Myexam - Phần mềm Toán | Đại học Sư phạm Hà Nội

Myexam - Phần mềm Toán | Đại học Sư phạm Hà Nội với những kiến thức và thông tin bổ ích giúp sinh viên tham khảo, ôn luyện và phục vụ nhu cầu học tập của mình cụ thể là có định hướng, ôn tập, nắm vững kiến thức môn học và làm bài tốt trong những bài kiểm tra, bài tiểu luận, bài tập kết thúc học phần, từ đó học tập tốt và có kết quả cao cũng như có thể vận dụng tốt những kiến thức mình đã học vào thực tiễn cuộc sống.

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Tác giả:

Profile Kim Oanh
HANOI NATIONAL UNIVERSITY OF EDUCATION FINAL EXAM
Faculty of Mathematics and Informatics Math Softwares (MATH 245E)
Duration: 90 minutes
Problem 1. Let f and g denote functions defined on some set . Prove thatA
sup
x A∈
( ( (f x) + g x)) sup≤
x∈A
f(x) + sup
x∈A
g(x).
Problem 2. Sketch the graph of the function
y
= f(x) =
x
2
x
2
− 1
.
Make sure that your graph clearly indicates the following:
The domain of definition of f( ).x
The behaviour of ) near the points where it is not defined (if any) and as .f(x x → ±∞
The exact coordinates of - and - intercepts and all minimas and maximas ofx y f( ).x
Problem 3. Compute the following integral:
I
=
Z
1
0
x
2
+ 1
x
+ 1
dx.
Solution.
By long division of polynomials, x
2
+ 1 = ( + 1)(x x − 1) + 2. Thus we can rewrite
our integral as a sum of two terms as follows:
I
=
Z
1
0
(x + 1)( 1)x −
x
+ 1
dx +
Z
1
0
2
x
+ 1
dx =
Z
1
0
(
x − 1)dx + 2
Z
1
0
1
x
+ 1
dx
=
x
2
− x + 2 ln(x + 1)
1
0
= 2 ln 2.
Problem 4. Use Geogebra to draw the diagram for the following problem:
Consider the circle (C), and let KL and KN be tangents to the circle, with lying onL and N
the circle. Take a point anywhere on the line on different sides of ).M KN (M and K N
Assume that the circle (C) intersects the circumcircle of triangle KLM at a second point .P
Let Q be the foot of the perpendicular dropped from N onto .M L
Prove that
\
MP Q = 2
\
KML.
Problem 5. For each of the piecewise-defined functions, (a) sketch the graph, and (b) evaluate
at the given values of the independent variable.
f
(x) =
x
2
− 3 x ≤ 0
4
x − 3 x > 0
. Find f( 4) (0) (2)− , f , f .
g
(x) =
x + 1 5x ≤
4
x > 5
. Find g(0) ( ) (5), g π , g .
Problem 6. (a) Evaluate piecewise function values from a graph. (b) Construct a piecewise
function corresponding to the graph.
Find f( 4) ( 2) ( 3) 2) (2)− , f − , f(0) Find g − , g(− , g
1
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HANOI NATIONAL UNIVERSITY OF EDUCATION FINAL EXAM
Faculty of Mathematics and Informatics Math Softwares (MATH 245E) Duration: 90 minutes
Problem 1. Let f and g denote functions defined on some set A. Prove that
sup (f (x) + g(x)) ≤ sup f (x) + sup g(x). x∈A x∈A x∈A
Problem 2. Sketch the graph of the function x2 y = f (x) = . x2 − 1
Make sure that your graph clearly indicates the following:
The domain of definition of f (x).
The behaviour of f (x) near the points where it is not defined (if any) and as x → ±∞.
The exact coordinates of x- and y- intercepts and all minimas and maximas of f (x).
Problem 3. Compute the following integral: Z 1 x2 + 1 I = dx. x + 1 0
Solution. By long division of polynomials, x2 + 1 = (x + 1)(x − 1) + 2. Thus we can rewrite
our integral as a sum of two terms as follows: Z 1 (x + 1)(x − 1) Z 1 2 Z 1 Z 1 1 I = dx + dx = (x − 1)dx + 2 dx x + 1 x + 1 x + 1 0 0 0 0 1
= x2 − x + 2 ln(x + 1) = 2 ln 2. 0
Problem 4. Use Geogebra to draw the diagram for the following problem:
Consider the circle (C), and let KL and KN be tangents to the circle, with L and N lying on
the circle. Take a point M anywhere on the line KN (M and K on different sides of N ).
Assume that the circle (C) intersects the circumcircle of triangle KLM at a second point P .
Let Q be the foot of the perpendicular dropped from N onto M L. Prove that \ M P Q = 2 \ KM L.
Problem 5. For each of the piecewise-defined functions, (a) sketch the graph, and (b) evaluate
at the given values of the independent variable.  x2 − 3 x ≤ 0 f (x) = . Find f (−4), f (0), f (2). 4x − 3 x > 0  x + 1 x ≤ 5 g(x) = . Find g(0), g(π), g(5). 4 x > 5
Problem 6. (a) Evaluate piecewise function values from a graph. (b) Construct a piecewise
function corresponding to the graph. Find f (−4), f (−2), f (0) Find g(−3), g(−2), g(2) 1