I. Fill in the blanks (12 points)
1. Zu Chongzhi, China mathematician, was born in the Southern and Northern Dynasties. His two results in pi are as follows: ① pi is between 3. 14 15926 and 3. 14 15927; ② The approximate velocity is 0 and the density is 0.
2. The necessary and sufficient condition for a function to have a limit at a point is that the left permission of the function at that point exists and the right limit is equal.
3. In short, the derivative is the limit of the average rate of change, and the definite integral is the limit of the integral summation formula.
4. The point with zero derivative is called stagnation point.
5. The Lagrange median formula of function y=f(x) is = ().
6. The variable upper bound definite integral is the original function of the integrand in the defined interval.
Second, multiple-choice questions (12 points)
Choose the correct answer from four conditions: ① Sufficient condition, ② Necessary condition, ③ Necessary and Sufficient condition, ④ Neither sufficient condition nor necessary condition, and fill the serial number in the brackets of the following questions:
1. The zero derivative is the extreme value of the derivative function (②)
2. The derivability is continuous (①)
3. Continuity is integrable (①)
4. For univariate functions, differentiability is differentiable (③)
5. Bounded is integrable (②)
6. The left and right derivatives of the function exist at one point and are differentiable (③).
Three, a brief description of dialectics in the process of seeking the limit (7 points)
The answer (1) embodies the contradictory law of unity of opposites.
Let the sequence {} be the limit. In the process of infinite increase, it is a variable, so there are infinite numbers ... This reflects the process of infinite change of variables, and the limit reflects the result of infinite change. Each one is not, reflecting the opposite of the change process and the change result, and transforming it into the unity of the process and the result; (2) Because {} can't be written completely, we use the changing state of the difference between = and a finite number to learn. If the difference tends to 0, the limit of the sequence is. Therefore, the limit is the unity of finite and infinite; (3) each is an approximation of a, the greater the n, the better the approximation. No matter how big N is, it is always an approximation of A. When N is used, the approximation is transformed into the exact value A, which embodies the unity of opposites between approximation and accuracy.
(2) It reflects the laws of quantitative change and qualitative change.
Four, the calculation problem (42 points)
1.
Solution = = (2x+ 1)
= 2x+ 1=-4+ 1=-3。
2.
Answer = =
= =
=e2 = e2 = e2
3.
Solution =
= = 1=- 1
4. Given the function y=, find.
Answer = =
= =
=- = .
5. Know and ask.
Solution, take the logarithm of both sides of the equation, and get
①
(1) take derivatives on both sides of the equation, and there are
=
∴ = +
∴ = + .
6.。
Answer = =
= = .
5. What is the definite integral of odd function on the interval? And prove it. (9 points)
When the solution (1) is odd function, the definite integral on the interval is zero, that is
=0
(2) Proof =+(*)
Where =-
Order, then when, t=0, when,
∴ =- =
Independent of integer symbols
F(x) is odd function.
- - .
Replace (*) to obtain
= + =- + =0.
6. Find the area of the figure surrounded by parabola and straight line. (9 points)
Draw a sketch according to the meaning of the question.
Solve simultaneous equations, and the intersection points are (-1, 1), (2,4).
∴ The area of the attached drawing is:
S= + -
= = - +4+2- = .
Seven, known function, continuous in point, value (9 points).
Solution 8
∴ .
=
=
=
= .
The function is continuous at this point.
∴ = = =
∴ .
I. Fill in the blanks (30 points)
Gauss is a great German mathematician at the turn of the century.
2. If it is correct, it always exists, making it timely.
3. The definition domain of the function is shown in the figure on the right.
4. The necessary condition for integrability on d is that the function is bounded on d. 。
5. If AB =, events A and B are mutually exclusive.
6. Determinant = 0.
Second, the basic operation (32 points)
1., find
solve
=
2. known d: calculation
solve
= .
3. A batch of products 100, including 90 genuine products and 0 defective products 10. Select 3 pieces from this batch of products and find out the probability of defective products.
Solution 1 Let A = {defective}, = {defective}, = 1, 2,3. So A = is mutually exclusive, so we can know from the classical probability.
P( )= P( )=
P( )=
From the addition formula, we get
P(A)= P(A 1+A2+A3)= P(A 1)+P(A2)+P(A3)
=0.24768+0.02505+0.00074=0.2735.
Solution 2 is calculated by inverse probability formula.
Because the opposite of thing A is = {all three products taken out are genuine}, so
P( )=
So p (a) =1-p () =1-0.7265 = 0.2735.
4. Find the area of the curve and the attached graph.
Such as right solution sketch. Solve equations.
(-3, -7), (1, 1).
As shown in the figure, projected on the X axis, we can see that the included figure is
D:-3≤x≤ 1，2x- 1≤y≤2-x2。
So the area of the drawing is:
= .
Three. Calculation (30 points)
1, 0, 0.
Solution rule z
=
2. Find the value of determinant
Add to column ① ② ③
(-1) × ④ column
Solve the original determinant
=x -2
=x
-
= =
3. Calculate the double integral:
Where d is surrounded by straight lines x=0, y=x and y=π.
Sketch, as shown on the right. Projecting the integration region d on the x axis, and expressing d by inequality:
D:0≤x≤π，x≤y≤π。
∴
(*)
In ...
Substitute into formula (*), Ⅷ
Step four, beg
Release instruction
Four, using matrix method to solve linear equations (8 points)
Elementary transformation of solution of augmented matrix.
① Line is added to ② line.
①× (-2) line is added to ③ line.
① Line is interchanged with ② line.
② Line is interchanged with ③ line.
(-1) × ③ line
(-4) × ② Row addition
Go to line ③.
The original equation of ∴ can be simplified as
Using the recursive method, the unknown number is obtained from bottom to top.
∴ the solution of the equation is
I. Fill in the blanks (18 mark)
The necessary and sufficient condition for the 1. function to have a limit at one point is that the left derivative and the right derivative exist and are equal.
2. The point with zero derivative is called stagnation point (stable point).
3. In short, the derivative is the limit of the average rate of change, and the definite integral is the limit of the integral summation formula.
4. The Lagrange median formula of the function on [a, b] is.
Zu Chongzhi, a mathematician in China, was born in the Southern and Northern Dynasties. His contribution to pi is that (1) pi is between 3. 14 15926 and 3. 14 15927; (2) Approximation rate is 0 and density rate is 0.
6. The variable upper bound definite integral is the original function of the integrand function.
Second, multiple-choice questions (12 points)
Choose the correct answer from four conditions: ① Sufficient condition, ② Necessary condition, ③ Necessary and Sufficient condition, ④ Neither sufficient condition nor necessary condition, and fill the serial number in the brackets of the following questions:
1 and the zero derivative are the extreme values of the derivative function (②).
2. The derivability is continuous (①).
3. Continuity is integrable (①).
4. For unary functions, differentiability is differentiable (③).
5. Bounded is integrable (②).
6. The left and right derivatives of the function exist at one point and are equally derivable (③).
Three, the calculation problem (42 points)
1、
solve
=
2、
solve
=
=
=
3. Know how to seek
The solution is lny = (x+ 1) ln (x+ 1) where the logarithm is taken on both sides of y = (x+ 1) and the derivative of x is taken on both sides:
4, known, find dy
Solve dy=y'dx and find y'
y′=
5、
solve
=
6、
solve
=
Four, find the area of the figure surrounded by parabola and straight line (12 points)
Solution ① Draw a graph surrounded by a parabola y = x2- 1 and a straight line y=x+2.
② Find the intersection of parabola y=x2 and straight line y = x+2 to get a (- 1,1); B (2,4)
(3) Find the area s of the attached figure:
=
Five, the known function is continuous at this point, and find the value of a (8 points).
The solution of the function f(x) is continuous when x = 0.
∴
but
∴
∴A=e.
Six, a brief description of dialectics in the process of seeking the limit of sequence (8 points)
The answer (1) embodies the contradictory law of unity of opposites.
Let the sequence {} be the limit. In the process of infinite increase, it is a variable, so there are infinite numbers ... This reflects the process of infinite change of variables, and the limit reflects the result of infinite change. Each one is not, reflecting the opposite of the change process and the change result, and transforming it into the unity of the process and the result; (2) Because {} can't be written completely, we use the changing state of the difference between = and a finite number to learn. If the difference tends to 0, the limit of the sequence is. Therefore, the limit is the unity of finite and infinite; (3) each is an approximation of a, the greater the n, the better the approximation. No matter how big N is, it is always an approximation of A. When N is used, the approximation is transformed into the exact value A, which embodies the unity of opposites between approximation and accuracy.
(2) It reflects the laws of quantitative change and qualitative change.
I. Fill in the blanks (18 mark)
1, in short, the derivative is the limit of the average rate of change, and the definite integral is the limit of the integral sum formula.
2. The point with zero derivative is called stagnation point.
3. The elementary row transformation of matrix refers to ① exchanging two rows of matrix; ② Multiply each element of a row in the matrix by a non-zero number; ③ Multiply each element in one row of the matrix by a number and add it to the corresponding element in another row.
4. Let A and B be squares of order n, then (ab)'=
5. The variable upper bound definite integral is the original function of the integrand function.
6、D(aξ+b)= .
Second, multiple-choice questions (12 points)
Choose the correct answer from four conditions: ① Sufficient condition, ② Necessary condition, ③ Necessary and Sufficient condition, ④ Neither sufficient condition nor necessary condition, and fill the serial number in the brackets of the following questions:
1 and the zero derivative are the extreme values of the derivative function (②).
2. For univariate functions, the derivability is continuous (①)
3. Continuity is integrable (①)
4. The determinant |A|≠0 of matrix A (③) is reversible.
5. For univariate functions, differentiability is differentiable (③)
6. The coefficient determinant δ ≠ 0 has a unique solution for the linear equations (①).
Three, a brief introduction to dialectics in the process of derivative (8 points)
The answer (1) embodies the contradictory law of unity of opposites.
The average change rate and instantaneous change rate, approximate value and exact value are contradictory before taking the limit, and the result of taking the limit unifies the two sides of the contradiction.
(2) It reflects the laws of quantitative change and qualitative change.
Four, the calculation problem (42 points)
1, known function y=lnsin (), find y'
solve
Step 2 seek the limit
solve
3. Given z=, find.
solve
4. Find indefinite integral
solve
5. Find indefinite integral
The solution is
=
=
6. Know, seek
solve
Five, the application problem (18)
Known curves and straight lines surround the plane area d,
1, find the area of d by definite integral.
Solution ① Draw a curve first, the area surrounded by the image in the rectangular coordinate system.
2 find the intersection.
③ Find the closed area s 。
.
2. Find the area of d by double integral.
When calculating the area of d by double integral, the integrand function should be 1.
Six, let random variables have probability density (8 points)
Find the (1) constant c
From the explanation, we can see that
Do you understand?
(2)
solve
(3) Distribution function
The solution distribution function is:
When,
When,
When,
=
∴
I. Fill in the blanks (15)
1, the density function of standard normal distribution is
2. Statistics are divided into descriptive statistics and inferential statistics.
3. The basic contents of statistical inference are parameter estimation and hypothesis testing.
4. For an n-order square matrix A, if there is an n-order square matrix B, let AB = BA = E, then A is a reversible moment connection, and B is called the inverse matrix of A, which is denoted as.
5. Write the definition of partial derivative of function about x at point.
Two. Calculation (20 points)
1, find the value of determinant.
2×① line addition
Go to line ②
Solution = 0
2, known and sought
Solution a+b =+=
AB= =
AT= =
3. Know, seek
Answer =, =
4. Know, seek
Issue orders.
∴
=
∴
=
∴ =
3. Calculate the double integral, where d is the area enclosed by X axis, Y axis and unit circle in the first quadrant (15 point).
The de-integration area is shown on the right.
D:0≤x≤ 1，0≤y≤
= .
Four, use double integral to find the area of the graph surrounded by curve and straight line (15 points)
Draw a single picture, as shown on the right. The integration area d is
D:-2≤x≤ 1，≤y≤
∴
5. A factory plans to recruit 420 people, and the number of people taking the recruitment examination is 2 100. The results of spot check show that the average score of the test is 120 and the standard deviation is 10. Strive for admission scores (note:,). (65,438+05)
As can be seen from the topic design, the score of this exam is x ~ n (120, 102).
Set the admission score to, and carry out standardized conversion:
(*)
Then z ~ n (0, 1)
The ratio of admission number P(z ≥= = 0.2.
∴P(- <z & lt)= 1-P(z≥ )= 1-0.2=0.8
Based on topic, knowledge = 0.84.
Substituting in the formula (*) has 0.84=,
Admission scores can be obtained as follows:
= 10×0.84+ 120= 128.4.
Six, a class of 36 students participated in the unified mathematics examination after the teaching reform experiment. It is known that the average math score of this class is 1 14, the average math score of the whole school is 1 10, and the standard deviation is 16. Is there any significant difference between the average math score of this class and that of the whole school? (15).
The solution (1) puts forward the hypothesis.
(2) Calculation and statistics
Known,
∴
Significance level =0.05, and
(3) Statistical decision-making
∴ Accept the original hypothesis of 150, and reject the alternative hypothesis, that is, there is no significant difference between the average math score of this class and that of the whole school.
7. What does the study of probability and statistics inspire your philosophy? (5 points)
A: There are a lot of random phenomena in the objective world. Although the results may not be known in advance, through a large number of experiments, we can find that some random phenomena have certain regularity, thus further clarifying the inevitable objective regularity of contingency in philosophy.
I. Known (14)
Seek AB
solve
Second, use gauss elimination to solve linear equations (12 points)
Elementary transformation of solving equations (exchange the first and second equations)
Add (1) × (-2) to (2) and (1) × (-3) to (3) to obtain:
Multiply -4 of the second equation by the third equation to get the trapezoidal equation.
Using the recursive method, from bottom to top, solve the unknown, and get
Third, it is known that
Find ( 1) | ( 1，0)； (2) (16)
Solve the rule z = sinu-lnv,
In the same way; In a similar way
∴dz =-2 cos 1dx+ody =-2 cos 1dx。
Four, a class of 50 people is known, and the results of a teaching exam are shown in the table below. Try to find the mathematical expectation of the score and write the formula for calculating the variance (16 score).
score
50
60
70
80
90
100
number of people
2
four
12
16
12
four
Note: Keep two digits after the decimal point.
solve
Verb (short for verb) is known.
( 1); (2) Calculation formula based on the definition of continuous random variable distribution function.
(3) Sketch (2 1)
Answer (1) =1-=1-0.8413 = 0.1587.
(2) = dt
The value of (3) is the area of the shaded part in the figure.
6. It is known that the plane area D is surrounded by straight lines and sums.
(1) Find the area s of d.
(2) Seek (16 points)
Draw a sketch. If it is on the right, the attached figure D is d: 0 ≤ x ≤ 1, -x ≤ y ≤ 2x.
( 1)
(2)
7. Briefly describe Descartes' contribution to the development of teaching. (5 points)
Descartes unified geometry and geometry by unifying curves (surfaces) and equations through coordinate system, and established a new mathematical discipline, namely analytic geometry. So variables enter mathematics, dialectics enters mathematics, and calculus naturally produces mathematics from constant mathematics to variable mathematics, which is a great milestone contribution in the history of mathematics!