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The solutions to a university-level mathematics exam covering topics such as calculus, trigonometry, logarithms, and integration. It includes both multiple-choice and free-response questions, with answers provided to the nearest decimal place.
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Mathematical Models 2 201-225 Instructor: Bob DeJean Winter 2009 Final Exam Answers to 4 decimal places please.
What is the Root Mean Square of the function y = 20 sin (60πt)?
Is y = 3x^2 – 5 a solution of xy’ – 2y = 10?
Find the derivatives y = 12 tan-1^ (2x+3)
y = x^3 ln(x - 2)
y = 173x-
1
1 2
2
− = (^) x
x
e
e y
Integrate
2 1 ( 4 x 3 )^2 dx
9 4
2
0
15 sin^4 3 cos 3
π x x dx
e x
x 3 1
What is this area?
Find the area under the curve y = 4x^3 + x – 1 between x = 2 and x = 3.
Here’s an area: Approximate the area using the Trapezoidal Rule with n = 5
Solve for y xy’ + 3y^2 = 0
Find y: y’ – y^2 = 16 When x = 0, y = 0
Find y: ′^ + = 2 x
y^ y
What is the equation of the tangent line to y = 6x^2 – 24x at the point where x = 5?
Solve ln (x + 3) - 2x = 0 using Newton’s Method. Your answer should be accurate to 3 decimal places. Show your work.
Here is a function
x if x
if x f x 0
and here is its graph:
I want the first few terms of its Fourier Series. What is a 0?
What is a 1?
What is b 1?
Answers
yes
( 2 3 ) 1
x +^2 +
3 ln( 2 ) 2 3 −
x
x x^ x
17 3x-5^ (ln 17) 3
2 2
2 ( 1 )
x +
x e
e
3x 4 + 2x-5^ + C
C
x
ln 4
4 3
1
3
1/5 sin 5x + C 1/3 (sin x) 3 – 1/5 (sin x)^5 + C x (^) + C
sin 1
10 tan -1^ (x + 3) + C
− x^ x + x + C 25
sin 5 5
cos 5
C x y
− 1 = − 3 ln or x C
y −
3 ln
y (^) = x
tan 4
(^1 1) or y = 4 tan( 4x)
y = x + C/x y = 36x – 150 x ≈ 0.
