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The questions and answers for the winter 2010 final examination of the mathematical models ii course. The examination covers various topics in calculus, including derivatives, integrals, and differential equations.
Typology: Exams
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Mathematical Models II 201-225-AB Final Examination Winter 2010
Instructor: Bob DeJean
Please answer any questions with decimal answers to 4 decimal places.
The current through an 20 H inductor is given by I = 25 sin(4πt). What is the Root-Mean-Square of this current?
Does
x
y x satisfy the equation x ( x + 1 ) y '= y?
Find the derivatives of the following functions:
x
y =sin^3^ x
y =tan(ln( 3 x ))
Find the derivatives y =sec x
y = tan −^1 ( x − 5 )
y = x^3 ln( x − 1 )
5 y = ex − 2
Differentiate implicitly: sin x – ln y = xy
Find the equation of the line that is normal to y = sin-1^ x at the point (0.6, 0.6435)
Integrate the following:
∫ + =
( )
∫ −
2
∫ (^) + xdx =
x 1 tan
sec 2
∫ −^ =
4 1
∫ x^ xe dx = sec tan sec x
Integrate
∫ 4 sin^2 (^3 x ) dx^ =
∫ (^9) − 4 x 2 =
dx
∫ x^ sin(^4 x ) dx =
Find the area of this region:
Consider the function that is 1 for 2
Here is its graph:
I am interested in its Fourier Expansion.
What is a 0 =
What is a 1 =
What is b 1 =
Use these values to write the beginning of the Fourier Expansion of the function.
Answers
yes
2
3 cos 3 sin 3 x
y ′= x x − x
x
y ′ =sec^2 (ln^3^ x )
x
y x x 2
′ =sec tan
x
y
3 ln( 1 ) 2 3 −
x
y x x^ x
y’ = 5 (e x^ – 2)^4 e x
x y
y x y
cos
y= - 0.8x + 1.
4 m by 8 m
x C x
2
ln(1 + tan x) + C
e secx^ + C
2x – 1/3 sin 6x + C x (^) + C
sin^2 2
− x^ x + sin 4 x + C 16
cos 4 1 4
