18.01 Section, September 23, 2015
Section leader: Eva Belmont (ebelmont@mit.edu, E18-401B)
.
More trig, polar coordinates, and chain rule
1. I am walking down a straight road; my position (distance from the starting point) as a
t
function of time is x(t) = 1+t
. Air temperature is a function of position: A(x) = x2 .
(a) What is my speed at t = 1?
(b) Calculate
dA
dt
at t = 1.
(c) Now change the problem so that x(t) and A(x) are no longer specified. Under what
circumstances (i.e., for what x(t) and A(x)) do I experience a constant temperature
over time?
2. Take the line y = 1 in the plane and rotate the plane clockwise around (0, 0) by
an equation in polar coordinates for the resulting line.
π
3.
Write
3. (Harder way of doing the above problem)
(a) The pre-rotated line contains the point (0, 1). What happens to that point after rotation?
(b) The rotated line crosses the x-axis. What is this intersection point?
(c) Using the points found in parts (a) and (b), write the equation for the line through
those points.
1
(d) Check that this agrees with your answer in problem 2.
4. Twenty milliliters of paint exactly covers the surface of a closed cube of side length 1 ft.
Assume thickness of paint is constant.
(a) If I have a cubical box of side length 1.1 ft, use linear approximation to estimate how
much paint is needed.
(b) If I have 21 milliliters of paint, use linear approximation to estimate the cube size this
can cover.
5. Suppose f (0) = 0. What can I say about limh→0
f (h)
h ?
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
• Chain rule:
d
f (g(x)) = f 0 (g(x)) · g 0 (x)
dx
• Converting rectangular coordinates to polar coordinates:
x = r cos θ
y = r sin θ
• Converting polar coordinates to rectangular coordinates:
p
r = x2 + y 2
θ = tan−1
2
y
x