D
M
epartment of
athematics
Final Examination
15 May 2015
14h–17h
Mathematical Models
201-225-AB
Instructor: R.Masters
Student name:
Student number:
Instructor:
Instructions
1. Do not open this booklet before the examination begins.
2. Check that this booklet contains 4 pages, excluding this cover page and the formula sheet.
3. Write all of your solutions in this booklet and show all supporting work.
4. If the space provided is not sufficient, continue the solution on the opposite page.
Final Examination
201-225-AB
15 May 2015
1. Find y 0 for the following. Do not simplify your answer.
(3)
(a) y = sin(arctan(3x)) + 72x
(3)
(b) arcsin(x + y) + 2y = x3
(3)
(c) y =
Z
Hint: Solve for y 0 .
1
(−2t + 3 sin t) dt
Hint: use part 1 of the F.T.C.
cos x
(3)
(d) y = log7 (tan x) + esec 5x
2. Given f (x) =
x2
x
−9
f 0 (x) = −
(x2 + 9)
(x2 − 9)2
(2x)(x2 + 27)
.
(x2 − 9)3
f 00 (x) =
Find (if any):
(1) (a) The x and y intercept(s).
(1) (b) The vertical and horizontal
asymptotes.
(1) (c) The critical numbers
(1) (d) The inflection points.
(1) (e) Local (relative) extrema.
(1) (f) Intervals of upward or downward
concavity.
(1) (g) Intervals on which f is increasing or
decreasing.
(3) (h) Sketch the graph of f .
Z
5
(4) 3. Use Simpson’s Rule with n = 4 to estimate the value of
x2
1
1
dx (give your answer to 3 decimals).
+x
(5) 4. The illumination of an object (O) by a light source is directly proportional to the strength of the source and
inversely proportional to the square of the distance from the source. If two light sources (S1 and S2 ), one 8
times as strong as the other, are placed 100 ft apart, how many feet from the brighter source should an object
be placed on the line between the sources so as to receive the least illumination (see diagram below)?
S1
S2
O
100f t
y
(4) 5. Find the area of the region between the two
graphs, f (x) = (x − 1)3 and g(x) = x − 1.
1
f (x)
0
−1
g
6. Evaluate the following limits.
Page 1 of 4
f
1
g(x)
2
x
Final Examination
(3)
201-225-AB
15 May 2015
(a) limπ (1 − sin x) · tan x
x→ 2
(3)
(4)
ex + e−x − 2
x→0 1 − cos 2x
sin x
1
(c) lim
x→0 x
(b) lim
(3) 7. Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the pressure P and
the volume V satisfy the equation P V = C, where C is a constant. Suppose that at a certain instant the
volume is 600 cm3 , the pressure is 150 kP a, and the pressure is increasing at a rate of 20 kP a/min. At what
rate is the volume decreasing at this instant?
8. Solve the following differential equations for y.
(3)
(a) ydt + tdy = 3ty dt
(3)
(b) (yx2 + y)
with initial condition y(1)=1.
dy
= tan−1 x
dx
(4) 9. Find the volume of the solid formed by rotating the region enclosed by
x = 0, x = 1, y = 0, y = 6 + x6
about the y-axis.
y
Hint: Split the region into two.
7
6
5
4
3
2
1
x
1
0
(3) 10. Determine if y = x3 + 2x2 − 5 is a solution to the differential equation −y 000 + y 00 −
11. Integrate the following integrals.
Z
cos(ln x)
dx
(4)
(a)
x
Z
(4)
(b)
tan3 x dx
Z
(4)
(c)
√
x2
1
dx
+ 6x + 13
Page 2 of 4
y0
= 3x − 6
x
Final Examination
Z
(4)
(d)
x2
Z
(4)
(e)
201-225-AB
15 May 2015
5
dx
−9
tan9 x sec4 x dx
Z
(4)
(f)
arcsin x dx
Z
(4)
(g)
sin2 (2x) dx
(5) 12. Solve the following first order linear differential equation for y.
dy
+ 2y cot x = 4 cos x
with initial condition x = π/2 when y = 1/3
dx
13. Given the function and its graph.
f (x)
3
(
2,
f (x) =
3,
if −π ≤ x < − π2 ,
if − π2 ≤ x < π2
π
2
2
≤x<π
1
x
−π
− π2
0
π
2
π
(2)
(a) Determine if the function is even or odd.
Show your work or Explain to obtain full marks
(4)
(b) Find the first three non-zero terms of the Fourier series for the function above. Hint: find a0 , a1 , a2 , a3 , . . . , b1 , b2 , . . .
and write the function expansion.
Answers
3
1. (a) y 0 = cos(arctan 3x) ·
+ 2 · 72x · ln 7
1 + (3x)2
p
3x2 1 − (x + y)2 − 1
0
p
(b) y =
1 + 2 1 − (x + y)2
sec2 x
+ esec 5x · 5 sec 5x tan 5x
tan x · ln 7
(d) y 0 = (2 cos x − 3 sin(cos x))(− sin x)
(c) y 0 =
2. A ≈ 0.522
3. (a) x-int = 0 and y-int = 0
(b) V.A. x = ±3 ; H.A. y = 0
(c) None
(d) x = 0
(e) None
(f) C.U. ] − 3, 0[∪]3, ∞[
C.D. ] − ∞, −3[∪]0, 3[
Page 3 of 4
Final Examination
201-225-AB
(g) Inc. No where
dec. ] − ∞, ∞[
4.
200
f t away from source 2
3
5. A = 1/2
6. (a) 0
(b) 1/2
(c) 1
7.
dV
= −80 cm3 /min
dt
e3(t−1)
t
p
(b) y = (arctan x)2 + 2c
8. (a) y =
9.
25
π
4
10. yes
11. (a) sin(ln x) + c
tan2 x
− ln | sec x| + c
2√
x2 + 6x + 13 + x + 3 (c) ln +c
2
x − 3 5/6
+c
(d) ln x + 3
(b)
tan12 x tan10 x
+
+c
12
10
p
(f) x · arcsin x + 1 − x2 + c
(e)
(g) 1/2 (x − 1/4 sin x) + c
12. y = 4/3 sin x − csc2 x
13. (a) f (x) is even
5 2
2
2
(b) f (x) = + cos x −
cos x +
cos x − · · ·
2 π
3π
5π
Page 4 of 4
15 May 2015