moocCALCULUS Ⅰ章节答案(mooc完整答案)
33 min readmoocCALCULUS Ⅰ章节答案(mooc完整答案)
Lecture 2. Sets and Mappings随堂测验
1、Ⅰ章整答Among the four sets of natural numbers,节答 integers, rational numbers and real numbers, which one includes the others?
A、The案m案 set of natural numbers
B、TheⅠ章整答 set of integers
C、The节答 set of rational numbers
D、The案m案 set of real numbers
2、Determine which one is Ⅰ章整答false?
A、
B、节答
C、案m案
D、Ⅰ章整答
3、节答Let ,案m案 then which one is an element of ?
A、
B、Ⅰ章整答
C、节答
D、案m案
4、The infimum of is
A、0
B、1
C、
D、Not exist
5、Let , which one is not the upper bound of ?
A、100
B、3
C、17
D、20
Lecture 2. Sets and Mappings随堂测验
1、The neighborhood of point is a closed interval whose length is .
2、The neighborhood of point is the set of such points that the distance to is less than .
3、The deleted neighborhood of point does not include the center .
Lecture 2. Sets and Mappings随堂测验
1、Let then the equation determines a mapping from A to B.
2、If two finite sets A and B are equivalent，then the numbers of their elements are same.
3、Let A and B be two sets and A is a subset of B, then the two sets are not equivalent.
Test for Lecture 2
1、Among the following alphabets, which represents the set of natural numbers?
A、
B、
C、
D、
2、Among the following alphabets, which representsthe set of integers?
A、
B、
C、
D、
3、Among the following alphabets, which represents the set of rational numbers?
A、
B、
C、
D、
4、Among the following alphabets, which represents the set of real numbers?
A、
B、
C、
D、
5、Which set does the number belong to?
A、
B、
C、
D、
6、Let and be two sets. and represents
A、
B、
C、
D、
7、Let and be two sets. or represents
A、
B、
C、
D、
8、Let and be two sets. and represents
A、
B、
C、
D、
9、Let and be two sets. represents
A、
B、
C、
D、
10、Let then the relation that represents a surjection from to is
A、
B、
C、
D、
Week Two
Lecture 3. Concepts of Functions and Their Properties随堂测验
1、Which of the two functions in each of the following groups are the same functions
A、
B、
C、
D、
2、The domain of the function is ( ).
A、
B、
C、
D、
3、The range of the function is ( ).
A、
B、
C、
D、
Lecture 3. Concepts of Functions and Their Properties随堂测验
1、The function and are the same functions.
2、Let represents the largest integer that is less than or equal to , then .
Lecture 3. Concepts of Functions and Their Properties随堂测验
1、If and then ( )
A、
B、
C、
D、
2、When ( )
A、
B、
C、
D、
Lecture 3. Concepts of Functions and Their Properties随堂测验
1、The function is strictly monotonically decreasing on .
2、The function is an even function, where and .
3、The function is a periodic function.
Lecture 4. Elementary Functions随堂测验
1、If the function then
2、Assume a triangle has two edges of length and , the angle formed by the two edges can be of any value, then the area of the triangle and the angle has the relationship .
3、Let , if function then
Lecture 4. Elementary Functions随堂测验
1、The function is a -periodic function.
2、Assume then .
3、The function is a composition function of and .
Lecture 4. Elementary Functions随堂测验
1、If ，assume ，then （ ）.
A、
B、
C、
D、
2、The function is a （ ）.
A、bounded function
B、periodic function
C、even function
D、odd function
3、Let ，if the real numbers satisfy ，then（ ）.
A、
B、
C、
D、the order of can not be determined
Test for Lecture 3
1、If ，then the domain of is ( ).
A、
B、
C、
D、
2、If ，then ( )
A、
B、
C、
D、
3、The inverse function of the hyperbolic sine function is ( )
A、
B、
C、
D、
4、If is an even function and is an odd function，then ( ).
A、All are even functions;
B、All are odd functions;
C、are odd functions and is an even function;
D、 are even functions and is an odd function.
5、If，thenis a function with ( )
A、period
B、period
C、period
D、no period
6、If，thenis a ( ).
A、unbounded function
B、even function
C、periodic function
D、monotone function
7、The domain of function is
8、If ，then .
9、If is the inverse of ，then the inverse of is
10、Any function defined on a symmetric interval can be expressed as the sum of an even function and an odd function．
Test for Lecture 4
1、Let then the value of is（ ）.
A、
B、
C、the larger number of
D、the smaller number of
2、Let ，then which function is symmetric to with respect to the line （ ）.
A、
B、
C、
D、
3、If ，then （ ）.
A、1
B、
C、4
D、
4、Assume ，then（ ）.
A、
B、
C、
D、
5、The inverse function of is（ ）.
A、
B、
C、
D、
6、If y = is given by ，then the analytical expression of is .
7、If ，then .
8、If is a linear function and ，then .
9、Assume ，if ，then .
10、If the graph of is symmetric with respect to ，then must be a periodic function.
Week Three
Lecture 5 Parametric Equations and Ploar Coordinats随堂测验
1、The parameter in the parametric equations can all be interpreted as time.
2、The parametric equations for the ellipse are .
3、The parametric equations for the curve are .
Lecture 5 Parametric Equations and Ploar Coordinats随堂测验
1、The point in polar coordinate system corresponds to in rectangular coordinate system.
2、Let the pole correspond to origin, the polar axis correspond to the positive -axis, then the polar equation for the circle is .
3、Let the pole correspond to origin, the polar axis correspond to the positive -axis, then the polar equation represents the line .
Lecture 5 Parametric Equations and Ploar Coordinats随堂测验
1、Which of the following curves does the equation represent ?（ ）.
A、Circle
B、Ellips
C、Hyperbola
D、Parabola
2、A conic with eccentricity is a parabola.
3、A conic with eccentricity is an ellipse.
Lecture 6 Concepts of limits for functions随堂测验
1、There are six possible cases of changes of continuously variable as approaches a finite point and infinity
2、 does not exist.
3、
Lecture 6 Concepts of limits for functions随堂测验
1、For any number ，there exists a corresponding number ，such that whenever ，then
2、 exists and exist
3、Since is not defined at ， does not exist.
Lecture 6 Concepts of limits for functions随堂测验
1、
2、If is defined in the neighborhood of ，and ，then .
Lecture 6 Concepts of limits for functions随堂测验
1、 exists and exist
2、 does not exist.
Test for Lecture 5
1、Which of the following parametric equations represent ?（ ）.
A、
B、
C、
D、
2、Which of the following point lies on the curve ( is a parameter)?（ ）
A、
B、
C、
D、
3、The intersection points of the curve ( is a parameter) and the coordinate axes are（ ）.
A、
B、
C、
D、
4、The polar equation for the circle is（ ）.
A、
B、
C、
D、
5、Which of the following equation represents a line tangent to the circle ?（ ）.
A、
B、
C、
D、
6、The parametric equations for the circle are ．
7、Let be a positive constant and be a parameter, then parametric equations can convert to rectangular coordinates equation ．
8、Let be a constant and be a parameter, then parametric equations represents a line geometrically．
9、The line ( is a parameter) is perpendicular to the line ( is a parameter), then .
10、In polar coordinate system, the curves intersect at ．
Test for Lecture 6
1、Which limit exists?
A、
B、
C、
D、
2、Which limit does not exist?
A、
B、
C、
D、
3、The function at has the property that
A、the limit is
B、the limit is
C、Both left-hand limit and right-hand limit do not exist
D、Both left-hand limit and right-hand limit exist
4、Suppose and ，then
A、
B、
C、
D、
5、Suppose and ，then
A、
B、
C、
D、
6、If ，then
A、 has no definition at
B、
C、
D、
7、If ，then
A、
B、 has no definition at
C、In the deleted neighborhood of point
D、In the deleted neighborhood of point
8、
9、
10、
Week Four
Lecture 7 Properties and Laws for Limits随堂测验
1、Let be a positive constant, which of the following statement is wrong? ( )
A、If ，then there exists such that whenever
B、If ，then there exists such that whenever
C、If ，then there exists such that whenever
D、If ，then there exists such that whenever
2、Which of the following statements is wrong ? ( )
A、If ，then there exists such that whenever .
B、If ，then there exists such that whenever .
C、If and there exists such that whenever ，then
D、If and there exists such that whenever , then
Lecture 7 Properties and Laws for Limits随堂测验
1、
A、
B、
C、
D、The limit does not exist
2、
A、
B、
C、
D、The limit does not exist
3、
A、
B、
C、
D、The limit does not exist
Lecture 7 Properties and Laws for Limits随堂测验
1、
A、
B、
C、
D、The limit does not exist
2、，where is a polynomial of degree .
Lecture 8 Tests for the Existence of a Limit随堂测验
1、
A、0
B、1
C、2
D、3
2、Based on the geometric description of the squeeze theorem, if the sequences approach the same horizontal line, then the sequence between these two sequences will approach the same horizontal line.
3、Assuming that for all positive integers, , and satisfy , and both and exist, then the sequence must be convergent.
Lecture 8 Tests for the Existence of a Limit随堂测验
1、
A、0
B、1
C、2
D、3
2、Which of the following sequences has a limit?
A、
B、
C、
D、
3、
A、0
B、1
C、
D、
Lecture 8 Tests for the Existence of a Limit随堂测验
1、When , which of the following sequences has a limit of e?
A、
B、
C、
D、
2、Given: , . Which of the following points belongs to all intervals ?
A、
B、
C、
D、
3、Given the following propositions: (1) If sequence satisfies , , and there is a constant that makes , then exists; (2) If sequence satisfies , , and there is a constant that makes , then exists; (3) If sequence satisfies , , and there is a constant that makes , then exists; (4) If sequence satisfies , , and there is a constant that makes , then exists;
A、0
B、1
C、2
D、3
Test for Lecture 8
1、
A、0
B、
C、1
D、2
2、
A、0
B、
C、1
D、2
3、Let be known positive numbers. Then :
A、1
B、
C、
D、e
4、Given: . Then the range of that makes is:
A、
B、
C、
D、
5、Given: . Then sequence has:
A、an upper bound of 1
B、a lower bound of
C、an upper bound of 2
D、a lower bound of 2
6、A monotone sequence must be convergent.
7、A convergent sequence must be monotone.
8、
9、Given: sequences and satisfy for all positive integers , and . Then .
10、Let the recurrence formula of sequence be . Then if .
Test for Lecture 7
1、
A、
B、
C、
D、The limit does not exist
2、
A、
B、
C、
D、The limit does not exist
3、
A、
B、
C、
D、The limit does not exist
4、Suppose ，where represents the largest integer that is less than or equal to ，then ．
A、both and do not exist.
B、 exists，but does not exist
C、 exists，but does not exist
D、both and exist.
5、Suppose is a polynomial and ，then
A、
B、
C、
D、
6、If both and exist，then also exists.
7、If exists and does not exist，then must not exist.
8、If exists and does not exist，then must not exist.
9、Suppose are define on and , if both and exist，then also exists.
10、If and ，then .
Week Five
Lecture 9 Infinitesimals and Infinite Limits随堂测验
1、Any very small constant except zero is not an infinitesimal.
2、Let (A is a finite number), then , where .
Lecture 9 Infinitesimals and Infinite Limits随堂测验
1、
2、The sum, difference, product and quotient of any finite number of non-zero infinitesimal functions are still infinitesimal functions in the same process of variation.
Lecture 9 Infinitesimals and Infinite Limits随堂测验
1、An infinity must be unbounded.
2、If , then the line is a horizontal asymptote of the curve .
Lecture 9 Infinitesimals and Infinite Limits随堂测验
1、As , the function is an infinitesimal of higher order than .
2、Let and , then and are equivalent infinitesimals as .
3、Given that and are equivalent infinitesimals as , then the constant .
Lecture 10 Concepts of Continuous Functions随堂测验
1、Which statement is not equivalent to“ is continuous at ”
A、The limit of as approaches is equal to
B、When the increment of approaches 0，the corresponding increment of approaches 0.
C、 is both right-continuous and left-continuous at
D、For any number , there exists a corresponding number such that whenever
2、What kind of condition is for the continuity of at （ is a constant）
A、Necessary condition
B、Sufficient condition
C、Sufficient and necessary conditions
D、Neither sufficient nor necessary condition
Lecture 10 Concepts of Continuous Functions随堂测验
1、 is continuous on the interval
A、
B、
C、
D、
2、Suppose that is continuous at ，then is continuous in the neighborhood of .
Lecture 10 Concepts of Continuous Functions随堂测验
1、 is a removable discontinuity point of
2、 is a removable discontinuity point of
3、 is jump discontinuous at
4、 is infinite discontinuous at ．
Test for Lecture 10
1、Suppose that is continuous at ，then
A、
B、
C、
D、
2、Suppose that is continuous at ，then
A、
B、
C、
D、
3、 is continuous on the interval（ ）．
A、
B、
C、
D、
4、How many discontinuous points does the function have?.
A、
B、
C、
D、Infinity
5、How many discontinuous points of the first kind does have?.
A、
B、
C、
D、Infinity
6、Which function is continuous at ?
A、
B、
C、
D、
7、Suppose that is continuous at ，then
A、
B、
C、
D、
8、If is continuous at , then is continuous at
9、If is continuous at , then is continuous at
10、If is continuous at , then is continuous at ; but the converse proposition is not true.
Test for Lecture 9
1、As , and are both infinitesimals, then which of the following is wrong?（ ）.
A、
B、
C、
D、
2、As , the function is（ ）.
A、an infinity
B、an infinitesimal
C、an unbounded function
D、a bounded function
3、As , the function is （ ）.
A、an equivalent infinitesimal with .
B、an infinitesimal of same order but not equivalent with .
C、an infinitesimal of higher order than .
D、an infinitesimal of lower order than .
4、As , which of the following function is an equivalent infinitesimal with （ ）.
A、
B、
C、
D、
5、As , which of the following function is not an equivalent infinitesimal with （ ）.
A、
B、
C、
D、
6、An infinitesimal is a very small number.
7、As , is an infinitesimal.
8、.
9、.
10、As , , so , then .
Week Six
Lecture 11 Properties of Continuous Functions随堂测验
1、Let ，then is continuous at every point of real axis
2、Let ，then is continuous at every point of real axis
3、
4、Let ，then is continuous on the interval .
Lecture 11 Properties of Continuous Functions随堂测验
1、If is continuous on the interval , then is invertible on the interval .
2、For , the inverse function of the continuous function is itself.
Lecture 11 Properties of Continuous Functions随堂测验
1、Basic elementary functions are continuous in their domains.
2、Suppose that is a point in the domain of the elementary function , then is continuous at .
Lecture 11 Properties of Continuous Functions随堂测验
1、If is continuous on a closed interval , then attains an absolute maximum value and an absolute minimum value in
2、If is continuous on a closed interval , then is bounded on
3、Every algebraic equation of odd degree with real coefficients must have real root.
4、A continuous function on the closed interval can take every intermediate value between the maximum value and the minimum value
Lecture 12 Concepts of Derivatives随堂测验
1、If denotes the relationship between the time and the distance of an object moving with a variable speed, then is its instantaneous speed at
2、The derivative is the limit of as approaches to 0
3、If is differentiable at , then has a tangent line at and the tangent line is
Lecture 12 Concepts of Derivatives随堂测验
1、If is differentiable at ，then .
2、If is continuous in a neighborhood of , then is differentiable at .
3、If is differentiable at , then is continuous in a neighborhood of .
4、If the right-hand and left-hand limits of at exist and are equal, then is differentiable at .
Lecture 12 Concepts of Derivatives随堂测验
1、Let and when ，then （ ）at .
A、is discontinuous
B、is continuous but not differentiable
C、is differentiable and
D、has a extreme point
2、If is a periodic function on ，then is also a periodic function on
Test for Lecture 11
1、Which one of the following functions has both maximum value and minimum value in their domains?
A、
B、
C、
D、
2、
A、
B、
C、
D、
3、Let , then
A、 has only maximum value on the closed interval.
B、 has only minimum value on the closed interval .
C、 has both maximum and minimum values on the closed interval .
D、 has neither maximum nor minimum value on the closed interval .
4、
A、
B、
C、
D、
5、The real roots of equation lie inside
A、
B、
C、
D、
6、If , then
A、
B、
C、
D、
7、 is discontinuous at
8、Let , then is continuous at .
9、For ,
10、If is continuous on the closed interval ,and then there are at least one point satisfying ．
Test for Lecture 12
1、Suppose that is differentiable and , then the slope of the tangent line of at is （ ）
A、
B、
C、
D、
2、Suppose that is continuous at and , what kind of condition is for has a derivative at （ ）.
A、Necessary condition
B、Sufficient condition
C、Sufficient and necessary conditions
D、Neither sufficient nor necessary condition
3、Determine the number of which is not differentiable.
A、
B、
C、
D、
4、If the tangent lines of and coincide at , where are constants, then （ ）.
A、
B、
C、
D、
5、If is continuous at ，then which one of the following conclusions is wrong?
A、If exists， then
B、If exists， then
C、If exists， then exists
D、If exists， then exists
6、If is differentiable at ，then
7、If is differentiable at ，then has a tangent line at
8、If has a tangent line at ，then is differentiable at
9、If the right-hand and left-hand derivatives of at exist and equal，then is differentiable at
10、If is continuous at ，then is differentiable at if and only if .
Week Seven
Lecture 13 Rules of Derivatives随堂测验
1、Let be a constant，then
2、
3、
Lecture 13 Rules of Derivatives随堂测验
1、
2、
3、Given ，then
Lecture 13 Rules of Derivatives随堂测验
1、
A、
B、
C、
D、
2、
Lecture 13 Rules of Derivatives随堂测验
1、Let ，then
A、
B、
C、
D、
2、
Lecture 14 Implicit and Parametric Differentiation随堂测验
1、If denotes the relationship between the time and the distance of an object moving with a variable speed, then is the instantaneous acceleration at
2、
3、Let be a positive integer，then
Lecture 14 Implicit and Parametric Differentiation随堂测验
1、If the function is determined by the implicit equation ，then（ ）．
A、
B、
C、
D、
2、The equation can not determine an implicit function between the variables and .
3、If the function is determined by the implicit equation ，then for ．
Lecture 14 Implicit and Parametric Differentiation随堂测验
1、If ，then
2、If ，then
Test for Lecture 13
1、
A、
B、
C、
D、
2、
A、
B、
C、
D、
3、
A、
B、
C、
D、
4、
A、
B、
C、
D、
5、
A、
B、
C、
D、
6、Let ，then
A、
B、
C、
D、
7、Suppose that the curve and have the same tangent line at origin，then （ ）．
A、
B、
C、
D、
8、If ，then （）.
A、
B、
C、
D、
9、
10、If is differentiable and ，then .
Test for Lecture 14
1、
A、
B、
C、
D、
2、
A、
B、
C、
D、
3、If ，then
A、
B、
C、
D、
4、If ，then
A、
B、
C、
D、
5、If ,then
A、
B、
C、
D、
6、If ,then
A、
B、
C、
D、
7、The tangent line of the ellipse at is
A、
B、
C、
D、
8、If ，then
A、
B、
C、
D、
9、If ，then
10、If the function is determined by the implicit equation ，where ，then
Week Eight
Lecture 15 Linear Approximations and Differentials随堂测验
1、The linearization of at is .
2、Let be defined in the neighborhood of ,if there exists a constant such that ， where is independent of , then we say that the function is differential at (or differentiable)， is called the differentiation of at ，which is denoted as or , i.e., ．
3、The term in must be very small.
Lecture 15 Linear Approximations and Differentials随堂测验
1、
2、When using the differential for approximate calculation, the error can be accurately known.
Lecture 15 Linear Approximations and Differentials随堂测验
1、If and are both differentiable functions of that satisfy the required conditions, then .
2、The invariance of first-order differential form of unary functions is also true for higher-order derivatives.
3、Let be a composite function，where and are both twice differentiable，then .
Lecture 16 Rates of Change in Physical Problems随堂测验
1、When ，the difference quotient is the average rate of over the interval or .
2、Let ,when changes from to , the corresponding increment of is , then
Lecture 16 Rates of Change in Physical Problems随堂测验
1、Let the area and radius of the circle be functions of time ，then ( )．
A、
B、
C、
D、
2、If and are both differentiable functions of and ,where is a constant，then .
Test for Lecture 15
1、Let ，then
A、
B、
C、
D、
2、The differential of at is ( )
A、
B、
C、
D、not exists.
3、Suppose that is defined in the neighborhood of and , where is a constant that independent of , then which of the following conclusions is wrong ?( )
A、
B、 is differentiable at
C、
D、 is not necessarily differentiable at .
4、Which of the following statements is right ? ( )．
A、The term in is a “sufficiently small quantity”
B、The term in is a “sufficiently large quantity”
C、In ，If is fixed，then is a linear function of .
D、The change of any function at a point can separate out the linear part.
5、Suppose that has a second-order derivative and ， represents the change of at , and represent the change and differential of at , respectively. If ，then ( )
A、
B、
C、
D、
6、The differential of at can be written as or .
7、Let be defined in the neighborhood of ，then is differentiable at if and only if is derivable at .
8、Suppose that is continuous at ，then the differential of at can be obtained in the following way： since ，thus .
9、Let be the inverse function of ，then .
10、Let be twice differentiable，then .
Test for Lecture 16
1、The position of a particle is given by the equation ，then the acceleration of the particle at is ( ) ．
A、
B、
C、
D、
2、Suppose a company has estimated that the cost (in dollars) of producing items is ，then the marginal cost of producing items is ( )．
A、
B、
C、
D、
3、If the length and width of rectangular iron pieces change according to the following rules: . When ，the rate of change of the iron area is ( )．
A、
B、
C、
D、
4、If and are both differentiable functions of , is the distance between and on , then ( )．
A、
B、
C、
D、
5、A stone falls into the calm water and creates concentric ripple, if the increasing rate of the radius of the outermost circle of wave is , then the increasing rate of the disturbance surface area at the end of is ( ) .
A、
B、
C、
D、
6、Suppose that is the position function of a particle that is moving in a straight line, when changes from to ，then the average velocity over this time period is
7、Suppose a company has estimated that the cost (in dollars) of producing items is , the income (in dollars) of producing items is , then the marginal profit is
8、If and are both differentiable functions of and ，then
9、If and are both differentiable functions of and ，then
10、If , and are differentiable functions of and ，then
Week Nine
Lecture 17 Antiderivatives and Indefinite Integrals随堂测验
1、If the curve through and the slope of the tangent line at any point is , then the equation of the curve is .
2、A continuous function on an interval must have an antiderivative.
3、
Lecture 17 Antiderivatives and Indefinite Integrals随堂测验
1、
A、
B、
C、
D、
2、
3、
Lecture 17 Antiderivatives and Indefinite Integrals随堂测验
1、If ，then is the integral curve of .
2、If is continuous on ，then
Lecture 18 Maximum and Minimum values and Fermat Theorem随堂测验
1、Let ，if is a local maximum value of ，then must be an absolute maximum value of on ．
2、Let ，if is an absolute maximum value of on ，then must be a local maximum value of .
Lecture 18 Maximum and Minimum values and Fermat Theorem随堂测验
1、If is continuous on ，then the absolute maximum point is ( ).
A、stationary point
B、endpoint of the interval
C、non-differentiable point
D、stationary point、endpoints of the interval or non-differentiable point
2、Let be an even function defined on ，if is a local maximum point of ，then is ( ) of .
A、an absolute minimum point
B、an absolute maximum point
C、a local minimum point
D、a local maximum point
3、If has a local extreme value at on ，then must be differentiable at ．
4、Suppose is twice differentiable at ，if has a local extreme value at ，then .
Test for Lecture 17
1、Which of the following statements is right ? ( )．
A、
B、
C、
D、
2、
A、
B、
C、
D、
3、Let , then
A、
B、
C、
D、
4、Let , then
A、
B、
C、
D、
5、If one of the integral curves of goes through the point ，then the equation of the integral curve is ( )．
A、
B、
C、
D、
6、 is an antiderivative of
7、
8、
9、
10、
Test for Lecture 18
1、Let be the local maximum point of ，then there must be ( ).
A、
B、
C、
D、 does not exist.
2、Let ，then ( ).
A、has a local extreme value
B、has no local extreme value
C、has a local maximum value but no local minimum value
D、has a local minimum value but no local maximum value
3、Suppose satisfies , then is the ( ) of .
A、removable discontinuity
B、infinite discontinuity
C、local maximum point
D、local minimum point
4、Use a fence with a total length of to enclose a rectangular land. To maximize the enclosed area, the length and the width of the rectangular should be ( ).
A、
B、
C、
D、
5、The minimum term of is ( ).
A、
B、
C、
D、
6、Let then ( ).
A、has no local minimum point
B、has a unique local minimum point
C、has two local minimum points
D、has three local minimum points
7、If is continuous on ，then attains an absolute maximum value and an absolute minimum value in ．
8、If is continuous on ，then attains an absolute maximum value and an absolute minimum value in ．
9、If is non-differentiable at in ，then the curve has no tangent at ．
10、If constants in satisfy , then has no local extreme value．
Week Ten
Lecture 19 Mean Value Theorem随堂测验
1、The formula shows that the average rate of on is equal to the instantaneous rate of at some point in under certain conditions．
2、Suppose that is continuous on and ，then there is at least one point such that .
3、Suppose that is continuous on and differentiable on . Suppose also that , then there is at least one point on the graph of where the tangent is horizontal.
Lecture 19 Mean Value Theorem随堂测验
1、Suppose that is continuous on and differentiable on ，then there is at least one point such that ．
2、Suppose that is continuous on and differentiable on ，if then there is at least one point such that .
Lecture 19 Mean Value Theorem随堂测验
1、Suppose and is differentiable on ，then by Cauchy’s theorem we have ( )．
A、
B、
C、
D、
2、The Cauchy’s theorem can not be used for to get the corresponding conclusion on the interval .
Lecture 20 Indeterminate Forms and L’Hospital’s Rule随堂测验
1、Let , during the same variation process of . Then how many infinitive limits are there among , , , and ?
A、1
B、2
C、3
D、4
2、Let , during the same variation process of . Then how many infinitive limits are there among , , , and ?
A、1
B、2
C、3
D、4
3、Functions cannot obtain corresponding conclusions within the interval based on Cauchy’s mean value theorem.
Lecture 20 Indeterminate Forms and L’Hospital’s Rule随堂测验
1、Let , and function is derivable within the interval . Then Cauchy’s mean value theorem concludes that:
A、
B、
C、
D、
2、Which of the following limits can be solved based on L’Hospital’s Rule?
A、
B、
C、
D、
Lecture 20 Indeterminate Forms and L’Hospital’s Rule随堂测验
1、
A、1
B、0
C、e
D、
2、
Test for Lecture 19
1、For , the points on satisfying are ( ).
A、
B、
C、
D、
2、Which of the following functions satisfies the hypotheses of Rolle’s Theorem on ?
A、
B、
C、
D、
3、Suppose is differentiable and the first-order derived function is strictly monotonically increasing, then ( ).
A、
B、
C、
D、
4、If , then
A、
B、
C、
D、
5、Let , the number of point on satisfies is ( ).
A、zero
B、one
C、two
D、three
6、Suppose is continuous on and differentiable on , if then satisfies the hypotheses of Rolle’s Theorem on .
7、The speed limit of the highway is , if a driver drives continuously for in , it can be concluded that the driver violates the regulations and overspeeds.
8、If the equation has a positive root , then has a positive root which is lease than .
9、Suppose is differentiable and then there is at least one point such that .
10、Suppose is continuous on and differentiable on , then by Cauchy’s theorem we have .
Test for Lecture 20
1、Given: function has derivatives within a certain neighborhood of , and . Then which of the following conclusions is incorrect (where range from to )?
A、
B、
C、
D、
2、Let and . Then how many satisfy within ?
A、0
B、1
C、2
D、3
3、
A、
B、2
C、1
D、0
4、
A、0
B、
C、1
D、No solution
5、
A、1
B、
C、0
D、
6、Let functions and be derivable within the interval , and . Then there must be a point that makes .
7、Let function be continuous within and derivable within , and . Then, according to Cauchy’s mean value theorem, .
8、If , then .
9、It can be seen from that since the equation does not have a limit on the right, does not exist.
10、
Week Eleven
Lecture 21 Taylor's Theorem随堂测验
1、 is the coefficient of the th-degree Taylor polynomial of function at .
2、Function shares the same value with its th-degree Taylor polynomial at within a specific neighborhood of .
Lecture 21 Taylor's Theorem随堂测验
1、Let be the th-degree Taylor polynomial of function , denoted by . Then is the absolute error for the th-degree Taylor polynomial to approximate function .
2、Given: function has th derivative at . Then when is approximated by its corresponding th-degree Taylor polynomial, the error generated is the equivalent infinitesimal of .
Lecture 21 Taylor's Theorem随堂测验
1、The th-order Maclaurin series with Peano remainder term of function is:
A、
B、
C、
D、
2、The Maclaurin series with Lagrange remainder term of function is .
Lecture 22 How Derivatives Affect the Shape of a Graph随堂测验
1、 is a function that strictly monotonically increases within the interval .
2、Point is an inflection point of curve:
Lecture 22 How Derivatives Affect the Shape of a Graph随堂测验
1、The rounding function is a monotonically increasing function within the interval .
2、The graph of is convex within the interval .
Lecture 22 How Derivatives Affect the Shape of a Graph随堂测验
1、Given: curve ( is a constant). For the curve to have the horizontal asymptote is:
A、a sufficient and necessary condition
B、a sufficient but not necessary condition
C、a necessary and insufficient condition
D、is neither a sufficient nor necessary condition
2、Both straight lines and are the vertical asymptotes of function .
Lecture 22 How Derivatives Affect the Shape of a Graph随堂测验
1、Given: function has a second-order derivative within the interval of , and . Then, within the interval of , the curve is:
A、increasing and convex upward
B、decreasing and convex upward
C、increasing and convex downward
D、decreasing and convex downward
2、Function has no extreme point within the interval (-2, 3).
Test for Lecture 21
1、Given: function has an th-order derivative at . If , then:
A、
B、
C、
D、
2、The Lagrange remainder term of the th-order Maclaurin series of function is:
A、
B、
C、
D、
3、The Maclaurin series with Peano remainder term of function is:
A、
B、
C、
D、
4、The Maclaurin series with Peano remainder term of function is:
A、
B、
C、
D、
5、The th-order Taylor’s formula with Peano remainder term of function at is:
A、
B、
C、
D、
6、Given: function has an th-order derivative at . Then when , the error generated when is approximated by its th-order Taylor polynomial is about the infinitesimal equivalent of
7、The Maclaurin series with Lagrange remainder term of the function is ..
8、The third-order Maclaurin series with Peano remainder term of function is .
9、The Maclaurin series with Peano remainder term of function is .
10、Taylor’s formula with the Peano remainder term of function at is .
Test for Lecture 22
1、Within the interval , function has:
A、neither a maximum nor a minimum
B、a maximum and a minimum
C、a maximum but no minimum
D、a minimum but no maximum
2、The coordinates of the inflection point of curve are:
A、(1, -1)
B、(0, 1)
C、(-1, 3)
D、(-1, -3)
3、How many asymptotes does the curve have
A、1
B、2
C、3
D、4
4、The graph of functionwithin the interval [0, 1] is:
A、increasing and convex upward
B、decreasing and convex upward
C、increasing and convex downward
D、decreasing and convex downward
5、The smallest term in sequence is ()
A、1
B、
C、
D、
6、Function is monotonically increasing within the interval [-1, 1].
7、If is the inflection point of curve , then .
8、Curve is convex upward within the interval .
9、Curve is convex upward within the interval .
10、Point is another inflection point of curve .
Week Twelve
Lecture 23 Curvatures of Plane Curves随堂测验
1、The arc differential of smooth curve is .
2、In the differential triangle of the smooth curve , the three sides are the differentials of x, y, and the arc length s.
Lecture 23 Curvatures of Plane Curves随堂测验
1、Which of the following points has the largest curvature?
A、Point on the circle with a radius of 1
B、Point on the circle with a radius of 2
C、Point on the circle with a radius of 3
D、Point on the straight line
2、Based on the definition of curvature , curvature is the rate of change of the rotational angle of the tangent on the curve to the arc length of the curve.
Lecture 23 Curvatures of Plane Curves随堂测验
1、The curvature circle of smooth curve and the curve itself have the same second-order derivative at point M.
2、The curvature of curve at is .
Lecture 24 Concepts of Definite Integrals随堂测验
1、.
2、.
Lecture 24 Concepts of Definite Integrals随堂测验
1、Given: S is a curved-side trapezoid enclosed by and , and divides the interval into four equal parts. Then its left sum and right sum .
2、Given: S is a curved-side trapezoid enclosed by and , and divides the interval into n equal parts. Its left sum and right sum are denoted by , respectively. Then .
Lecture 24 Concepts of Definite Integrals随堂测验
1、The area of the curved-side triangle enclosed by curve and straight line can be expressed as
2、Given: the speed of an object for variable rectilinear motion. The motion starts at . Then after 10 seconds, the distance traveled by the object can be expressed as .
Lecture 24 Concepts of Definite Integrals随堂测验
1、Let function be continuous within the interval . Then the area of the curved-side trapezoid enclosed by curve and staight lines is .
2、Let function be continuous within the interval , and . Then the area of the curved-side trapezoid enclosed by curve and staight lines is .
Test for Lecture 23
1、The curvature of curve at point is:
A、
B、
C、
D、
2、The horizontal axis value of the point with the largest curvature on sine curve , is:
A、
B、
C、
D、
3、The curvature of sine curve at point , is:
A、
B、
C、
D、
4、The curvature K of oval at , is:
A、
B、
C、
D、
5、The point with the largest curvature on oval is
A、and
B、 and
C、and
D、and
6、The arc differential form of the curve under the parametric equation is .
7、 , the ratio between the angle that the tangent line rotates on the arc AB, to the arc length , is referred to as the mean curvature of this arc section.
8、The curvature circle on the curve has the same tangent line and curvature as the curve at point , and then have the same concave direction near this point.
9、Let a curve be given by the parametric equation . Then the curvature formula at the corresponding point is .
10、Parabola has the largest curvature at its vertex.
Test for Lecture 24
1、Given: S is a curved-side trapezoid enclosed by curve and straight lines , and division points are inserted to divide the interval into n equal parts. Then the length of each small interval is:
A、
B、
C、
D、
2、Given: is the area of the graph enclosed by curve and straight lines , and division points are inserted to divide the interval into equal parts, the left sum of which is denoted by and the right sum by . Then which of the following relational expressions about is incorrect?
A、
B、
C、
D、
3、If , , and , then :
A、10
B、6
C、4
D、0
4、Based on the nature of definite integrals, the relationship between definite integrals , , and is:
A、
B、
C、
D、
5、Based on the nature of definite integrals, the relationship between definite integrals , , and is:
A、
B、
C、
D、
6、Based on the definition of a definite integral, the division of the integral interval [a, b] means inserting n-1 division points to divide the interval [a, b] into n equal parts.
7、Let the variable current intensity be a function of time and . Then the amount of electricity that passes through the cross section of the conductor after time since the beginning of the experiment is .
8、If function is integrable within the interval , then .
9、If functions are integrable within the interval , then for any real constants , .
10、If function is integrable within the interval , function is integrable within the interval , and , then .
Week Thirteen
Lecture 25 Properties of Definite Integrals随堂测验
1、If the function only has limited discontinuity points in the interval , then the function is integrable in the interval .
2、If the function is monotonically increasing or monotonically decreasing in the interval , then the function is integrable in the interval .
Lecture 25 Properties of Definite Integrals随堂测验
1、
A、
B、
C、
D、
2、
A、
B、
C、
D、
Lecture 25 Properties of Definite Integrals随堂测验
1、If the function is a monotonically increasing continuous function, then .
2、The average value of the function in the interval is .
Lecture 26 The Fundamental Theorem of Calculus随堂测验
1、
2、
Lecture 26 The Fundamental Theorem of Calculus随堂测验
1、
A、
B、
C、
D、
2、
A、
B、
C、
D、
Lecture 26 The Fundamental Theorem of Calculus随堂测验
1、Suppose , let , then ( ).
A、
B、
C、
D、
2、Suppose , when , .
Lecture 26 The Fundamental Theorem of Calculus随堂测验
1、When , the function and are equivalent to infinitesimal, then ( ).
A、
B、
C、
D、
2、The maximum value of the function at can be obtained.
Test for Lecture 25
1、Integrable requirements of the function in the interval is ( ).
A、The function is monotonic in the interval
B、The function is continuous in the interval
C、The function only has limited discontinuity points in the interval
D、The function is bounded in the interval
2、The integrable sufficiency of the function in the interval is ( ).
A、The function is bounded in the interval
B、The function is continuous in the interval
C、The function only has limited discontinuity points in the interval
D、All discontinuity points of the function in the interval are the discontinuity points of first class
3、If the function is integrable in the interval , then the definite integral is equal to ( )
A、
B、
C、
D、
4、
A、
B、
C、
D、
5、
A、
B、
C、
D、
6、If the function is integrable in the interval , the function is bounded in the interval.
7、If the function is bounded in the interval , the function is integrable in the interval.
8、If the function only has limited discontinuity points of first class in the interval , the function is integrable in the interval .
9、If the limit exists, the function is integrable in the interval and .
10、If the function and are continuous on the closed interval , there must have a point in the closed interval , so as to .
Test for Lecture 26
1、
A、
B、
C、
D、
2、The function is represented by the parametric equation , then ( ).
A、
B、
C、
D、
3、Suppose is the implicit function given by the equation , then ( ).
A、
B、
C、
D、
4、
A、
B、
C、
D、
5、Suppose the function is continuous in , when , ( ).
A、
B、
C、
D、
6、Suppose the function is continuous in , when , ( ).
A、
B、
C、
D、
7、Suppose the function is continuous in , and , , when , is a ( )
A、monotone decreasing function
B、Upward convex function
C、Monotone increasing function
D、Downward convex function
8、Suppose an object moving in a straight line, and the square between its speed and its time is proportional, is the relationship between the distance and the time traveled by the object. Suppose the object moving from for 18 cm after 3 seconds, then ( ).
A、
B、
C、
D、
9、Suppose , when , .
10、The function is monotonically decreasing in the interval .
Week Fourteen
Lecture 27 Integration by Substitution随堂测验
1、If , then .
2、
3、
Lecture 27 Integration by Substitution随堂测验
1、
A、
B、
C、
D、
2、
Lecture 28 Integration by Parts随堂测验
1、
2、
Lecture 28 Integration by Parts随堂测验
1、
2、
Test for Lecture 27
1、The following indefinite integral with miscalculation is ( ).
A、
B、
C、
D、
2、The following indefinite integral with correct calculation is ( ).
A、
B、
C、
D、
3、
A、
B、
C、
D、
4、
A、
B、
C、
D、
5、The following indefinite integral with correct calculation is ( ).
A、
B、
C、
D、
6、
7、
8、
9、
10、
Test for Lecture 28
1、, during the calculation, the indefinite integral with wrong equals sign is ( ).
A、a
B、b
C、c
D、None
2、, during the calculation, the indefinite integral with wrong equals sign is ( ).
A、c
B、a
C、b
D、None
3、, during the calculation, the indefinite integral with wrong equals sign is ( )
A、b
B、a
C、c
D、None
4、, during the calculation, the indefinite integral with wrong equals sign is ( ).
A、None
B、c
C、b
D、a
5、, during the calculation, the indefinite integral with wrong equals sign is ( ).
A、c
B、d
C、b
D、a
6、It is known that the original function of is , then ( ).
A、
B、
C、
D、
7、Suppose that is continuous, known by integration by parts of definite integral, ( ).
A、
B、
C、
D、
8、Because , so .
9、
10、
Week Fifteen
Lecture 29 Techniques of Integration随堂测验
1、Indefinite integral ( ).
A、
B、
C、
D、
2、
3、Suppose , indefinite integral
Lecture 29 Techniques of Integration随堂测验
1、( )．
A、
B、
C、
D、
2、Suppose the function is continuous in the interval and is satisfying , then .
Lecture 29 Techniques of Integration随堂测验
1、Suppose that is the maximum integer that does not exceed , then the value of definite integral is ( ).
A、2014
B、1007
C、-1007
D、-2014
2、Suppose as a positive integer, then .
Lecture 30 Improper Integrals随堂测验
1、
2、2、If is an odd function，then ．
Lecture 30 Improper Integrals随堂测验
1、The integral diverges when , and converges when .
2、The integral diverges for all ．
Lecture 30 Improper Integrals随堂测验
1、By the Comparison Test，the integral converges.
2、For the Gamma function ，．
Test for Lecture 29
1、Indefinite integral ( )．
A、
B、
C、
D、
2、Indefinite integral ( )．
A、
B、
C、
D、
3、Suppose as a positive integer, then the value of definite integral is ( ).
A、
B、
C、
D、
4、Suppose , , , then ( )．
A、
B、
C、
D、
5、Suppose , then ( )．
A、Normal number
B、Negative constant
C、Always 0
D、Not constant
6、
7、
8、
9、, known by the property of definite integral of the parity function.
10、Suppose the function is continuous in , then .
Test for Lecture 30
1、，the wrong step would be （ ）.
A、c
B、a
C、b
D、none
2、Let ，then ， the wrong step would be（ ）.
A、b
B、a
C、none
D、c
3、，the wrong step would be（ ）.
A、a
B、b
C、c
D、none
4、The integral （ ）.
A、divergence
B、-2
C、2
D、0
5、If the integral converges，then（ ）.
A、
B、
C、
D、
6、
7、If is continuous on and ，， for a constant , then the integral maybe converge .
8、Let be two continuous functions on ，and ，then if converges，then converges too.
9、For , ，and ，then converges.
10、The area enclosed by 、 and -axis is .
Week Sixteen
Lecture 31 Applications to Area and Volume随堂测验
1、The area of the plane figure enclosed by and is（ ）．
A、
B、
C、
D、
2、Let be the area of plane figure enclosed by two curves , , and the lines ，then .
3、The area of plane figure enclosed by and is .
4、The area of plane figure enclosed by 、 and is .
Lecture 31 Applications to Area and Volume随堂测验
1、Assume that the height of a tower is 80m, and the horizontal section away from top m is a square with side length ，then its volume is .
2、If a body of revolution is obtained when the plane figure, enclosed by , and axis, rotates around the axis for one cycle, then its volume is .
3、If a body of revolution is obtained when the plane figure, enclosed by , and axis, rotates around the axis for one cycle, then its volume is .
4、If a body of revolution is obtained when the plane figure, enclosed by , and , rotates around the axis for one cycle, then its volume is .
Lecture 32 Applications to Physics随堂测验
1、The force required to lengthen the rod from to is , where is a constant, then the work done to lengthen the rod from to is （ ）.
A、
B、
C、
D、
2、If the force of 1 kg can make the spring elongate by 1cm, then the work required to make the spring elongate by 10cm is 0.5 .
Lecture 32 Applications to Physics随堂测验
1、When a upward conical container with bottom radius and height is filled with water, and the specific gravity of water is , the work needed to be done to lift all water meters above the top surface of the container is ．
2、If the radius of the bottom of the lying bucket is and the specific gravity of the water is , then the pressure on one end face of the bucket is ．
Lecture 32 Applications to Physics随堂测验
1、There is a uniform straight rod with linear density and length . If the mass of the rod is , and a particle with mass is placed on its extension line. Assume the distance between the particle and the nearest end point of the rod is , then the gravitational force of the rod to the particle is .
Test for Lecture 31
1、The area of plane figure enclosed by and is（ ）．
A、
B、
C、
D、
2、The area of plane figure enclosed by , the tangent line of passing origin point and axis is（ ）．
A、
B、
C、1
D、
3、Let D be a plane figure enclosed by , its tangent line passing the origin point, and axis，then the volume of the revolution, generated by D rotating around the for one cycle, is（ ）．
A、
B、
C、
D、
4、The volume of the body of revolution generated by the enclosed figure and rotating around axis one circle is （ ）．
A、
B、
C、
D、
5、Let be the volume of the revolution generated by the circle rotating around one circle , then the value of is（ ）．
A、
B、
C、
D、
6、Let be a continuous on ，then the integral denotes the area of the plane figure enclosed by 、 and axis.
7、Let be the area of the plane figure enclosed by , and ，then or .
8、If a body of revolution is obtained when the plane figure, enclosed by , and axis, rotates around the axis for one cycle, then its volume is .
9、The volume of the body of revolution generated by the figure rotating around axis one circle is .
10、Assume that the base of a solid is a circle with radius of 5, and the sections of a fixed diameter perpendicular to the bottom circle are equilateral triangles, then the volume of the solid is .
Test for Lecture 32
1、When a body moves straightly with , and the resistance it receives is directly proportional to the square of the velocity, then the work done by the body to overcome the resistance from to is （ ）
A、
B、
C、
D、
2、Assume that the cylinder with a diameter 20 cm and a height 80 cm is filled with steam with a pressure of 10 . If the temperature remains constant, when the piston moves in compression m, by the law of Bohr-Marquis, the pressure in the cylinder can be determined as （ ）.
A、
B、
C、
D、
3、Assume that the cylinder with a diameter 20 cm and a height 80 cm is filled with steam with a pressure of 10 . If the temperature remains constant, and the volume of steam is reduced by half, the work is （ ）J
A、
B、
C、
D、
4、There is a rectangular gate with a width 10 meters and a height 6 meters. When the upper boundary of the gate is parallel to the water surface and perpendicular to the water, the pressure on the gate is （ ）.
A、
B、
C、
D、
5、There is a rectangular gate with a width 10 meters and a height 6 meters. When the upper boundary of the gate is meters under the water surface, the water pressure on the gate is just twice that of the initial pressure. Then （ ）meters.
A、4
B、2
C、1
D、3
6、Assume that the decomposition rate of a radioactive object is given as , represents the decomposed mass from the time to , then ．
7、If the intensity I of the current is given as , the electric quantity passing through the cross section of the conductor in the interval [0, ] is ．
8、Assume that the intensity I of the current passing through the resistor is , then the work done from time to is ．
9、If the density of a non-uniform rod with a length cm at cm from its end point is g/cm, then the mass of the rod is determined by .
10、When a body moves in a straight line at the speed , then the distance traveled by the object in the time interval is .
Week Seventeen
Lecture 33 Modeling with Differential Equations随堂测验学习通CALCULUS Ⅰ
Calculus是高等数学的重要分支之一，是研究变化率和积分的数学学科。学习通CALCULUS Ⅰ是学习高等数学和工程学科的必备课程。
课程内容
学习通CALCULUS Ⅰ的课程内容主要包括：
- 导数和微分
- 函数的极限和连续性
- 微分中值定理
- 泰勒公式
- 积分
学习方法
学习通CALCULUS Ⅰ需要有一定的数学基础，建议学生先学习微积分的基础知识再来学习本课程。在学习过程中，可以采取以下方法：
- 多做习题，加深对知识的理解。
- 多看例题，掌握解题方法和技巧。
- 积极参加讨论，与同学交流，共同进步。
- 结合实际问题，理解知识的应用。
学习成果
学习通CALCULUS Ⅰ后，学生可以掌握以下能力：
- 掌握导数和微分的概念和计算方法。
- 了解函数的极限和连续性。
- 掌握微分中值定理和泰勒公式的应用。
- 掌握积分的概念和计算方法。
- 了解积分的应用，如计算面积和体积等。
学习时间
学习通CALCULUS Ⅰ的学习时间根据学生的基础和学习进度不同而有所差异。一般建议学生每周投入5-10小时左右的时间来学习和做习题。
总结
学习通CALCULUS Ⅰ的学习对于学习高等数学和工程学科都非常重要。通过学习该课程，可以掌握微积分的基本概念和计算方法，提高数学素养和思维能力。
中国大学CALCULUS Ⅰ
在中国大学的数学课程中，CALCULUS Ⅰ是一门必修课程。它是数学分析的基础，涵盖了微积分的基本概念、定理、方法和应用。学生需要掌握导数、积分、微分方程等基本知识，以及它们在实际问题中的应用。
导数
导数是微积分中最重要的概念之一。学生需要从定义出发，理解导数的本质和意义。导数可以理解为一个函数在某一点处的瞬时变化率，也可以理解为曲线在某一点处的切线斜率。导数具有一系列重要的性质，如可导性、导数的四则运算、链式法则、反函数求导等。
积分
积分是微积分中另一个重要的概念。学生需要掌握不定积分、定积分、换元积分法、分部积分法等基本方法。积分可以理解为曲线下方的面积，也可以理解为一个函数的反导数。积分具有一系列重要的性质，如可积性、积分的线性性、换元积分法、分部积分法等。
微分方程
微分方程是微积分中应用最广泛的领域之一。学生需要掌握一阶和二阶线性微分方程、变量分离法、常系数齐次线性微分方程、常系数非齐次线性微分方程等基本知识。微分方程可以理解为一个函数及其导数构成的关系式，它在很多自然科学和社会科学中都有广泛的应用。
应用
微积分在科学、工程、经济等领域中具有广泛的应用。学生需要学会将微积分的知识应用到实际问题中，如瞬时速度、加速度、优化问题、微分方程建模等。同时，学生需要学会使用计算机工具，如MATLAB、Mathematica等，来解决更加复杂的问题。
总结
CALCULUS Ⅰ是中国大学数学课程中不可或缺的一门课程。通过学习这门课程，学生将掌握微积分的基本概念、定理、方法和应用，为进一步的数学、物理等领域的学习打下坚实的基础。
