A)169 B)144 C)121 D)100
⇣ Math
⇣ p ⌘
⇣ p ⌘
⇣ p ⌘
p ⌘ 1431
45
13
58
35
13
19
19
17
A) LAB
;
B)
;
C)
;
D)
;
session
12
2
2
2
2
2
2
2
2
⇣
⌘
⇣
⌘
⇣
⌘
⇣
⌘
1
5
1
5
1
3
1
3
p ;p
p ;p
A) p6 ; pe and
B) p7 ; pe and
e
e
6
7
⇣
⌘
⇣
⌘
⇣
⌘
⇣
⌘
1
3
1
3
1
5
1
5
p ;p
p ;p
C) p4 ; pe and
D) p8 ; pe and
e
e
4
8
Quiz 20
Example 1:
Use differentials to estimate the value (81.5) /4 sin °(31)
3
Example 2:
A)169 B)144 C)121 D)100
⇣ p ⌘
⇣ p ⌘
⇣ p ⌘
⇣ p ⌘
A) 13
; 245
B) 19
; 213
C) 19
; 258
D) 17
; 235
2
2
2
2
⇣
⌘
⇣
⌘
⇣
⌘
⇣
⌘
1
5
1
5
1
3
1
3
p ; p
p ; p
A) p6 ; pe and
B) p7 ; pe and
e
e
6
7
⇣
⌘
⇣
⌘
⇣
⌘
⇣
⌘
1
3
1
3
1
5
1
5
p ; p
p ; p
C) p4 ; pe and
D) p8 ; pe and
e
e
4
8
3
(81.5)
Use differentials to estimate the
value/4
sin °(31)
12
dy
Example 3:
Find the differential
(81.5)3/4 sin °(31)
dy
for
y = 5x3 cos (x)
y = 5x3 cos (x)
Example 4:
A spherical ball bearing will be coated by 0.03 cm of protective coating. If the radius of this ball bearing is 7
cm, approximately how much coating will be required? (use π≈3.14)
12
12
y = 5x cos (x)
1
10
Example 5:
y = 5x3 cos (x)
earcsin (5x)
Give the derivative of
at the point where x =
.
arcsin (5x)
e
A)169 B)144 C)121 D)100
⇣ p ⌘
⇣ p ⌘
⇣ p ⌘
⇣ p ⌘
45
13
58
13
19
19
A)169 B)144 C)121 D)100
A) 2 ; 2
B) 2 ; 2
C) 2 ; 2
D) 17
; 235
2
⇣ p ⌘
⇣ p ⌘
⇣⇣ p ⌘⌘
⇣⇣ p ⌘ ⌘
⇣
⌘
⇣
⌘
45
13
13
19
191 58
17 135 5
5
1
3
1
3
A) 2 ; 2
B) 2 ; 2
C)
D) 2 ; p2 ; p
p ;p
A) 2p;6 ; 2pe and
B) p7 ; pe and
e
e
6
7
⇣
⌘
⇣
⌘
⇣
⌘
⇣
⌘
⇣ p1 p3 ⌘
⇣ p1 p3 ⌘
⇣
⌘
⇣
⌘
p1 ; p5
A) p16 ; p5e and
B)
;
and
;
1
3
1
3
1
5
1
5
e
6
7 ; ep
p
p ;p
C) p47 ; pee and
D) p8 ; pe and
e
e
4
8
C)
⇣
p1 ; p3
e
4
⌘
and
(81.5)3/4 sin °(31)
⇣
p1 ; p3
e
4
⌘
D)
⇣
p1 ; p5
83/4 e
(81.5)
⌘
and
⇣
sin °(31)
p1 ; p5
e
8
⌘
dy
dy
A)169 B)144 C)121 D)100
⇣ p ⌘
⇣ p ⌘
⇣ p y⌘= 5x3 cos
⇣ (x)p ⌘
45
13
58
13
19
19
A) 2 ; 2
B) 2 ; 2
C) 2 ; 2 p x =
D) 1 17
; 235
2
1
2
⇣
⌘ x =⇣ 10
⌘
⇣lim 4 ⌘1 + x10
⇣
⌘
1
5
1
5
1
3
1
3
2
p ; p
p ; p 3x
p ; p
A) p6Question
; pe and#
B) x!1
and
e
e
e
6
7
arcsin (5x) 7
e
⇣
⌘
⇣
⌘
⇣
⌘
⇣
⌘
1 p3
1 p3
15xp5 4 tan (x) p1 p5
p
p
p
C) 4Taking
; e and
; e,lnuse
and
; e 4.9
⇡ 1.609
ln 5 ⇡ 1.609
4.9D)
differentials
to5estimate
lim 8 ; e ln
4
8 ln
x!0
81.5)A)1.574
sin °(31)B)1.629
3/4
2 sin (x)
C)1.589
dy
x=
1
10
y = 5x3 cos (x)
earcsin (5x)
10x
D)1.614
12
12
y = 5x3 cos (x)
1
10
e
p
4 1 + x2
lim
x!1
3x2
arcsin (5x)
5x 4 tan (x)
x!0 2 sin (x)
10x
n 5 ⇡ 1.609 ln 4.9
Question #
lim
p
A)1.574 the B)1.629
C)1.589
D)1.614 the change in f as x changes from 8 to 9
= x . Approximate
Consider
function f (x)
A) 56
B) 16
C) 23
D) 15
12
12
D)100 p
(x) ⇣= xp ⌘
⌘ fExample
⇣ p ⌘
6:
3
C) 19
; 258
D) 17
; 235
2
2
x 3
⌘ lim 2⇣
⌘
⇣
⌘
x!3 x
9
5
1
3
1
3
p
p ; p
B) p7 ; pe and
e
e
7
x
10
1
⌘ lim ⇣
⌘
⇣
⌘
3
1
5
1
5
x!0
x
p
p ; p
D) p8 ; pe and
e
e
8
3 + 3x 3ex
lim
x!0 7x(ex
1)
✓ ✓ ◆◆2x
11
lim cos
x!1
x
✓
◆
5
5
im
!0
x ln3(1 + x)
y = 5x cos (x)
6
lim (ex + 1) /x
x!1
(5x)
earcsin
12
f (x) =
p
x
x 3
x!3 x2 7: 9
Example
lim
10x 1
lim
x!0
x
3 + 3x 3ex
lim
x!0 7x(ex
1)
✓
✓ ◆◆2x
11
lim cos
x!1
x
✓
◆
5
5
im
!0
x ln (1 + x)
6
lim (ex + 1) /x
x!1
12
x
9
10x 1
lim 8:
Example
x!0
x
3 + 3x 3ex
lim
x!0 7x(ex
1)
✓ ✓ ◆◆2x
11
lim cos
x!1
x
✓
◆
5
5
lim
x!0
x ln (1 + x)
6
lim (ex + 1) /x
x!1
12
3 + 3x 3ex
lim
x!0 7x(e
Example
9: x 1)
✓ ✓ ◆◆2x
11
lim cos
x!1
x
✓
◆
5
5
lim
x!0
x ln (1 + x)
6
lim (ex + 1) /x
x!1
12
✓
✓
◆◆2x
11
lim cos
Example
10: x
x!1
✓
◆
5
5
lim
x!0 x
ln (1 + x)
6
lim (ex + 1) /x
x!1
12
✓
5
5
im
x!0 Example
x ln11:
(1 + x)
6
lim (ex + 1) /x
x!1
12
◆
x!1
x 3
lim
Question
x!3 x2 #9
x
9
A)1.574
lim
5x
x!0
A) 56
5x 4 tan (x)
x!0 2 sin (x)
10x
lim
1B)1.629
B) 16
3x2
2
C)
x 3
C)1.589
D)1.614
D) 15
3 + 3x 3e
ln (9) x
ln (5)
x!0A) 7x(e
B)1)
C)ln (9)
5
9
✓ ✓ ◆◆2x
11
lim A)5cos B) 1 C)3 D)0
x!1
x
1
✓A) 8 B) 15 ◆C) 18 D)
5
5
lim
x!0
x ln (1 + x)
lim
D) ln 5(5)
5
8
p
lim (ex + 1) /x
4 1 + x2
lim
x!1
3x2
12
5x 4 tan (x)
x!0 2 sin (x)
10x
6
x!1
lim
A)1.574 B)1.629
Question #
p
2
A) 56 4 B)116 + xC)
2 3
lim
x!1
ln (9)
A)
5
C)1.589
D) 15
2 ln (5)
3x
B)
1
8
B)
1
5
D) ln 5(5)
C)ln (9)
9
5x 4 tan (x)
A)5 B) 1 C)3
lim
x!0 2 sin (x)
10x
A)
D)1.614
C) 18
D)0
D)
5
8
p
4 1 + x2
lim
x!1
3x2
5x 4 tan (x)
x!0 2 sin (x)
10x
lim
A)1.574
B)1.629 C)1.589 13D)1.614
p
4 1 + x2
A) 56limB) 16 C) 23 D) 15
x!1 # 3x2
Question
ln (5)
ln (5)
A) ln 5(9)
B)
C)ln
(9)
D)
9
5
5x 4 tan (x)
lim
x!0 2 sin (x)
10x D)0
A)5 B) 1 C)3
A)
1
8
B)
1
5
C) 18
D)
5
8
13