I. Multiple choice questions
The domain of 1. function is ().
Asian Development Bank and
Analysis; The key to finding a domain name is to remember three principles for finding a domain name!
Answer d, the first small question in homework 4. This type should do.
2. In the following function pairs, the two functions in () are equal.
A.，b .，+ 1
C.，d，
Analysis: The key to solve this problem is to look at the definition domain first, and then at the corresponding relationship. Only when the definition fields are the same can they be simplified, and then look at the corresponding relationship. Only the two are the same, and the two functions guess that it is the same function.
3. If, then ().
A.B. C. D。
、
4. In the following functions, odd function is ().
A.B. C. D。
Analysis: pay attention to the operational nature of parity function (see handout), and then know the answer is C by exclusion.
5. When () is known, it is infinitesimal.
A.B. C. D。
Analysis: therefore, choosing a. exam can of course be changed to
This problem involves the important limit 1.
6. When, the following variables are infinitesimal ().
A.B. C. D。
Analysis: the result is obtained from the property that the product of infinitesimal and bounded variable produces infinitesimal, and the answer is D.
7. If the function is continuous at x = 0, then k = (c).
A.-2 B.- 1 C. 1 D.2
8. The tangent slope of the curve at point (0, 1) is ().
A.B. C. D。
Analysis: this question examines the geometric meaning of derivative, which is the slope of the tangent of the curve, and finding the slope of the tangent is to find the derivative.
9. The tangent equation of the curve at point (0,0) is ().
A.y = x B. y = 2x C. y = x D. y = -x
Analysis:
Remember the point-diagonal equation: homework 1 There is a kind of problem to do.
10. If set, then ().
A.B. C. D。
1 1. The monotonic increase of the following function in the specified interval is ().
a . sinx b . e . x c . x 2d . 3-x
12. Let the function of demand q to price p be, then the demand elasticity is EP = ().
A.B. C. D。
Second, fill in the blanks
The domain of the 1. function is.
Analysis: the domain of piecewise function is the value of combined continuous segment X.
2. The domain of the function is.
Analysis:
3. If the function, then.
This question is the key question type.
4. If, then the graph of the function is symmetrical.
Analysis: To know the image characteristics of even-odd function (see handout), this is called even function.
5.。
Analysis: pay attention to the difference between homework and homework.
6. When it is known, it is infinitely small.
Analysis: Same as the previous multiple-choice question 5.
7. The tangent slope of the curve at this point is.
Analysis: Finding the slope is finding the derivative.
8. The stagnation point of the function is.
Analysis: the point with zero derivative is called the stagnation point of the function.
9. If the function of demand q to price is, the demand elasticity is.
Third, the calculation problem (through the calculation of the following questions, you have to master the basic formula of derivative and the derivative rule of compound function! This is the 10 question in the exam)
1. Yes, ask. 2. Know and ask.
3. Know and ask. 4. Know and ask.
5. Know and seek; Step 6 set up and ask
7. set and ask. 8. set and ask.
Fourth, the application problem (the following application problems must be mastered! This is a 20-point type question in the exam)
1. Let the cost function of producing a certain product be: (ten thousand yuan),
Find: (1) current total cost, average cost and marginal cost;
(2) When the output is what, the average cost is the smallest?
2. A factory produces a batch of products, the fixed cost is 2000 yuan, and the cost per ton of products is 60 yuan. The law of market demand for this product is (demand, price). Try asking:
(1) cost function, revenue function; (2) What is the maximum profit?
3. The total cost function of a factory producing q pieces of a certain product is C(q) = 20+4q+0.0 1q2 (yuan), and the unit selling price is p = 14-0.0 1q (yuan/piece). Try to find out:
(1) What is the maximum profit output? (2) What is the maximum profit?
4. The cost function of a product produced by a factory every day is (yuan). What should be the daily output in order to minimize the average cost? At this time, what is the average cost of each product?
5. It is known that the cost of producing a product in a factory is (ten thousand yuan). Q: How many products should be produced to minimize the average cost? What is the lowest average cost?
Reference solution
I. Multiple choice questions
1.D 2。 D 3。 C 4 explosive C 5。 A six. D 7。 C 8。 A nine. A 10。 B 1 1。 B 12。 B
Second, fill in the blanks
1.2.(-5,2) 3.4.y axis 5.16.7.8.9.
Third, the calculation problem
1. solution:
Step 2: Solutions
Step 3: Solution
4. Solution:
5. Solution: Because
therefore
6. Solution: Because of this.
7. Solution: Because
therefore
8. Solution: Because
therefore
Fourth, the application questions
1. Solution (1) Because the total cost, average cost and marginal cost are:
,
So,
,
(2) Ordering, obtaining (giving up)
Because it is the only stagnation point in the definition domain, and the problem does have a minimum value, the average cost is the minimum at 20.
2. Solve the (1) cost function = 60+2000.
Because, that is,
So the income function = = () =.
(2) Profit function =-(60+2000) = 40-2000
And = (40-2000 = 40-0.2.
Let = 0, that is, 40- 0.2 = 0 and get = 200, which is the only stagnation point in its domain.
Therefore, = 200 is the maximum point of the profit function, that is, the profit is the largest when the output is 200 tons.
3. The solution (1) is known as
Profit function
Then, sort and solve the unique stagnation point.
Because the profit function has a maximum value, the profit can be maximized when the output is 250 pieces.
(2) The maximum profit is
(yuan)
Step 4 explain why
Order, i.e. =0, get = 140, =-140 (discarded).
= 140 is the only stagnation point in its domain, and the problem does have a minimum value.
So = 140 is the minimum point of the average cost function, that is, to minimize the average cost, the daily output should be 140 pieces. At this time, the average cost is (yuan/piece).
5. Solution (1) because = =, = =
Order =0, that is, get, =-50 (discard),
=50 is the only stagnation point in its domain.
Therefore, =50 is the minimum point, that is, to minimize the average cost, it is necessary to produce 50 products.
(2)
Comprehensive exercises
First, multiple choice questions
1. The following equation is incorrect (). Correct answer: a.
A.B.
C.D.
Analysis; The key to solve this problem is to remember several common integrals and differentials (see lecture notes)
2. If yes, then = (). Correct answer: D.
A.B. C. D。
Note: The original function and the second derivative are mainly investigated, but the key to the exam is to know how to find f(x), that is, the indefinite integral of f(x) is all the original functions of f(x), as shown in the fourth question below.
3. The following indefinite integral, commonly used to calculate the partial integral (). Correct answer: C.
A.B.
C.D.
4. If yes, then f (x) = (). Correct answer: C.
A.b .-c . d-
5. If it is the original function, then the following equation holds (). Correct answer: B.
A.B.
C.D.
6. In the following definite integral, the integral value of 0 is (). Correct answer: a.
A.B.
C.D.
7. The following definite integral calculation is correct (). Correct answer: D.
A.B.
C.D.
Analysis: The above two questions mainly revolve around "odd function's definite integral in the symmetrical interval is known as 0", so remember!
8. The convergence of the following infinite integrals is (). Correct answer: C.
A.B. C. D。
9. Infinite integral = (). Correct answer: C.
0 BC
Second, fill in the blanks
1 .. should be filled in:
Note: We mainly study the reciprocal operation of indefinite integral and derivative (differential). Be sure to pay attention to whether to integrate first and then derivative (differential) or derivative (differential) and then integrate. This problem is to integrate first and then differentiate. Don't forget dx.
2. The original function of the function is. It should be:-cos2x+c.
3. If it exists and continues to exist, it should be filled in:
Note: this problem is to differentiate first, then integrate, and finally take the derivative.
4. If yes, please fill in:
5. If yes, then = Should fill in:
note:
6 .. Should be filled in: 0
Note: the result of definite integral is "numerical value", while the derivative of constant is 0.
7. Integral. Should be filled in: 0
Note: The definite integral of odd function in the symmetrical interval is 0.
8. Infinite integral is. Should be filled in: convergence
, so the infinite integral converges.
Third, the calculation questions (the following calculation questions should be mastered! This is the 10 question in the exam)
1. Answer: = =
2. Computing solutions:
3. Computing solutions:
4. Computing solutions:
calculate
Answer: = = =
6. Calculation solution: =
7. Answer: = = =
8. Answer: =-= =
9.
Solution:
= = = = 1
Note: To answer the above questions skillfully, you need to pay attention to the following two points.
(1) Common differential types must be remembered.
(2) Local integration: Three types should be defined in the routine examination.
Fourth, the application problem (the following application problems must be mastered! This is a 20-point type question in the exam)
1. The fixed cost of putting a product into production is 360 (ten thousand yuan), and the marginal cost =2x+40 (ten thousand yuan/hundred units). Try to find the increment of total cost when the output increases from 400 units to 600 units, and the average cost is the smallest when the output is what.
Solution: When the output is increased from 400 units to 600 units, the increment of total cost is
= = 100 (ten thousand yuan)
and
It was found that. X = 6 is the only stagnation point, and the problem does have a value that minimizes the average cost. Therefore, when the output is 600 units, the average cost can be minimized.
2. Given that the marginal cost (x) of a product is 2 yuan/piece, the fixed cost is 0, and the marginal revenue (x) is 12-0.02x, what is the maximum profit? If 50 pieces are produced on the basis of maximum profit output, what will happen to the profit?
Solution: Because the marginal profit =12-0.02x–2 =10-0.02x.
Order = 0, x = 500；; X = 500 is the only stagnation point, and the problem does have a maximum.
Therefore, when the output is 500 pieces, the profit is the largest.
When the output increases from 500 pieces to 550 pieces, the change of profit is
=500-525 =-25 yuan
That is to say, profits will decrease, 25 yuan.
3. The marginal cost of producing a product is (x)=8x (ten thousand yuan/100 pieces), and the marginal revenue is (x)= 100-2x (ten thousand yuan/100 pieces), where x is the output. What is the maximum profit? When the output reaches the maximum profit, what will happen to the profit when 200 units are produced?
Solution: (x) = (x)-(x) = (100–2x)-8x =100–10x.
Let (x)=0 and x = 10 (100 units); X = 10 is the only stagnation point of L(x), and the problem does have a maximum.
Therefore, x = 10 is the maximum point of L(x), that is, when the output is 10 (100 units), the profit is the largest.
Delta again
That is, 200 units will be produced from the maximum profit, and the profit will be reduced by 200,000 yuan.
4. Given that the marginal cost of a product is (ten thousand yuan/100 sets), the output is (100 sets) and the fixed cost is 18 (ten thousand yuan), find the lowest average cost.
Solution: Because the total cost function is =
When = 0, C(0) = 18, c =18; ; That is, C( )=
The average cost function is
Order, the solution is = 3 (100 units), there is indeed the problem of making the average cost the lowest.
So when q = 3, the average cost is the lowest. The average cost is the lowest (ten thousand yuan/100 sets).
5. Let the total cost function of producing a product be (10,000 yuan), where X is the output and the unit is100t. When selling x tons, the marginal profit is (ten thousand yuan/100 tons), and the output when the profit is maximum is (1);
(2) If 100t is produced on the basis of maximum profit, what will be the profit?
Solution: (1) Because the marginal cost is 0, the marginal profit =14–2x.
Order, x = 7；; According to the practical significance of this problem, x = 7 is the maximum point of the profit function L(x) and also the maximum point. Therefore, the profit is the largest when the output is 700 tons.
(2) When the output increases from 700 tons to 800 tons, the change of profit =- 1 (ten thousand yuan).
That is, the profit will be reduced by 6,543,800 yuan.
Comprehensive exercises of linear algebra
I. Multiple choice questions
1. Let A be a matrix and B be a matrix, then the following operation () can be performed.
Correct answer: a
A.AB B ABT C A+B D bat
Analysis: the number of columns in the left matrix is equal to the number of rows in the right matrix, and the multiplication is meaningful.
2. Let it be an invertible matrix of the same order, then the following equation holds () Correct answer: B.
A.B.
C.D.
Note: Remember the properties of transposed matrix and inverse matrix.
3. The following conclusion or equation is correct (). Correct answer: C.
A. if all matrices are zero, then there is B. If and, then
C. the diagonal matrix is a symmetric matrix D. If, then
4. Let it be a reversible matrix, and then (). Correct answer: C.
University of California, USA.
Note: Because A(I+B)=I, I+B.
5. Let identity matrix be, then = ().
Correct answer: D.
A.B. C. D。
6. If, then R (a) = (). Correct answer: C.
A.4 B.3 C.2 D. 1
, so rank (A)=2.
7. Suppose that the augmented matrix of a linear system of equations is transformed into by elementary row transformation, then the number of free unknowns in the general solution of this linear system of equations is () Correct answer: A.
A. 1
Analysis: number of free unknowns =n (number of unknowns)-rank (A)=4-3= 1,
The inspection should be directly visible to the naked eye.
8. The solution of linear equations is (). Correct answer: a.
A. there is no solution B. there is only 0 solution C. there is a unique solution D. there are infinite solutions
Note: After being transformed into a trapezoidal matrix, the contradictory equation "0=K" appears in the last row and has no solution.
9. The necessary and sufficient condition for a system of linear equations to have infinite solutions is ().
Correct answer: D.
A.B. C. D。
Note: The judgment theorem of the solution of linear equations should be memorized on the basis of understanding.
10. Let the linear equations have a unique solution, then the corresponding homogeneous equations ().
A. there is no solution B. there is a non-zero solution C. there is only a zero solution D. the solution cannot be determined
Correct answer: C.
Note: there is a unique solution that shows that
However, it should be noted that if AX=0 only has a unique zero solution, while AX=b may have no solution (or the solution is uncertain).
Second, fill in the blanks
1. If the matrix A = and B =, then ATB =. You should fill in:
2. If all matrices are hierarchical, then the necessary and sufficient condition for the equation to be established is to fill in: exchange matrix or AB=BA.
3. Let, if, be a symmetric matrix. Should be filled in: 0.
Note: The distribution of elements in a symmetric matrix is symmetric about the main diagonal, so you can see the symmetric matrix.
4. Let all matrices be orderly and reversible, then the solution of matrix x =.
Should fill in:
5. If the linear equations have non-zero solutions, it should be:-1.
, has a nonzero solution.
6. Homogeneous linear equations with rank (a) = r
Note: the key is to see that the number of unknowns is n.
7. If the coefficient matrix of homogeneous linear equations is 0, the general solution of the equations is 0.
The general solution of the equation is (where is the free unknown quantity)
Third, the calculation questions (the following questions should be mastered skillfully! This is the 15 sub-question type of the exam)
1. Let the matrix A = and find the inverse matrix.
Solution: Because (A I)= 1
So A- 1=
Note: This question can also be changed to the following form:
For example, solve the matrix equation AX=B, where
Answer:
Another example is: knowing and seeking.
2. Let the matrix A = and find the inverse matrix.
Solution: Because, and
therefore
3. Let the matrices A = and B = and calculate (ba)- 1.
Solution: Because BA= =
(BA I)= 1
So (BA)- 1=
4. Set the matrix and solve the matrix equation.
Solution: because, that is,
So X = = =
5. Find the general solution of linear equations.
Solution: Because
So the general solution is (where is the free unknown)
6. Find the general solution of linear equations.
So the general solution is (where is the unknown freedom)
7. Set the homogeneous linear equations, find out what value L takes, and find the general solution if the equations have non-zero solutions.
Solution: Because the coefficient matrix A =
Therefore, when l = 5, the system of equations has a non-zero solution, and the general solution is (here is a free unknown).
8. When what value is taken, is there a solution to the system of linear equations? And find the general solution.
Solution: Because augmented matrix
So when =0, the system of linear equations has infinite solutions.
And the general solution is a free unknown]
There are also the following test methods for this kind of problems: when the value is, the linear equations
There is a solution, find a general solution.
9. When is a value, an equation?
There is a unique solution, infinitely many solutions, and no solution?
When and when, the equations have no solution;
When, when the equation has a unique solution;
When sum, the equation has infinite solutions.