Download n2 + 6n − √ n2 + 1 - skalan

MÄLARDALEN UNIVERSITY
School of Education, Culture and Communication
Department of Applied Mathematics
Examiner: Lars-Göran Larsson
EXAMINATION IN MATHEMATICS
MAA151 Single Variable Calculus, TEN1
Date: 2015-08-10
Write time: 3 hours
Aid: Writing materials
This examination is intended for the examination part TEN1. The examination consists of eight randomly ordered problems
each of which is worth at maximum 3 points. The pass-marks 3, 4 and 5 require a minimum of 11, 16 and 21 points respectively.
The minimum points for the ECTS-marks E, D, C, B and A are 11, 13, 16, 20 and 23 respectively. If the obtained sum of points
is denoted S1 , and that obtained at examination TEN2 S2 , the mark for a completed course is according to the following
S1 ≥ 11, S2 ≥ 9
and
S1 + 2S2 ≤ 41
→
3
S1 ≥ 11, S2 ≥ 9
and
42 ≤ S1 + 2S2 ≤ 53
→
4
54 ≤ S1 + 2S2
→
5
S1 ≥ 11, S2 ≥ 9
and
S1 + 2S2 ≤ 32
→
E
S1 ≥ 11, S2 ≥ 9
and
33 ≤ S1 + 2S2 ≤ 41
→
D
S1 ≥ 11, S2 ≥ 9
and
42 ≤ S1 + 2S2 ≤ 51
→
C
52 ≤ S1 + 2S2 ≤ 60
→
B
61 ≤ S1 + 2S2
→
A
Solutions are supposed to include rigorous justifications and clear answers. All sheets of solutions must be sorted in the order
the problems are given in.
1.
Determine whether lim
n→∞
√
√
n2 + 6n − n2 + 1 exists or not.
If the answer is no: Give an explanation of why!
If the answer is yes: Give an explanation of why and find the limit!
2.
Explain why 3a2 − 2a + 4/3 + . . . is a geometric series for a 6= 0 . Also, find
those a for which the series converges, and determine the corresponding sums.
3.
Find to the function x y f (x) =
2
the antiderivative F that have
(x + 2)(x + 4)
the value 1 at the point −3.
4.
Determine to the differential equation y 00 − 6y 0 + 9y = 0 the solution that
satisfies the initial conditions y(0) = 1 , y 0 (0) = 5.
5.
Let f (x) = e−4( x−2) . Find an equation for the tangent line to the curve
y = f (x) at the point P whose x-coordinate is equal to 4.
6.
Let f (x) =
7.
The product of two positive real numbers a and b equals 8. Which are the
numbers if the weighted sum 2a + 9b of them is a minimum? Prove your
conclusion!
√
x+1
. Find the function expression, the domain and the range of
x−1
the composition f ◦ f .
Z
8.
π
| sin(2x)| dx .
Evaluate the integral
0
Om du föredrar uppgifterna formulerade på svenska, var god vänd på bladet.
MÄLARDALENS HÖGSKOLA
Akademin för utbildning, kultur och kommunikation
Avdelningen för tillämpad matematik
Examinator: Lars-Göran Larsson
TENTAMEN I MATEMATIK
MAA151 Envariabelkalkyl, TEN1
Datum: 2015-08-10
Skrivtid: 3 timmar
Hjälpmedel: Skrivdon
Denna tentamen är avsedd för examinationsmomentet TEN1. Provet består av åtta stycken om varannat slumpmässigt ordnade
uppgifter som vardera kan ge maximalt 3 poäng. För godkänd-betygen 3, 4 och 5 krävs erhållna poängsummor om minst 11,
16 respektive 21 poäng. Om den erhållna poängen benämns S1 , och den vid tentamen TEN2 erhållna S2 , bestäms graden av
sammanfattningsbetyg på en slutförd kurs enligt
S1 ≥ 11, S2 ≥ 9
och
S1 + 2S2 ≤ 41
→
3
S1 ≥ 11, S2 ≥ 9
och
42 ≤ S1 + 2S2 ≤ 53
→
4
54 ≤ S1 + 2S2
→
5
Lösningar förutsätts innefatta ordentliga motiveringar och tydliga svar. Samtliga lösningsblad skall vid inlämning vara sorterade
i den ordning som uppgifterna är givna i.
1.
Avgör om lim
n→∞
√
√
n2 + 6n − n2 + 1 existerar eller ej.
Om svaret är nej: Ge en förklaring till varför!
Om svaret är ja: Ge en förklaring till varför och bestäm gränsvärdet!
2.
Förklara varför 3a2 − 2a + 4/3 + . . . är en geometrisk serie för a 6= 0 . Bestäm
även de a för vilka serien konvergerar, och bestäm motsvarande summor.
3.
Bestäm till funktionen x y f (x) =
2
den primitiv F som har
(x + 2)(x + 4)
värdet 1 i punkten −3 .
4.
Bestäm till differentialekvationen y 00 − 6y 0 + 9y = 0 den lösning som satisfierar
begynnelsevillkoren y(0) = 1 , y 0 (0) = 5.
5.
Låt f (x) = e−4( x−2) . Bestäm en ekvation för tangenten till funktionskurvan
y = f (x) i den punkt P vars x-koordinat är lika med 4.
6.
Låt f (x) =
7.
Produkten av två positiva reella tal a och b är lika med 8. Vilka är talen om
den viktade summan 2a + 9b av dem är minimal? Bevisa din slutsats!
√
x+1
. Bestäm funktionsuttrycket, definitionsmängden och värdex−1
mängden för sammansättningen f ◦ f .
Z
8.
π
| sin(2x)| dx .
Beräkna integralen
0
If you prefer the problems formulated in English, please turn the page.
MÄLARDALEN UNIVERSITY
School of Education, Culture and Communication
Department of Applied Mathematics
Examiner: Lars-Göran Larsson
EXAMINATION IN MATHEMATICS
MAA151 Single Variable Calculus
EVALUATION PRINCIPLES with POINT RANGES
Academic Year: 2014/15
Examination TEN1 – 2015-08-10
Maximum points for subparts of the problems in the final examination
1.
The limit exists and is equal to 3
1p: Correctly extended the difference of square roots with the
conjugate, and correctly simplified the numerator
1p: Correctly algebraically prepared for determining the limit
1p: Correctly determined the limit
2.
It is a geometric series since the each
quotient of two terms following each
other has the common value  2 (3a) .
The series converges for a  23 , i.e.
for (a   23 )  (a  23 ) .
3
The sum of the series equals 9a
1p: Correctly explained why the series is a geometric series
1p: Correctly determined the a for which the series converges
1p: Correctly determined the sum of the series
2  3a
x2
x4
3.
F ( x)  1  ln
4.
y  (1  2 x) e 3 x
Note: The student who has stated that
y  Ae 3x  Be 3x is the general solution of the
differential equation, and who has not found
any explanation to the impossible conditions
occuring when adapting to the initial values,
can not obtain any other sum of points than 0p.
1p: Correctly found the partial fractions of f (x)
1p: Correctly found the general antiderivative of f
1p: Correctly adapted the antiderivative to the value at  3
1p: Correctly found one solution of the DE
1p: Correctly found the general solution of the DE
1p: Correctly adapted the general solution to the initial
values, and correctly summarized the solution of the IVP
5.
x y 5
1p: Correctly found the derivative of the the function f , all
with the purpose of determining the slope at the point P
1p: Correctly evaluated the function f and the derivative f 
at the point P
1p: Correctly found an equation for the tangent line to the
curve y  f (x) at the point P
6.
f  f ( x)  x
1p: Correctly found the expression for f  f (x)
1p: Correctly found the domain of the composition f  f
1p: Correctly found the range of the composition f  f
where D f  f  V f  f  ( ,1)  (1, )
7.
a  6 and b 
8.
2
4
3
1p: Correctly for the optimization problem formulated a
function of one variable, and correctly found the
derivative of the function
1p: Correctly worked out a first derivative test
1p: Correctly found the numbers a and b which give the
minimum value of the weighted sum 2a  9b
1p: Correctly divided the integral in two integrals, each in
which the absolute value bars can be removed
1p: Correctly determined an antiderivative for each integrand
1p: Correctly determined the value of the integral
1 (1)