18.01 Section, September 30, 2015
Section leader: Eva Belmont (ebelmont@mit.edu, E18-401B)
Midterm 1 review
Miscellaneous trig facts and polar coordinates
• Trig functions of common angles
√
3
π
cos =
6
2
π
1
cos =
3
2
π
1
sin =
6
2√
3
π
sin =
3
2
π
=1
4
√
2
π
sin =
4
2
tan
• Sum formulas:
sin(A + B) = sin A cos B + cos A sin B
cos(A + B) = cos A cos B − sin A sin B
• Double angle formulæ:
sin 2θ = 2 sin θ cos θ
cos 2θ = cos2 θ − sin2 θ
= 1 − 2 sin2 θ = 2 cos2 θ − 1
2 tan θ
tan 2θ =
1 − tan2 θ
• Half-angle formulæ:
r
sin( 12 θ) = ±
r
1 − cos θ
2
r
cos( 21 θ) = ±
1 + cos θ
2
1 − cos θ
1 − cos θ
sin θ
=
=
1 + cos θ
sin θ
1 + cos θ
• Converting rectangular coordinates to polar coordinates:
tan( 12 θ) = ±
x = r cos θ
y = r sin θ
• Converting polar coordinates to rectangular coordinates:
p
r = x2 + y 2
(try problem 1.)
θ = tan−1
• Converting an equation in rectangular coordinates to polar (and vice versa)
y
x
(try problem 2.)
• To rotate a curve an angle of θ0 counterclockwise, convert to polar and replace θ with θ − θ0 .
(review the HW problem about rotating x = 1)
Exponentials
• Exponential decay formula:
y = y0 e−kt
where
half life =
• e = limn→∞ 1 +
ln 2
.
k
1 n
n
How to differentiate all the things
• Product rule:
d
dx f (x)g(x)
= f 0 (x)g(x) + f (x)g 0 (x)
d
• Chain rule: dx
f (g(x)) = f 0 (g(x)) · g 0 (x)
◦ Chain rule in Leibniz notation: e.g.
d
dt R(x(t))
=
1
dR dx
dx dt
(try problem 3.)
• Quotient rule:
d f (x)
dx g(x)
=
f 0 (x)g(x)−g 0 (x)f (x)
g(x)2
• Derivatives of trig functions:
d
sin x = cos x
dx
d
tan x = sec2 x
dx
d
1
sin−1 x = √
dx
1 − x2
•
•
d x
d x
x
dx e = e and dx b
1
d
dx ln x = x
(try problem 4.)
d
cos x = − sin x
dx
d
cot x = − csc2 x
dx
d
−1
cos−1 x = √
dx
1 − x2
d
1
tan−1 x =
dx
1 + x2
= (ln b) · bx
• Use the chain rule to differentiate inverse functions
(try problem 5.)
Applications of derivatives
• Tangent line to a graph at a point a: T (x) = f (a) + f 0 (a)(x − a)
• Linear approximation at a point a: f (x) ≈ f (a) + f 0 (a)(x − a)
(try problem 6.)
f 0 (x)
• Critical points: when
=0
◦ To find max/min of f (x) in a range, solve f 0 (x) = 0 and check if the solutions are max or min (or neither).
Also check endpoints!
(try problem 7.)
• Newton’s method for approximating roots
√ of g(x)
Use Newton’s method to approximate 2. (This was done in lecture 2.)
−g(xn )
+ xn
g 0 (xn )
◦ This is a special case of fixed point iteration for the function f (x) = −g(x)
g 0 (x) + x
x0 = initial guess
xn+1 =
• Fixed point iteration method for approximating fixed points of f (x)
x0 = initial guess
x1 = f (x0 )
x2 = f (x1 )
x3 = f (x2 )
◦ If α is a fixed point and |f 0 (α)| < 1, then α is an attracting fixed point.
I Means that, for close enough guess x0 , the xn ’s converge to the fixed point.
I
...
Doesn’t mean all guesses work. (But if f is a line, then all guesses work.)
◦ If α is a fixed point and |f 0 (α)| > 1, then α is a repelling fixed point.
◦ If |f 0 (α)| = 1, test is inconclusive.
• Implicit differentiation
◦ Given f (x, y) = 0, be able to find
dy
dx .
(try problem 8.)
◦ Be able to maximize/minimize g(x, y) subject to a constraint f (x, y) = 0.
(try problem 9.)
• Related rates
Steps:
(1) Name all the quantities you care about.
(try problem 10.)
(2) Write an equation relating said quantities.
(3) Apply
d
dt
to step (2), remembering the chain rule.
(4) Plug in rates you are told and solve for rates you’re trying to find.
2
Problems
√
1. Convert (
√
2
2
,
2
2 )
to polar coordinates. Convert (θ = π3 , r = 2) to rectangular coordinates.
2. Convert x2 − y 2 = 1 to polar coordinates.
3. A particle is travelling counterclockwise in a unit circle at a rate of 1 radian per second. What is the rate
of change of the particle’s x-coordinate at t = π2 ?
4. Derive the quotient rule using the product rule and the chain rule.
5. Derive the formula for
d
dx
sin−1 (x). (Remember that this is the inverse function to sin x, not
3
1
sin x .)
6. To what accuracy would one need to measure the circumference of a circle of radius 2 cm to get an estimate
(using linear approximation) of its area good to within 0.2π cm2 ?
7. Find the point on y 2 = 2x closest to (1, 4).
What quantity are we minimizing here?
8. Find the tangent line to
x2
9
+
9. Maximize 3y + 2x subject to
y2
4
x2
9
√
= 1 at the point ( 3 2 3 , 1).
+
y2
4
= 1.
10. A 5-foot ladder is initially propped up in the picture, and then starts slipping such that the height of the
top of the ladder is decreasing at a constant rate of 12 ft/s. When it has slipped 1 foot (i.e. the top of the
ladder is one foot lower than it started), how fast is the distance between the corner and the bottom of
the ladder increasing?
4