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Kim Oanh

1 năm trước
Danh sách Quiz
/1
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

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