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5 5 Aalto University
Aalto University School of Science
Department of Mathematics and Systems Analysis
Exam / Final Exam 7.3.2017 Malinen
MS-A0205 Differential and Integral Calculation 2 (ENG1)
MS-A0206 Differential and Integral Calculation 2 (ENG2)
Final exam: 1, 3, 4, 6, 8
Exam: 2, 4, 5, 7, 9.
The calculator and the tables must not be used in the test. Fill in all heading information
To answer letters.
Let's look at the spiral: i: = e "cos (t), y = e" sin (t) when t 6 R.
A) Define the use of the tangeut torch corresponding to the paramctric value t = 0.
B) Calculate the arc length of the spiral round 0 S t S 271 '.
2. Determine the surface zr3 + 3xzy + y2 + 2 cos z = 13 with a normal vector at point
(1, 2, O).
3. Consider the function f, jollc
sin (1? 2)
F (r.y) = w,
When a:> 0 and y> 0. lvl set the function f limits with positive coordinates
(220, O) and (O, yo) when: co> 0 and yo> O.
Hint: The polynomial approximation of the sin function / L'Hospital's rule
Can be useful.
TURN!
Page two
Define the function
2
f (= v, y) = I + 2y +%
The largest and smallest value on the ball 9: 2 + 3/2 + 22 = 5 using Lagrange method.
Make the parabola y = 10 to 1022 between the parabola and the flat bow D between the axis r.
The y coordinate of the center
_ L
Calculate the cross-sectional diameter of the passing ball in the polar coordinate system,
The area of r = vs 0, O s 0 s 1r of the cut.
\
u
I
of
-u -cu -n
Figure 1: Falling on a countdown pulley.
A = // DDA.
And integral
// I = YDA.
D
What is the physical significance of 37 = I fA?
. Draw a pattern from the set D = {(n, y) | -1 s a: s 2,: E2 s y s x + 2}.
Lower its surface area
Calculate the smoothing g of the function g (x, y) = sin 1: + y) beyond the triangle T by which.
The tips are at points (0,0), (1r, 0) and (1r, 7r).
. The temperature T = T (r) of the circumferential R-radial plate D drops the mean
As a function of the distance measured by the point r as a function of the center point value of 100 lines »
Rewind edge to O. Form temperature expression T ('r) and lgske plate
Average temperature Turn!!!
_ 1
T-Z / DTdA
It's not that hard just try& focus.
You're welcome.
Page one translation
5 5 Aalto University
Aalto University School of Science
Department of Mathematics and Systems Analysis
Exam / Final Exam 7.3.2017 Malinen
MS-A0205 Differential and Integral Calculation 2 (ENG1)
MS-A0206 Differential and Integral Calculation 2 (ENG2)
Final exam: 1, 3, 4, 6, 8
Exam: 2, 4, 5, 7, 9.
The calculator and the tables must not be used in the test. Fill in all heading information
To answer letters.
Let's look at the spiral: i: = e "cos (t), y = e" sin (t) when t 6 R.
A) Define the use of the tangeut torch corresponding to the paramctric value t = 0.
B) Calculate the arc length of the spiral round 0 S t S 271 '.
2. Determine the surface zr3 + 3xzy + y2 + 2 cos z = 13 with a normal vector at point
(1, 2, O).
3. Consider the function f, jollc
sin (1? 2)
F (r.y) = w,
When a:> 0 and y> 0. lvl set the function f limits with positive coordinates
(220, O) and (O, yo) when: co> 0 and yo> O.
Hint: The polynomial approximation of the sin function / L'Hospital's rule
Can be useful.
TURN!
Page two
Define the function
2
f (= v, y) = I + 2y +%
The largest and smallest value on the ball 9: 2 + 3/2 + 22 = 5 using Lagrange method.
Make the parabola y = 10 to 1022 between the parabola and the flat bow D between the axis r.
The y coordinate of the center
_ L
Calculate the cross-sectional diameter of the passing ball in the polar coordinate system,
The area of r = vs 0, O s 0 s 1r of the cut.
\
u
I
of
-u -cu -n
Figure 1: Falling on a countdown pulley.
A = // DDA.
And integral
// I = YDA.
D
What is the physical significance of 37 = I fA?
. Draw a pattern from the set D = {(n, y) | -1 s a: s 2,: E2 s y s x + 2}.
Lower its surface area
Calculate the smoothing g of the function g (x, y) = sin 1: + y) beyond the triangle T by which.
The tips are at points (0,0), (1r, 0) and (1r, 7r).
. The temperature T = T (r) of the circumferential R-radial plate D drops the mean
As a function of the distance measured by the point r as a function of the center point value of 100 lines »
Rewind edge to O. Form temperature expression T ('r) and lgske plate
Average temperature Turn!!!
_ 1
T-Z / DTdA
It's not that hard just try& focus.
You're welcome.
Last edited:
