Study Guides (380,000)
US (220,000)
Rutgers (3,000)
All (30)
Final

# 01:640:135 Final: 01:640:135 Final Exam 2006 SolutionsExam

Department
Mathematics
Course Code
01:640:135
Professor
All
Study Guide
Final

This preview shows pages 1-3. to view the full 9 pages of the document.
1A
(16) 1. Calculate the following limits. Give a brief justiî€Ścation of your answers without
reference to calculator computations or graphing.
(a) lim
x!0
4x
sin3x
lim
x!0
4x
sin3x
= lim
x!0
4
3
sin 3x
3x
=
4
3
1
=
4
3
(b) lim
x!î€€1
3x
4
+x
3
î€€2x
2
+10
7x
5
î€€4x
3
+2x+1
lim
x!î€€1
3x
4
+x
3
î€€2x
2
+10
7x
5
î€€4x
3
+2x+1
= lim
x!î€€1
3
x
+
1
x
2
î€€
2
x
3
+
10
x
5
7î€€
4
x
2
+
2
x
4
+
1
x
5
=
0
7
=0
(c) lim
x!0
cos(2x)î€€1
x
2
Using l'H^opital's Rule twice,
lim
x!0
cos(2x)î€€1
x
2
= lim
x!0
î€€2sin(2x)
2x
= lim
x!0
î€€4cos(2x)
2
=
î€€4
2
=î€€2
(d) lim
x!î€€3
î€€
j2x+6j
x+3
If xis slightly less than î€€3, then 2x+3<0, so j2x+6j=î€€2xî€€6. Thus
lim
x!î€€3
î€€
j2x+6j
x+3
= lim
x!î€€3
î€€
î€€2(x+3)
x+3
= lim
x!î€€3
î€€
î€€2=î€€2
(10) 2. Compute the derivative of
2
xî€€1
directly from the deî€Śnition.
î€’
2
xî€€1
î€“
0
= lim
h!0
2
x+hî€€1
î€€
2
xî€€1
h
= lim
h!0
2(xî€€1) î€€2(x+hî€€1)
h(x+hî€€1)(xî€€1)
=
lim
h!0
î€€2h
h(x+hî€€1)(xî€€1)
= lim
h!0
î€€2
(x+hî€€1)(xî€€1)
=
î€€2
(xî€€1)
2
bvnbvnbv

Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

2A
(16) 3. Compute the derivatives with respect to xof the following functions. Algebraic
simpliî€Ścation of the answers need not be performed.
(a) e
x
ln(2x)
e
x
î€’
1
2x
î€“
2+e
x
ln(2x)
(b)
sinx
x
3
+2x
(x
3
+2x)cos(x)î€€sin(x)(3x
2
+2)
(x
3
+2x)
2
(c)
Z
0
x
sectdt
î€’
Z
0
x
sectdt
î€“
0
=
î€’
î€€
Z
x
0
sectdt
î€“
0
=î€€secx
(d)
Z
x
2
+x
0
sin(2t)dt
sin(2(x
2
+x))(2x+1)
(10) 4. Suppose that fis a function with î€Śrst and second derivatives. Suppose in addition
that the following values are known: f(1) = 0, f
0
(1) = 3, and f
00
(1) = 5. If g(x)=e
f(x)
,
what are g
0
(1) and g
00
(1)?
g
0
(x)=e
f(x)
f
0
(x)
so g
0
(1) = e
0
(3) = 3.
g
00
(x)=e
f(x)
f
00
(x)+e
f(x)
f
0
(x)
2
Thus g
00
(1) = e
0
(5) + e
0
(3)
2
=14

Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

3A
(15) 5. Find the following indeî€Śnite integrals:
(a)
Z
î€’
x
4
+ sec(x)tan(x)î€€
2
x
î€“
dx
x
5
5
+ secxî€€2lnjxj+C
(b)
Z
3x
2
sin(x
3
+1)dx
If u=x
3
+1,then du =3x
2
dx and
Z
3x
2
sin(x
3
+1)dx =
Z
sin(u)du =î€€cos(u)+C=î€€cos(x
3
+1)+C
(c)
Z
cos(x)e
sin(x)
dx
If u= sin(x), then du = cos(x)dx and
Z
cos(x)e
sin(x)
dx =
Z
e
u
du =e
u
+C=e
sin(x)
+C
(18) 6. Compute the following:
(a)
Z
2
1
x
3
+5
x
dx
Z
2
1
x
3
+5
x
dx =
Z
2
1
î€’
x
2
+
5
x
î€“
dx =
î€’
x
3
3
+5lnx
î€“
2
1
=
8
3
+ 5ln2 î€€
î€’
1
3
+ 5ln1
î€“
=
7
3
+ 5ln2
(b) The area under the graph of y=4x+e
x
on the interval [0;2].
The area is
Z
2
0
(4x+e
x
)dx =(2x
2
+e
x
)j
2
0
=8+e
2
î€€1=7+e
2