Engineering Mathematics Composite Function Problems. Solving... It s urgent

3.Let u=(x 3+y 3) (x 2+y 2) ,z≠0,f(z)=u+iu,z≠0,du/dx=du/dy;du dx -du dy=0 satisfies the r-c condition, f(z) is intermittent at z=0, and is not differentiable.

4. f(z)=u+iv

1) v=0, r-c condition ==> du dx=du dy=0, u=constant.

2)f(z),f('(z) Analysis, f'(z)=du/dx+idv/dx

f'(z)=du/dy-idu/dy

r-c condition ==>f(z)=constant.

3) U=constant, r-c condition ==>v=constant.

5. z=x+iy

z^2=x^2+2ixy-y^2

x z 2=x 3-x(y 2) +2ix 2 y ==> x(y 2) cannot be unique real numbers.

exp(ix)=cos(x)+isin(x)

sinx=(1/2)i(exp(ix)-exp(-ix))

cosx=(1/2)i(exp(ix)+exp(-ix))

sinhx=(1/2)(exp(x)-exp(-x))

coshx=(1/2)(exp(x)+exp(-x)).

lim(z->z0) f(z)/g(z)=lim(z->z0) [f(z)-f(z0)]/[g(z)-g(z0)]

lim(z->z0) =f'(z0)/g'(z0)

lim [sin(u+iv)]/(u+iv)=lim (u->0,v->0)cos(u+iv)=lim (u->0,v->0)[cosu cosiv-sinu siniv]=1

Brother, I still can't see clearly after zooming in.

Can someone who doesn't have the same academic ability as a university do these questions?? You'd better study hard.

The analytic function can be Taylor, and then it can be reduced.

The first problem is to find the partial derivative to see if the Cauchy-Riemann equation is satisfied.

The last problem can be calculated using Lopida's law, where the numerator and denominator are derived at the same time.

Analysis: g(x)=f(u)=8+2u-u2, u= is a composite function, only need to find the monotonic interval of f(u)=8+2u-u2 and u(x)=2-x2, and then solve it according to the determination theorem of the monotonicity of the composite stool slip function, that is, the coarse can be solved.

Answer: Let f(u)=-u2+2u+8, u(x)=2-x2, from u(x)=2-x2, we can see that x 0 is decreasing, x<0 is increasing and u 2

From f(u)=-u2+2u+8, it can be seen that when u 1 increases, when 11) when u 1, 2-x2 1, that is, x 1 or x -1, so when x 1, g(x) decreases monotonically, and when x finches -1, g(x) increases monotonically.

2) When 1 is -1 combined, the monotonically increasing interval of g(x) is (- 1 , 0,1).

The monotonically decreasing interval of g(x) is (-1,0), 1,+

y=1/√[x-1)(x+3)].1)--x<-3 or x>1.

1/√[x+1)^2-4]

.2) The axis of symmetry is x=-1, so the denominator is a subtraction function at x<-3 and an increasing function when the denominator is >1. The function y=1 x is a subtraction function when the denominator x>0 is scattered.

According to the law of compound functions, "the same increases if there is a difference, and the difference decreases", he knows that the function is a subtraction function at x>1.

So the single subtraction interval of the function is (1,+

When x > boy sings 0

LNX has a definition.

e^lnx =x

Because y=lnx is the inverse of y=e x.

The cherry blossom model to. The answer is Nambi: a

Pick A. Option b, e (2lnx) =e lnx) 2 = x 2, option tung bend c, e [(1 xunlun destroy2)lnx] =e lnx) (1 mu 2) =x

Option d, e (-lnx) =1 (e lnx) =1 x

1。Substituting x 2 directly into f(x) is 2x 2-1

2。Substituting 1 (x 2+1) into f(x) is 2*(1 (x 2+1))-1

3。Think of the independent variable as 2x-1+2, i.e., 2x+1 is substituted for g(x) to get 1 (4*x 2+4*x+2).

First, consider defining the domain, judging monotonicity according to the same increase and difference and subtraction, and what you are talking about is of course to find the intersection of the two.

Definition of composite function: If y=f( ) and =g(x), and the intersection of the g(x) range and the f( ) definition domain is not empty, then the function f[g(x)] is called the composite function, where y=f( ) is called the outer function, =g(x) is called the inner function, in short, the so-called composite function is a function composed of some elementary functions. For example, y=log(1 2) (x +4x+4), so that y= log(1 2) u(outer function), u= x +4x+4(inner function) ps:

1 2 is the base.

The steps to determine the monotonicity of a composite function are as follows: (1) find the domain of the composite function definition; (2) decomposing the composite function into several common functions (primary, quadratic, exponential, finger, and pair functions); (3) Use the definition method or the derivative method to judge the monotonicity of each common function (f'(x) 0, the obtained x range is a monotonically increasing interval; f'(x) 0, the range of x is a monotonically decreasing interval,); 4) convert the value range of intermediate variables into the value range of independent variables; (5) Find the monotonicity of the composite function (the inner and outer functions are "the same increase and different decrease").

1. Definition of composite function.

Let y=f(u) and u=g(x), when x changes in the definition domain dg of u=g(x), the value of u=g(x) changes in the definition domain df of y=f(u), so a functional relationship between the variable x and y formed by the variable u is denoted as.

y=f(u)=f[g(x)] is called a composite function, where x is called the independent variable, u is the intermediate variable, and y is the dependent variable (i.e., the function).

2. Generation conditions.

Not any two functions can be compounded into a composite function, only if there is a non-empty subset of the domain of = (x) z which is a subset of the defined domain df of y=f( ).

3. Define the domain.

If the domain of the function y=f(u) is b u=g(x) and the domain is a, then the domain of the composite function y=f[g(x)] is .

The derivative of the composite function is d=

Fourth, periodicity.

Let y=f(u), the minimum positive period be t1, and the minimum positive period of = (x) is t2, then the minimum positive period of y=f( ) is t1*t2, and any period can be expressed as k*t1*t2 (k belongs to r+).

5. Monotonicity The monotonicity of the composite function is determined by the increase or decrease of y=f(u), = (x). That is, "increase and increase, decrease and decrease, increase and decrease", which can be simplified to "increase and decrease with difference".

The steps to determine the monotonicity of a composite function are as follows: (1) find the domain of the composite function definition;

(2) decomposing the composite function into several common functions (primary, quadratic, exponential, finger, and pair functions);

3) judge the monotonicity of each common function;

4) convert the value range of intermediate variables into the value range of independent variables;

5) Find the monotonicity of the composite function.

For example, discuss the monotonicity of the function y=. Derivative solution of a composite function: The function defines the domain as r.

Let u=x 2-4x+3,y=.

The exponential function y = is a subtraction function on (-, u=x 2-4x+3 is a subtraction function on (- 2], an increasing function on [2,+, function y=an increasing function on (- 2], and a subtraction function on [2,+.

The composite function is used to find the range of parameter values.

Finding the range of values of a parameter is an important problem, and the key to solving the problem is to establish an inequality group about this parameter.

Transform all known conditions.

Composite Functions:

Let y=f(u), u=g(x), when x changes in the definition domain dg of u=g(x), the value of u=g(x) changes in the definition domain df of y=f(u), so a functional relationship between the variable x and y formed by the variable u, denoted as: y=f(u)=f[g(x)] is called the composite function, where x is called the independent variable, u is the intermediate variable, and y is the dependent variable (i.e., the function).

The inner functions and outer functions will not be listed and defined for you, for example:

Let's say y=(3x+5).

This is the composite function.

It can be seen as y=x and y=3x+5

That is, a composite function of a primary function and an exponential function.

How to find monotonicity, you can find the monotonicity of the inner and outer functions separately The monotonicity of the composite function is generally to look at the monotonicity of the two functions contained in the function (1) If both are increasing, then the function is an increasing function (2) One is subtracting and the other is increasing, that is the subtraction function.

3) Both are subtractive, which is the increase function.

Take a simple example: f(x)=x2+x+1, g(x)=x+2

The so-called composite function f(g(x)) is to use g(x) instead of the x in f(x), that is.

f(g(x))=[g(x)]^2+g(x)+1

x+2)^2+(x+2)+1

In this case, f( ) is the outer function, and g( ) is the inner function.

Monotonicity Problem:

Let's assume that f(x) is monotonically increased, and as the value of x increases, the value of f( ) increases.

As an outer function of the composite function f(g(x)), the value of f( ) increases as the value of g( ) increases.

If g(x) also increases monotonically, then as the value of x increases, the value of g( ) increases. And since the value of g( ) increases, the value of f( ) increases, so when x increases, the value of f(g( ) increases, that is, f(g(x)) is an increase function.

If g(x) is monotonically decreasing, then as the value of x increases, the value of g( ) decreases. And since the value of g( ) decreases, the value of f( ) decreases, so when x increases, the value of f(g( ) decreases, and f(g(x)) is the reduction function.

The inner layer, the outer layer, and the same monotonicity increases, and the heteromonotonicity decreases.

For example: y=2 (x 2+1).

The outer layer is, y=2 u is an increasing function on the defined domain, and the inner layer is, u=x 2+1 is decreasing on (-infinity, 0) and increasing on (0, +infinity) The function y(-infinity, 0) is subtracting on (0, +infinity).

Let y=f(u) and u=g(x), when x changes in the definition domain of u=g(x), the value of u=g(x) changes in the definition domain of y=f(u), so a functional relationship between the variable x and y formed by the variable u, denoted as: y=f(u)=f[g(x)] is called a composite function, where x is called the independent variable, u is the intermediate variable, and y is the dependent variable (i.e., the function). u=g(x) is called the inner function, and y=f(u) is the outer function.

Monotonicity can be borrowed from the law of multiplication, the same sign is positive, and the different sign is negative. That is, u=g(x) is an increasing function, and y=f(u) is also an increasing function, then y=f[g(x)]

is an increment function. For example, y=-u+1, u=x 2, when x<0, u monotonically decreases, and when u increases, y monotonically decreases, so x<0 is the monotonically increasing interval of y; When x > 0, u increases monotonically, and when u increases, y decreases monotonically, so x>0 is a monotonically decreasing interval of y.

That is, increase or increase, subtraction and compound are also increase, increase and decrease, and decrease and increase are all subtraction.

The composite function is y=f(t), t=g(x), and its monotonicity is the same increase and different decrease, that is, f(t) and g(x) have the same monotonicity, then the composite function increases, and vice versa. I don't know if you understand.

The domain is defined as modulus silver x 2-5x+6 0, i.e., (x-2)(x-3) 0, i.e., x3 or x2

The value range is x 2-5x+6=(x-5 2) 2+6-25 4.

x^2-5x+6=(x-5/2)^2+6-25/4≥6-25/4=-1/4<0

The minimum value should be 0 of the defined domain, i.e., the value range is [0,+

Monotonicity: f(x) is derived from the root number of x 2-5x+6, so its monotonicity is the same as x 2-5x+6, that is, when x 5 2, monotonically decreases, and when x 5 2, monotonically increases.

However, because the definition domain is limited to x 3 or x 2, there is no definition in the middle paragraph, so its actual monotonic interval is:

When x 2, it decreases monotonically; When x 3, monotonically incremented.

I hope it helps you.