Name:
Student number:
Math 110-003
Quiz 1 (Jan 21)
Write full solutions to the questions below. It is to your own benefit to write as clearly and
legibly as possible.
1. A particle moves around the circle
x2 + y 2 = 25.
As it reaches the point (3, 4), its x-component is increasing at a rate of 3mm/s.
(a) Use implicit differentiation to find
Solution.
dy
.
dx
d 2
d
dy
(x + y 2 ) =
25 ⇒ 2x + 2y
=0
dx
dx
dx
dy
⇒ 2y
= −2x
dx
dy
−2x
x
⇒
=
=− .
dx
2y
y
(b) Find the rate of change of the x-component as the particle reaches (3, 4).
There was a typo in this question so it will not be graded. Instead of “x-component”,
the question should ask about the y-component. The solution below finds the rate
of change of the y-component.
Solution.
By the chain rule and the result of (a),
dy
dy dx
x dx
=
=−
.
dt
dx dt
y dt
When the particle reaches (3, 4), we have −
(1)
x
3
dx
= − and
= 3, so
y
4
dt
dy
3
9
= − (3) = − .
dt
4
4
(2)
9
mm/s.
4
(c) What direction is the particle moving in at this moment? (clockwise or counterclockwise)
Solution.
Since the circle is centered at the origin and the x-component is increasing while
the y-component decreases, the particle is moving clockwise.
Note: it is not necessary to complete (b) in order to see this. One can alternately
argue that since the circle is centered at the origin, the particle is moving clockwise
since its x-component is increasing while it is located in the upper-right quadrant.
Thus, the y-component decreases at
Name:
Student number:
Math 110-003
Quiz 1 (Jan 21)
2. The surface area A of a spherical snowball of radius r is given by A = 4πr2 . The
snowball melts so that its area decreases at 2cm2 /min.
(a) Use the chain rule to find a relationship between
Solution.
By the chain rule,
dA dr
dA
=
.
dt
dr dt
dA
dt
and
dr
.
dt
(3)
(b) How fast is the radius decreasing when the radius is 1cm?
Solution.
dA
By (a) and the fact that
= −2,
dt
−2 =
dA dr
dA
=
.
dt
dr dt
(4)
Since A = 4πr2 ,
dA
= 8πr.
dr
(5)
dr
dt
(6)
Thus,
−2 = 8πr
and it follows that
−2
1
dr
=
=−
.
dt
8πr
4πr
(7)
dr
1
1
=−
=− .
dt
4π(1)
4π
(8)
When r = 1, we get
Thus, the radius is decreasing at
1
cm/min.
4π