Test 1
Math 126
Name:
Summer 2007
Score:
/53
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 answer. Use proper notation.
Problem 1: Integrate the following indefinite integrals analytically.
Z
5x
√
a.
dx
Score:
/3
3
1 − 2x
3
15
(1 − 2x)5/3 − (1 − 2x)2/3 + c
4
8
Z
b.
cos(x)
dx
1 + sin2 (x)
Score:
/3
Score:
/3
Score:
/3
arctan(sin(x)) + c
Z
t2 et dt
c.
t2 et − 2tet + 2et + c
Z
d.
sin(x) cos(cos(x)) dx
− sin(cos(x)) + c
Z
3x − 1
dx
−x−6
e.
Score:
x2
/3
7
8
ln|x + 2| + ln|x − 3| + c
5
5
Problem 2: Evaluate the following definite integrals exactly using the Fundamental Theorem of Calculus.
Z 1
a.
x3 (2 − 3x4 )4 dx
Score:
/3
0
11/20
Z
e6
b.
e
dx
√
x ln x
Score:
/3
√
sin( x) dx
Score:
/3
√
2 6−2
Z
π2
c.
π 2 /4
2π − 2
Page 2
Math 126
Test 1
Math 126
Name:
Summer 2007
Show all your work
Dr. Lily Yen
Calculators permitted from here on.
Problem 3: Use the graph of the function f to evaluate the following exactly.
6
1
-
1
2
3
4
5
6
6
semi-circle
−1
Z
2
a.
f (x) dx =
1 π
− ≈ 0.1073
2 8
f (x) dx =
3
2
0
Z
6
b.
2
Z
6
c.
|f (x)| dx = 3 +
0
¯Z
¯
d. ¯¯
6
0
π
≈ 3.3927
8
¯ ¯
¯ ¯
π
π ¯¯
f (x) dx¯¯ = ¯2 − ¯ = 2 − ≈ 1.6073
8
8
µ
¶
Z 2
n
X
i 1
π
e. lim
f 1+
=
f (x) dx = − ≈ −0.3927
n→∞
n n
8
1
i=1
f. Calculate L4 for f on [2, 6]. Write out the sum in terms of specific function evaluations
(i.e., f (xk )) to show which xk ’s you use for each interval before showing the actual
function values and final sum.
L4 = (f (2) + f (3) + f (4) + f (5)) · 1 = 0 + 1 + 1 + 0 = 2.
g. Suppose that f gives the velocity in metres per second of a particle as it travels along
a straight line. Assuming time is measured in seconds, what is the displacement of the
particle between t = 0 and t = 6.
R6
The displacement is 0 f (t) dt = 2 − π8 ≈ 1.6073 metres.
Score:
Page 3
/10
Math 126
R∞
2
Problem 4: Show 0 e−x dx is convergent.
Score:
/3
¯
R∞ 1
−1 ¯∞
x2
2
−x2
2
Since e > x for all x, you have that e
< 1/x . Since 1 x2 dx = x 1 = 1 < ∞, it
R ∞ −x2
R1
R∞
2
2
follows by comparison that 1 e
dx converges. Clearly 0 e−x dx exists, so 0 e−x dx
exists.
R 10
Problem 5: Estimate 0 f (x)g 0 (x) dx if f (x) = x2 and g has the values in the following
table. Specify which approximation you use.
x
0
2
4
6
8 10
g(x) 2.3 3.1 4.1 5.5 5.9 6.1
Score:
/3
By
integration
by
parts,
R 10
R 10 0
R 10
10
2
f (x)g 0 (x) dx = f (x)g(x)|10
0 − 0 f (x)g(x) dx = x g(x)|0 − 0 2xg(x) dx. Now
0
10
x2 g(x)|0 = 100 · 6.1 − 0 = 610, and using the left Riemann sum,
R 10
P
2xg(x) dx ≈ 4j=0 4j g(2j) · 2 = (0 + 4 · 3.1 + 8 · 4.1 + 12 · 5.5 + 16 · 5.9) · 2 = 411.2.
0
(This
underestimate since 2xg(x) is increasing.) Thus
R 10 is an
0
f (x)g (x) dx ≈ 610 − 411.2 = 198.8.
0
R9
R3
Problem 6: If f is continuous and 0 f (x) dx = 4, find 0 xf (x2 ) dx.
= 2x, and by substitution,
Let u = x2 . Then du
dx
R3
R x=3
R9
1
2
xf (x ) dx = x=0 xf (u) 2x
du = 0 12 f (u) du = 12 · 4 = 2.
0
Score:
/3
Problem 7: State both parts of the Fundamental Theorem of Calculus.
Theorem Let f be a continuous function in [a, b], then
Z x
d
a.
f (t) dt = f (x).
dx a
Z
b
f (x) dx = F (b) − F (a), where F is an anti-derivative of f , F 0 (x) = f (x).
b.
a
Score:
Page 4
/2
Math 126
Problem 8: Evaluate exactly:
¶¶
µ
µ
Z
d
1
6
a.
sin √
dx
dx
x
µ
sin
d
b.
dx
d
c.
du
Z
7
Z
e
Score:
/1
Score:
/2
Score:
/1
¶
+c
x + ln x
dx
cos(x2 )
cos(u)
d
d.
dx
/1
³
´
√
3
−2 (2x)7 − 2 2x + csc(2x)
sin(u)
eu
1
√
x
√
3
t7 − 2 t + csc(t) dt
2x
Z
6
Score:
u
sin(u) + ln(sin(u))
u e +u
−
e
cos(e2u )
cos(sin2 (u))
3u2 + 9 du
−π
0
Rx
Problem 9: An integral function F is given by F (x) = 0 f (t) dt. Draw a possible graph of
the function y = f (t) (with f continuous) on the interval [0, 10] so that F has the following
properties:
a. F is decreasing on [3, 8], and increasing on both [0, 3] and [8, 10].
b. F 0 (5) = −4.
c. F is concave down on [1, 4], otherwise concave up.
6
5
-
5
10
−5
Score:
Page 5
/3
Math 126