1、0;
2、 1;
3、x-2y+ 1 = 0；
4、2x；
5、- ;
Second, multiple-choice questions:
1、D；
2、B；
3、B；
4、B；
5、B；
Third, answer questions.
1, calculation limit
(1) solution: original formula =
=
=
(2) Solution: Original formula =
=
=-
(3) Solution: Original formula =
=
=-
(4) Solution: Original formula =
=
(5) When the solution is ∵x,
∴ =
=
(6) Solution: =
= (x+2)
=4
2, set the function:
Solution: f(x)= (sin +b)=b
f(x)= 1
(1) To make f(x) have a limit at x=0, as long as b= 1,
(2) Let f(x) be continuous at x=0, then
f(x)= =f(0)=a
That is, when a=b= 1, f(x) is continuous at x=0.
3. Calculate the derivative or differential of the function:
(1) solution: y'=2x+2xlog2+
(2) Solution: y'=
=
(3) Solution: y'=[]'
=- (3x-5)'
=-
(4) Solution: y' =-(ex+xex)
= -ex-xex
(5) solution: ∫y' = aeaxsinbx+beaxcosbx
=eax(asmbx+bcosbx)
∴dy=eax(asmbx+bcosbx)dx
(6) Solution: ∫y' =-+
∴dy=(- + )dx
(7) solution: ∫y' =-sin+
∴dy=(-symplectic) dx
(solution: ∫y' = nsinn- 1x+ncos NX.
∴dy=n(nsinn- 1+ cosnx)dx
(9) Solution: ∫y' =
=
∴
(10) solution:
4.( 1) Solution: Derive x on both sides of the equation.
2x+2yy'-y-xy'+3=0
(2y-x)y'=y-2x-3
y'=
∴dy=
(2) Solution: Derive x on both sides of the equation:
cos(x+y)( 1+y ')+exy(y+xy ')= 4
[cos(x+y)+xexy]y ' = 4-cos(x+y)-yexy
y'=
5.( 1) Solution: ∫y' =
=
(2) Solution:
=
Basic homework of economic mathematics II
First, fill in the blanks:
1、2xln2+2
2、sinx+C
3、-
4、ln( 1+x2)
5、-
Second, multiple-choice questions:
1、D
2、C
3、C
4、D
5、B
Third, answer questions:
1, calculate the following indefinite integral:
(1) solution: original formula =
=
=
(2) Solution: Original formula =
=
(3) Solution: Original formula =
=
=
(4) solution: the original formula =-
=- +C
(5) Solve the original formula =
=
=
(6) solution: original formula =Z
=-2cos
(7) Solution: Original formula =-2
=-2xcos
=-2xcos
(Solution: Original formula =
=(x+ 1)ln(x+ 1)-
=(x+ 1)ln(x+ 1)-x+c
2. Calculate the following integral
(1) solution: original formula =
=(x-
=2+
=
(2) Solution: Original formula =
=
=
(3) Solution: Original formula =
=
=
=4-2
=2
(4) Solution: Original formula =
=
=
=
(5) Solution: Original formula =
=
=
=
=
=
(6) Solution: Original formula =
=4+
=
=
=
=
Basic homework of economic mathematics 3
First, fill in the blanks:
1.3
2.-72
3.a and B are interchangeable.
4.(I-B)- 1A
5.
Second, multiple-choice questions:
1.C 2。 A 3。 C 4 explosive A 5。 B
Third, answer questions.
1, solution: original formula =
=
2. Solution: Original formula =
=
3. Solution: Original formula =
=
2. Calculation:
Solution: Original formula =
=
=
3. Set the matrix: Solution:
4. Set the matrix: Solution: A= Minimize r(A).
only
5. Find the matrix A=
∴r(A)=3
6. Find the inverse of the following matrix:
(1) solution: [A 1]=
∴A- 1=
(2) solution: [A 1]=
∴A- 1=
7. Build a matrix
Solution: Suppose.
that is
∴X=
Fourth, the problem of proof:
1, certificate: B 1 and B2 are interchangeable with a, that is,
B 1A=AB 1 B2A=AB2
(b 1+B2)A = b 1A+B2A = ab 1+AB2
AA(B 1+B2)=AB 1+AB2
∴(B 1+B2)A=A(B 1+B2)
(b 1 B2)A = b 1(B2A)= b 1(AB2)=(B2A)B2 = ab 1 B2
That is to say, B 1+B2, B 1B2 and a are interchangeable.
2. certificate: (A+AT)T=AT+(AT)T=AT+A=A+AT.
Therefore, A+AT is a symmetric matrix.
(AAT)AAT
(AAT)T=AT(AT)T=ATA
3. Prove that if AB is a matching matrix, then (AB)T=BTAT=BA=AB.
∵AB is a geometrically symmetric matrix.
Knowing that AT=A BT=B means AB=BA.
On the other hand, if AB=BA (AB)T=BTAT=BA=AB.
That is, (AB)T=AB.
∴AB is a symmetric matrix.
4. let a be a geometrically symmetric matrix, that is, at = a.
(B- 1AB)T=BTAT(B- 1)T
=BTAT(BT)T (∵B- 1=BT)
=B- 1AB
∴B- 1AB is a symmetric matrix.
Basic homework of economic mathematics 4
First, fill in the blanks:
1, 1 < x ≤ 4 and x≠2
2, x= 1, x= 1, small value.
3、
4、 4
5、 ≠- 1
Second, multiple-choice questions:
1、B
2、C
3. A.
4、C
5、C
Third, answer questions.
1, (1) solution:
-e-y=ex+C means ex+e-y = c.
(2) Solution: 3y2dy=xexdx
y3=xex-ex+C
2.( 1) solution: the solution of the equation corresponding to the homogeneous linear equation is y=C(X+ 1)2.
Let the solution of the equation be: y=C(x)(x+ 1)2. Use the constant high easy method.
Substituting the original equation, we get c' (x) (x+1) 2 = (x+1) 3.
C'(x)=x+ 1
c(x)= 1
Therefore, the general solution of the equation is: (
(2) solution: from the general solution formula
Where P(x)=-
Y=e
=elnx
=x
=cx-xcos2x
3 、( 1)y'=e2x/ey
That is, eydy=e2xdx
ey=
Substitute x = 0 and y = 0 to get C=
∴ey=
(2) Solution: Equation deformation
y'+
Substitution mode
Y=e
=
=
= substitute X = 1 and Y = 0 to get c =-e.
∴y= is a special solution satisfying y( 1)=0.
4. Solve the general solution of the following linear equation:
(1) solution: coefficient matrix:
A2=
∴ The general solution of the equation is:
Where x3 and x4 are free unknowns.
(2) Solution: The augmented matrix is transformed into trapezoid by elementary row transformation.
A (& mdash =
So the general solution of the equation is:
X 1=
X2=, where x3 and x4 are free unknowns.
(5) Solution: A (&; mdash=
In order to make the equation have a solution, then
At this time, the general solution is that x3 and x4 are free unknowns.
(6) solving: transforming the augmented matrix of the equations into a trapezoidal matrix;
A (& mdash =
From the judgment theorem of the solution of the equation, we can get
When a =-3 and b ≠ 3, rank (a): 0, q2<0, c ˊ (q) < 0.
When q=20, the function has a minimum value.
That is, when the output q=20, the average cost is the smallest.
(2) Solution: the total income function R(q)=P%=( 14-0. 0 1q)q = 14q-0.0 1q2
Profit function l (q) = r (q)-c (q) =-0.02q2+10q-20,10.
Let's find the maximum value of the profit function
L'(q)=-0.0 1q+ 10=0，q=250。
When q> is 250, l' (q) < 0, and l '(q)> 0 at q & lt250.
Therefore, the maximum value of L(q) is L(250)= 1230 when q=250.
That is, the profit reaches the maximum when the output is 250, and the maximum value is 1230.
(3) Solution: C'(x)=2x+40
C (x) = x2+40x+C. When x=0, (cx)=36, so C=36.
Total cost function: C(x)=x2+40x+36.
C(4)=42+40×4+36=252 (ten thousand yuan)
C(6)=62+40×6+36=3 12 (ten thousand yuan)
Total cost increment: △ c (x) = 312-212 =100 (ten thousand yuan)
Average cost C(x)=x+40+
When the minimum value is obtained only when x=, that is, the output is 600 units, the average cost can be minimized.
Solution: income function R(x)= 1
When x=0, R(0)=0, that is, C=0.
Income function r (x) =12x-0.01x2 (0
When the cost function C(x)=2x+C x=0, C(x)=0, so C 1=0.
Cost function C(x)=2x
Profit function l (x) = r (x)-l (x) =1x-0.01x.
L′(x)= 10-0.02 x x = 500，L′(x)>0
Therefore, when x=500, L(x) reaches the maximum.
The maximum profit is 2500 yuan when the output is 500 pieces.
If 50 pieces are produced on this basis, that is, when the output is 550 pieces, the profit will be L(550)=2475, and the profit will be reduced by 25 yuan.