Math 126
Name:
Summer 2015
Score:
/50
Show all your work
Dr. Lily Yen
No Calculator permitted in this part. Read the questions carefully. Show all your work
and clearly indicate your final
R x answer. Use proper notation.
Problem 1: Let A(x) = 0 f (t) dt, where f is the function graphed below (and x is in the
interval shown).
Test 1
y
R
T
O
Q
x
S
P
a. A(x) has a local maximum at the x-coordinates of (circle all that apply):
O
P
Q
R
S
T
b. A(x) has a local minimum at the x-coordinates of (circle all that apply):
O
P
Q
R
S
T
c. A(x) has points of inflection at the x-coordinates of (circle all that apply):
O
P
Q
R
S
T
d. Which inequality is true for all x in the interval over which the graph of f is shown?
(Circle all that apply.)
A(x) ≤ 0
A(x) ≥ 0
neither
both
e. A(x) has an absolute minimum at the x-coordinates of (circle all that apply):
O
P
Q
R
S
T
Score:
/5
Problem 2: Describe the area represented by the following limit as an integral. Do not
evaluate.
N
π X π
π
jπ lim
−
+
cos
N →∞ 2N
3 4N
2N
j=1
This is the middle-point Riemann sum for
Z
0
π/2
√
1
−
3
cos( π3 + x) dx =
2
That is, the net area between the x-axis and y = cos( π3 + x) for 0 ≤ x ≤ π2 .
Score:
/3
Problem 3: A particle begins at the origin at time t = 0 and moves with velocity v(t) as
shown below. (The graph consists of semicircles and line segments.)
v(t) (m/s)
Areas of parts of the original graph in red
2π
2
M5
1
1
2
1
t (s)
2
4
−1
6
8
− 12
−π/2
a. Find the distance travelled by the particle after 6 seconds.
b. Find the displacement of the particle after 8 seconds.
2.5π metres
1.5π + 1 metres
c. Find the time the particle is farthest from the origin.
4 seconds
d. Directly on the graph, draw in and shade the approximating rectangles representing
M5 .
Score:
/5
Problem 4: Evaluate the following integrals exactly using the Fundamental Theorem of
Calculus.
Z
d t
sec(5x − 9) dx
a.
dt 100
Z
b.
d
(23x2 − 8ex + cos(x2 )) dx
dx
√
Z
c.
u
sec(5t − 9)
23x2 − 8ex + cos(x2 ) + C
d
1
dx
du −u
1 + x2
Rt
1
1
0
Let f be the function defined by f (t) = 0 tan( 1+x
2 ) dx. Then f (t) = tan( 1+t2 ) and
the integral
you have to find
√ is 1
√
d
1
1
0
(f ( u) − f (−u)) = f ( u) · 2√u − f 0 (−u) · (−1) = tan( 1+u
) · 2√1 u + tan( 1+u
2 ).
du
tan
Problem 5: Evaluate exactly
R3
1
Score:
/4
Score:
/2
|2x − 4| dx. Sketch the graph.
y
The area under the graph is clearly 2.
2
x
1
Page 2
2
3
Math 126
Problem 6: Find an expression of a function f and a corresponding interval [a, b] such that
Z b
Z b
1
3
|f (x)| dx = .
and
f (x) dx =
2
2
a
a
√
For example f (x)(= x on [−1, 2].
x − 1,
x≤2
Another: f (x) =
on [0, 3].
−x + 3, x > 2
Score:
/3
Score:
/3
Problem 7: Integrate the following analytically.
√
Z
2x2 ex − 3 x + 6
a.
dx
x2
R 2x2 ex −3√x+6
R
dx = 2ex − 3x−3/2 + 6x−2 dx = 2ex + 6x−1/2 − 6x−1 + C
x2
Z
b.
(x + 1)(x2 + 2x)3/4 dx
1
= 2x + 2 = 2(x + 1), so dx = 2(x+1)
du. Thus
If u = x2 + 2x, then du
dx
R
R
R
1
1
2
3/4
3/4
3/4
(x + 1)(x + 2x) dx = (x + 1)u 2(x+1) du = 2 u du = 12 · 47 u7/4 + C =
2
(x2 + 2x)7/4 + C
7
Score:
/2
Z
c.
tan(ln(x))
dx
x
If u = ln(x), then du
= x1 , so dx = x du. Thus
dx
R tan(ln(x))
R tan(u)
R
dx
=
x
du
=
tan(u) du = − ln|cos(u)| + C = − ln|cos(ln(x))| + C
x
x
Score:
Page 3
/3
Math 126
Math 126
Name:
Summer 2015
Show all your work
Dr. Lily Yen
Calculators permitted from here on.
Problem 8: Express the following integral as a limit of sums.
Z 4
ex sin(x) dx
Test 1
1
Using the left Riemann sum,
n−1
3 X 1+3j/n
lim
e
sin(1 +
n→∞ n
j=0
3j
)
n
Score:
/3
Problem 9: Graph x = 3y − y and find the exact area bounded by the curve and y-axis.
2
y
3
2
1
x
−4
−3
−2
−1
−1
1
2
The area is
27
2
−9=
R3
0
2
3y − y dy =
3 2
y
2
−
3
1 3
y 3
9
.
2
=
0
Score:
/4
Problem 10: Evaluate the following integral exactly.
Z 10
1
dx
2
−3 4 + 25x
If u = 52 x, then du
= 5 , so dx = 52 du. Thus
R 25
R 10 1
Rdxx=10 2 1
R 25
1
2
1
1
dx
=
·
du
=
·
25
du
=
, du =
2
2
2
10 −15/2 1+u2
−3 4+25x
−15/2 4(1+u )
25 x=−3 4+4u 5
1
1
1
1
1
arctan(u)
= 10
arctan(25) − 10
arctan(− 15
) = 10
arctan(25) + 10
arctan( 15
)
10
2
2
−15/2
Score:
Page 4
/4
Math 126
Problem 11: The rate (in litres per minute) at which water drains from a tank is recorded
at half-minute intervals. Compute the average left- and right-endpoint approximations to
estimate the total amount of water drained during the first 3 minutes.
t (min)
r (L/min)
0
50
0.5 1 1.5
48 46 44
2
42
2.5 3
40 38
The left-sum is 12 (50 + 48 + 46 + 44 + 42 + 40) = 135 and the right-sum is
1
(48 + 46 + 44 + 42 + 40 + 38) = 129, so the average is 132 litres.
2
Score:
/5
Problem 12: Wind engineers have found that wind speed v (in metres per second) at a
given location follows a Rayleigh distribution of the type
W (v) =
1 −v2 /64
ve
.
32
This means that at a given moment in time, the probability that v lies between a and b is
equal to the shaded area under W in the interval [a, b].
a. Find analytically the probability that v ∈ [0, b].
Rb
2
v
You have to find 0 W (v) dv. Now, if u = −v
, then du
= − 32
, so dv = − 32
, du. Thus
64
dv
vv=b
Rb
R b 1 −v2 /64
R v=b 1 u 32
R v=b u
=
W (v) dv = 0 32 ve
dv = v=0 32 ve (− v ) du = − v=0 e du = −eu 0
v=0
b
2
2
2
−e−v /64 = −e−b /64 − (−1) = 1 − e−b /64 .
0
Score:
/3
b. Compute the probability that v ∈ [1, 3] to 6 decimal place accuracy. Show the shaded
region on the graph of W .
W
0.09
0.06
0.03
(1 − e−9/64 ) − (1 − e−1/64 ) ≈ 0.116
v
2
4
6
8
10 12 14 16 18
Score:
Page 5
/1
Math 126