Name:
Section:
Math 10550, Quiz 6
October 28, 2014
•
•
•
•
The Honor Code is in e↵ect for this quiz. All work is to be your own.
Please turn o↵ all cellphones and electronic devices.
Calculators are NOT allowed
The quiz lasts for 10 min.
PLEASE MARK YOUR ANSWERS WITH AN X, not a circle!
1. (a)
(b)
(c)
(d)
(e)
2. (a)
(b)
(c)
(d)
(e)
........................................................................................................................
Name:
Section:
Multiple Choice
1.(2 pts.) A beetle is moving along a straight line, with position given by s(t) = sin(t) +
cos(t). How much distance does it travel from t = ⇡ to t = 2⇡?
Solution:
We need to find the points where the beetle might change direction, and look at its
position at those times. The direction of travel is governed by the sign of the velocity,
i.e. the sign of s0 (t). Calculate s0 (t):
s0 (t) = cos(t)
sin(t).
Since s0 (t) is continuous, s0 (t) can only change sign by crossing through zero. Therefore,
we need to find the points where s0 (t) = 0, i.e., the points where sin(t) = cos(t). Thinking
geometrically, this occurs on a right triangle exactly when the lengths of the legs (nonhypotenuse sides) are equal. In the interval (⇡, 2⇡), we have this equality exactly at an
5⇡
angle of
.
4
We therefore compute the total distance travelled as follows:
total distance
✓ ◆
5⇡
s
s(⇡)
4
s (2⇡)
s
✓
5⇡
4
◆
total distance
✓
◆
✓ ◆
5⇡
5⇡
= s
s(⇡) + s (2⇡) s
4
4
✓ ◆
✓ ◆
5⇡
5⇡
= sin
+ cos
sin(⇡) cos(⇡)
4
4
1
1
p
= p
0 ( 1)
2
2
p
=
2+1
p
= 2 1
✓ ◆
✓ ◆
5⇡
5⇡
= sin (2⇡) + cos (2⇡) sin
cos
4
4
1
1
= 0+1+ p + p
2
2
p
= 1+ 2
p
=1+ 2
p
p
= 2 1+1+ 2
p
=2 2
2
Name:
Section:
(a)
2
(d)
p
2 2 2
p
2 2
p
1+ 2
(e)
None of the above.
(b)
(c)
2.(2 pts.) Find the linearization L(x) of the function f (x) = sec(x) at
(a)
(b)
(c)
p ⇣
L(x) = 2 + 2 3 x
⇡
.
3
⇡⌘
3
⇡⌘
2
2⇣
L(x) = p +
x
3
3 3
⇣
⌘
⇡
L(x) = 2 4 x
3
⇣⇡
(d)
L(x) = sec(x) + sec(x) tan(x)
(e)
Does not exist; sec(x) is not di↵erentiable at
3
x
⌘
⇡
.
3
Solution:
⇡
We need to find the value and derivative of f at , then use the formula
3
⇣⇡ ⌘
⇣⇡ ⌘ ⇣
⇡⌘
L(x) = f
+ f0
x
.
3
3
3
Observe:
f (x) = sec(x)
1
=
cos(x)
1
f 0 (x) =
( sin(x))
2
cos (x)
sin(x)
=
.
cos2 (x)
3
Name:
Section:
p
⇡
3
⇡
1
Recall that sin( ) =
and cos( ) = . Substituting, we obtain:
3
2
3
2
⇣⇡ ⌘
1
f
=
1
3
2
=2
p
3
⇣⇡ ⌘
f0
= ✓ 2◆2
3
1
2
p
= 2 3.
Finally, we obtain that
p ⇣
L(x) = 2 + 2 3 x
4
⇡⌘
.
3
Name:
Section: ANSWERS
Math 10550, Quiz 6
October 28, 2014
•
•
•
•
The Honor Code is in e↵ect for this quiz. All work is to be your own.
Please turn o↵ all cellphones and electronic devices.
Calculators are NOT allowed
The quiz lasts for 10 min.
PLEASE MARK YOUR ANSWERS WITH AN X, not a circle!
1. (a)
(b)
(•)
(d)
(e)
2. (•)
(b)
(c)
(d)
(e)
........................................................................................................................