18.01 Section, September 28, 2015
Section leader: Eva Belmont (ebelmont@mit.edu, E18-401B)
.
Related rates and implicit differentiation
Bonus question: Consider f (x) = 1 − 12 (x − 1)2 . This has a fixed point of x = 1. Find
an interval [a, b] containing 1 such that any number in that interval is a “good guess” for
the fixed point iteration method (i.e. f (. . . (f (f (x0 )))) ≈ 1 for x0 in the interval).
Hint: Use the idea of the proof of the criterion for attracting fixed points.
1. (Done by Eva) Water is draining out of the bottom of a cone at a rate of 0.1 m3 /s. The
angle of the cone is 90◦ . At t = 0, the water goes up to a height of 2 m. When the water
has drained to a height of 1 m, what is the rate of change of the radius of the surface of the
water?
2. (Done by Eva) Maximize x + y subject to the constraint x2 + y 2 = 1.
3. A spherical balloon is being inflated at a rate of 8 ft3 /min. When the balloon has a radius
of 1 ft., what is the rate of change of its radius?
1
4. Consider the curve defined by x2 + y 3 + y = 1. Find the (local) maximum in the picture
below.
Hint: In the parlance of the problem I did on the board, you are maximizing y.
5. Suppose ln f (x) = x2 . Find f 0 (x) in terms of f (x).
6. Sketch the graph of y = x sin x on the interval [−100, 100]. (I’m just asking for the general
shape.)
2
7. Suppose f is some function such that |f (x) − f (y)| ≤ (x − y)2 . Show that f is constant.
Review
• Sum formulas:
sin(A + B) = sin A cos B + cos A sin B
cos(A + B) = cos A cos B − sin A sin B
• Exponential decay formula:
y = y0 e−kt
where
half life =
•
•
d x
d x
x
dx e = e and dx b
1
d
dx ln x = x
ln 2
.
k
= (ln b) · bx
• How to solve related rates problems:
1. Write an equation relating the quantities (areas, volumes, lengths,. . . ) you care about:
e.g. if the problem is about area and radius of a circle, the equation is A = πr2 .
2. Do
d
dt
of step (1): e.g.
dA
dt
= π · 2r dr
dt .
3. Plug in given rates and solve for unknown rates.
• How to do implicit differentiation: given f (x, y) = 0, do
dy
d 2
is a function of x. E.g. dx
y = 2y dx
(chain rule!).
3
d
dx
to everything, remembering that y