PECHAKUCHA REVIEW FOR FIRST MIDTERMpeople.math.harvard.edu/~knill/teaching/math1a... · PECHAKUCHA...
20
PECHAKUCHA REVIEW FOR FIRST MIDTERM Oliver Knill, March 6, 2013, Pecha Kucha Day 20 20 x Wednesday, March 6, 13
Transcript of PECHAKUCHA REVIEW FOR FIRST MIDTERMpeople.math.harvard.edu/~knill/teaching/math1a... · PECHAKUCHA...
PECHAKUCHA REVIEWFOR FIRST MIDTERM
Oliver Knill, March 6, 2013, Pecha Kucha Day
20 20xWednesday, March 6, 13
Functions
y=f(x)
x
Wednesday, March 6, 13
Limits
limx a
f(x)=b
a
Wednesday, March 6, 13
Continuity
-0.3 -0.2 -0.1 0.1 0.2 0.3
-1.0
-0.5
0.5
1.0
-3 -2 -1 1 2 3
0.5
1.0
1.5
2.0
-3 -2 -1 1 2 3
-5
5
-0.3 -0.2 -0.1 0.1 0.2 0.3
-0.20
-0.15
-0.10
-0.05
0.05
0.10
Wednesday, March 6, 13
Derivative
lim h 0
f(x+h)-f(x) h
f '(x) =
ddx
xn = x n-1n
ddx
exp(x) = exp(x)
Wednesday, March 6, 13
Roots
-1.5 -1.0 -0.5 0.5 1.0 1.5
-1.5
-1.0
-0.5
0.5
1.0
1.5
-1.5 -1.0 -0.5 0.5 1.0 1.5
-3
-2
-1
f(x)=x -x3 f(x)=log|x|
Wednesday, March 6, 13
Critical points
-4 -2 2 4
-2
-1
1
2
Wednesday, March 6, 13
Concavity
-1.0 -0.5 0.5 1.0
-1.20
-1.15
-1.10
-1.05
-1.00
Wednesday, March 6, 13
Hopital
lim h a
f(x)g(x)
00
= lim h a
f '(x)g '(x)
0000
lim h 0
sin(x)x
lim h 0
cos(x)1= =1
lim h 0
log(x)x
lim h 0
1/x-1/x= =02
Wednesday, March 6, 13
Product rule
(f g)'(x) = f'(x) g(x) + f(x) g'(x)
Wednesday, March 6, 13
Quotient rule
(f/ g)'(x) = f'(x) g(x) - f(x) g'(x) g(x)2
Wednesday, March 6, 13
Chain rule
f(g(x))' = f'(g(x)) g'(x)
Wednesday, March 6, 13
Intermediate value theorem
-1.5 -1.0 -0.5 0.5 1.0 1.5
-0.6
-0.4
-0.2
0.2
0.4
0.6
Wednesday, March 6, 13
Newton Method
-2 -1 1 2
-15
-10
-5
5
10
15
T(x) = x-f(x)/f'(x)
Wednesday, March 6, 13
Second Derivative test
-2 -1 1 2
-2
-1
1
2- -
+ +
- -
++00
Wednesday, March 6, 13
Global Extrema
-2 -1 1 2
-2
-1
1
2
Wednesday, March 6, 13
Deriving the Inverse
sin(arccos(x)) arccos'(x) = 1
cos( arccos(x)) = x
arccos'(x) = 1/√1-cos (arccos(x))2
Wednesday, March 6, 13
Logs
log(a b) = log(a) + log(b)
log(a ) = b log(a) b
log(a/b) = log(a) - log(b)
log(1) =0 exp(log(x)) =x
Wednesday, March 6, 13
Trig
cos (x) + sin (x) = 1 2 2
60
30
45
45
1 1
1/2
√2
1
√3/2
90
90
Wednesday, March 6, 13
Exp
exp(a +b) =exp(a) * exp(b)
ab = exp(b log(a))
ab =c
a bc
Wednesday, March 6, 13
