All Chained Up
P's and Q's
Think Critically
Auntie Derivative
Area
100
It's y' if y = (2x3+1)10
What is y' = 60x2(2x3+1)9
100
It's the product rule to find y' of y = f times g
What is f' g + g' f?
100
We look at when the first derivative of f(x) is THIS to find when f(x) has a critical number.
What is when the first derivative zero or undefined?
100
∫
9dx
What is 9x + C
100
It's the shaded area.
What is 24?
200
It's y' if
y=
e
4x
What is 4e^(4x)?
200
It's the quotient rule to find y ' if
y=
hi
lo
What is
y'=
loDeehi−hiDeelo
lolo
200
We look at THIS to find when f(x) has a point of inflection.
What is where f '' (x) = 0 or undefined.
200
∫
20xdx
What is 10x^2 + C?
200
It's the shaded area between the x-axis and the parabola.
y=20−
x
2
20
What is 108.3?
300
It's y ' if
y=ln
x
229
What is
y'=
229
x
300
It's y' if
y=
x
4
lnx
What is
y'=
x
3
+4
x
3
lnx
300
It is x for which f(x) climbs of falls the steepest.
What is the x-coordinate of the point of inflection of f(x).
It is the value of x for which f ' (x) has a max or a min.
It is the value of x for which f ' (x) has a max or a min.
300
It's
∫
(20
x
9
+11)dx
What is 2x10 +11x + C
300
It's the area bounded by x = 1, x = -1, and the two parabolas.
What is 12?
∫ −1 1 ((3 x 2 +6)−(−3 x 2 +2) )dx = ∫ −1 1 (6 x 2 +4 )dx = 2 x 3 +4x| −1 1 =6−(−6)=12 or =2 ∫ 0 1 (6 x 2 +4 )dx=2⋅6=12
∫ −1 1 ((3 x 2 +6)−(−3 x 2 +2) )dx = ∫ −1 1 (6 x 2 +4 )dx = 2 x 3 +4x| −1 1 =6−(−6)=12 or =2 ∫ 0 1 (6 x 2 +4 )dx=2⋅6=12
400
It's y ' if
y=ln(229x)
y'=
1
x
y=ln(229x) =ln(229) + ln(x) y'=0+ 1 x
y=ln(229x) =ln(229) + ln(x) y'=0+ 1 x
400
It's the numerator of the derivative of
y=
x
2
x
2
+3
when simplified.
What is 6x?
y= x 2 ( x 2 +3) y'= ( x 2 +3)2x−x (2x) 2 ( x 2 +1) 2 = 2 x 3 +6x−2 x 3 ( x 2 +1) 2 = 6x ( x 2 +1) 2
y= x 2 ( x 2 +3) y'= ( x 2 +3)2x−x (2x) 2 ( x 2 +1) 2 = 2 x 3 +6x−2 x 3 ( x 2 +1) 2 = 6x ( x 2 +1) 2
400
It is the critical value of y = xekx.
What is 1/k?
400
It's
d
dx
(
∫
3
x
dx
)
What is 3x?
400
It's the larger of the shaded areas bounded by the two curves.
What is 33.75?
A 1 = ∫ 0 3 (3 x 3 −18 x 2 +36x−3 x 2 )dx = ∫ 0 3 (3 x 3 −21 x 2 +36x )dx =33.75
A 1 = ∫ 0 3 (3 x 3 −18 x 2 +36x−3 x 2 )dx = ∫ 0 3 (3 x 3 −21 x 2 +36x )dx =33.75
500
It's y' if
y=
(lnx)
230
What is
y'=
230
(lnx)
229
x
500
It is the linear factor multiplied by e^(kx) of y ' if
y=x
e
kx
What is (kx + 1)?
y'=kx e kx + e kx =(kx+1) e kx
y'=kx e kx + e kx =(kx+1) e kx
500
It is the value of x of the point of inflection of y = xe^(kx).
What is x = 2/k?
500
It's
∫
1
1+4
x
2
(8
x
3
)dx
What is ln (1+4x2) + C
500
It's the total of the shaded area bounded by the two curves.
What is 35.5?
A 1 = ∫ 0 3 (3 x 3 −18 x 2 +36x−3 x 2 )dx = ∫ 0 3 (3 x 3 −21 x 2 +36x )dx =33.75
A 2 = ∫ 3 4 (3 x 2 −(3 x 3 −18 x 2 +36x ))dx = ∫ 3 4 (−3 x 3 +21 x 2 −36x )dx =1.75
A 1 + A 2 =33.75+1.75=35.5
A 1 = ∫ 0 3 (3 x 3 −18 x 2 +36x−3 x 2 )dx = ∫ 0 3 (3 x 3 −21 x 2 +36x )dx =33.75
A 2 = ∫ 3 4 (3 x 2 −(3 x 3 −18 x 2 +36x ))dx = ∫ 3 4 (−3 x 3 +21 x 2 −36x )dx =1.75
A 1 + A 2 =33.75+1.75=35.5