα= 1
sinα cscα= 1
cosα secα= 1
Relationship between businesses:
sinα/cosα=tanα=secα/cscα
cosα/sinα=cotα=cscα/secα
In the right triangle ABC,
The sine value of angle a is equal to the ratio of the opposite side to the hypotenuse of angle a,
Cosine is equal to the adjacent side of angle a than the hypotenuse.
The tangent is equal to the opposite side of the adjacent side,
Constant deformation formula of trigonometric function
Trigonometric function of sum and difference of two angles;
cos(α+β)=cosα cosβ-sinα sinβ
cos(α-β)=cosα cosβ+sinα sinβ
sin(α β)=sinα cosβ cosα sinβ
tan(α+β)=(tanα+tanβ)/( 1-tanαtanβ)
tan(α-β)=(tanα-tanβ)/( 1+tanαtanβ)
Trigonometric function of trigonometric sum:
sin(α+β+γ)= sinαcosβcosγ+cosαsinβcosγ+cosαcosβsinγ-sinαsinβsinγ
cos(α+β+γ)= cosαcosβcosγ-cosαsinβsinγ-sinαcosβsinγ-sinαsinαsinβcosγ-sinαsinβcosγ
tan(α+β+γ)=(tanα+tanβ+tanγ-tanαtanβtanγ)/( 1-tanαtanβ-tanβtanγ-tanγtanα)
Auxiliary angle formula:
Asinα+Bcosα=√(A？ +B？ ) sin(α+arctan(B/A)), where
sint=B/√(A？ +B？ )
Cost =A/√(A? +B？ )
tant=B/A
Asinα-Bcosα=√(A？ +B？ )cos(α-t)，tant=A/B
Double angle formula:
sin(2α)=2sinα cosα=2/(tanα+cotα)
cos(2α)=cos？ α-sin？ α=2cos？ α- 1= 1-2sin？ α
tan(2α)=2tanα/( 1-tan？ α)
Triple angle formula:
sin(3α) = 3sinα-4sin？ α = 4sinα sin(60 +α)sin(60 -α)
cos(3α) = 4cos？ α-3cosα = 4cosα cos(60 +α)cos(60 -α)
tan(3α) = (3tanα-tan？ α)/( 1-3tan？ α) = tanαtan(π/3+α)tan(π/3-α)
Half-angle formula:
sin(α/2)= √(( 1-cosα)/2)
cos(α/2)= √(( 1+cosα)/2)
tan(α/2)=√(( 1-cosα)/( 1+cosα))= sinα/( 1+cosα)=( 1-cosα)/sinα
Power reduction formula
Sin? α=( 1-cos(2α))/2 = versin(2α)/2
Because? α=( 1+cos(2α))/2 = covers(2α)/2
Tan? α=( 1-cos(2α))/( 1+cos(2α))
General formula:
sinα=2tan(α/2)/[ 1+tan？ (α/2)]
cosα=[ 1-tan？ (α/2)]/[ 1+tan？ (α/2)]
tanα=2tan(α/2)/[ 1-tan？ (α/2)]
Product sum and difference formula:
sinαcosβ=( 1/2)[sin(α+β)+sin(α-β)]
cosαsinβ=( 1/2)[sin(α+β)-sin(α-β)]
cosαcosβ=( 1/2)[cos(α+β)+cos(α-β)]
sinαsinβ=-( 1/2)[cos(α+β)-cos(α-β)]
Sum-difference product formula:
sinα+sinβ= 2 sin[(α+β)/2]cos[(α-β)/2]
sinα-sinβ= 2cos[(α+β)/2]sin[(α-β)/2]
cosα+cosβ= 2cos[(α+β)/2]cos[(α-β)/2]
cosα-cosβ=-2 sin[(α+β)/2]sin[(α-β)/2]
Derived formula
tanα+cotα=2/sin2α
tanα-cotα=-2cot2α
1+cos2α=2cos？ α
1-cos2α=2sin？ α
1+sinα=[sin(α/2)+cos(α/2)]？
Ⅲ logarithmic function
Generally speaking, if the power of a (a is greater than 0, and a is not equal to 1) is equal to n, then this number b is called the logarithm of n with the base of a, and it is recorded as log aN=b, where a is called the base of logarithm and n is called a real number.
Axiomatic definition of logarithmic function
If the real number formula has no root number, then as long as the real number formula is greater than zero, if there is a root number, the real number is required to be greater than zero, and the formula in the root number is greater than zero.
Cardinality is greater than 0 instead of 1.
Why is the base of logarithmic function greater than 0 instead of 1?
In the ordinary logarithmic formula, a
The general form of logarithmic function is y=log(a)x, which is actually the inverse function of exponential function and can be expressed as x = a y, so the stipulation of a in exponential function is also applicable to logarithmic function.
The figure on the right shows the function diagram of different size A:
You can see that the graphs of logarithmic functions are only symmetric graphs of exponential functions about the straight line y=x, because they are reciprocal functions.
The domain of (1) logarithmic function is a set of real numbers greater than 0.
(2) The range of logarithmic function is the set of all real numbers.
(3) The function image always passes through the (1, 0) point.
(4) When a is greater than 1, it is monotone increasing function and convex; When a is less than 1 and greater than 0, the function is monotonically decreasing and concave.
(5) Obviously, the logarithmic function is unbounded.
Common abbreviations for logarithmic functions:
( 1)log(a)(b)=log(a)(b)
(2)lg(b)=log( 10)(b)
(3)ln(b)=log(e)(b)
Operational properties of logarithmic function;
If a > 0 and a is not equal to 1, m >；; 0, N>0, then:
( 1)log(a)(MN)= log(a)(M)+log(a)(N)；
(2)log(a)(M/N)= log(a)(M)-log(a)(N)；
(3) log (a) (m n) = nlog (a) (m) (n belongs to r)
(4) log (a k) (m n) = (n/k) log (a) (m) (n belongs to r)
Relationship between logarithm and exponent
When a is greater than 0 and a is not equal to 1, the x power of a =N is equivalent to log (a) n.
Log (a k) (m n) = (n/k) log (a) (m) (n belongs to r)
Bottom-changing formula (very important)
log(a)(N)= log(b)(N)/log(b)(a)= lnN/lna = lgN/LGA
The natural logarithm of ln is based on e.
The common logarithm of Lg is based on 10.
Generally speaking, if the power of a (a is greater than 0, and a is not equal to 1) is equal to n, then this number b is called the logarithm of n with the base of a, and it is recorded as log(a)(N)=b, where a is called the base of logarithm and n is called a real number.
Cardinality is greater than 0 instead of 1.
Operational properties of logarithm:
When a>0 and a≠ 1, m >；; 0, N>0, then:
( 1)log(a)(MN)= log(a)(M)+log(a)(N)；
(2)log(a)(M/N)= log(a)(M)-log(a)(N)；
(3)log(a)(M^n)=nlog(a)(M)
(4) the formula of bottoming: log (a) m = log (b) m/log (b) a (b >); 0 and b≠ 1)
Relationship between logarithm and exponent
When a>0 and a≠ 1, a x = n x = ㏒ (a) n (logarithmic identity).
Common abbreviations for logarithmic functions:
( 1)log(a)(b)=log(a)(b)
(2) common logarithm: lg(b)=log( 10)(b)
(3) natural logarithm: ln(b)=log(e)(b)
E=2.7 1828 1828 ... usually only the definition of logarithmic function is adopted.
The general form of logarithmic function is y=㏒(a)x, which is actually the inverse function of exponential function (Y = x = a y inverse function of two functions symmetrical about a straight line), and can be expressed as x = A Y. Therefore, the adjustment of A in exponential function (a >；); 0 and a≠ 1) are also applicable to logarithmic functions.
The figure on the right shows the function diagram of different size A:
You can see that the graphs of logarithmic functions are only symmetric graphs of exponential functions about the straight line y=x, because they are reciprocal functions.
[Edit this paragraph] Naturally
Definition domain: (0, +∞) Value domain: real number set r
Fixed point: The function image always passes through the fixed point (1, 0).
Monotonicity: when a> is 1, it is monotone increasing function and convex on the domain;
When 0<a< 1, it is a monotonic decreasing function in the definition domain and is concave.
Parity: Non-odd and non-even functions, or no parity.
Periodicity: Not a periodic function.
Zero: x= 1
Note: Negative numbers and 0 have no logarithm.
Two classic words: the bottom is true and the logarithm is positive.
True heteronegativity at the bottom
Tired! Adopt me. Sorry, I can't take pictures.