activity: teacher-directed instruction
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Activity: Teacher-Directed Instruction. 2013 Implicit Differentiation. Calculus AB. Objective. C: The swbat differentiate implicitly equations in more than one variable. L: the swbat explain to others how to find derivatives of multiple types of problems verbally and demonstratively. - PowerPoint PPT PresentationTRANSCRIPT
C CONVERSATION: Voice level 0. No talking!
HHELP: Raise your hand and wait to be called on.
AACTIVITY: Whole class instruction; students in seats.
M MOVEMENT: Remain in seat during instruction.
P PARTICIPATION: Look at teacher or materials being discussed. Raise hand to contribute; respond to questions, write or perform other actions as directed.NO SLEEPING OR PUTTING HEAD DOWN, TEXTING, DOING OTHER WORK.
S
Activity: Teacher-Directed Instruction
Calculus AB
2013 Implicit Differentiation
Objective
โข C: The swbat differentiate implicitly equations in more than one variable.
โข L: the sw explain to others how to find derivatives of multiple types of problems verbally and demonstratively
Implicit Differentiation
Equation for a line:
Explicit Form
<One variable given explicitly in terms of the other>
Implicit Form
<Function implied by the equation>
Differentiate the Explicit
< Explicit: , y is function of x >
Differentiation taking place with respect to x. The derivative is explicit also.
y mx b
Ax By C
24 3 4y x x
8 3dy xdx
Implicit Differentiation
Equation of circle:
To work explicitly; must work two equations
2 2 9y x
29y x
Implicit Differentiation is a Short Cut - A method to handle equations that are not easily written explicitly.
( Usually non-functions)
29y x ๐๐ฆ๐๐ฅ =1
2(9โ๐ฅ2 )
โ12 (โ2 ๐ฅ )
๐๐ฆ๐๐ฅ =
โ๐ฅโ9โ๐ฅ2
๐๐ฆ๐๐ฅ=โ 1
2( 9โ ๐ฅ2 )
โ 12 (โ2๐ฅ )
๐๐ฆ๐๐ฅ =
๐ฅโ9โ๐ฅ2
๐ฅ2๐ฆ+2 ๐ฆ2๐ฅ+3 ๐ฆ3=7Donโt want to solve for y
Implicit Differentiation
Chain Rule Pretend y is some function like
so becomes
(A)
(B)
(C)
Note: Use the Leibniz form. Leads to Parametric and Related Rates.
2 2 3y x x 2 4( 2 3)x x 4y
Find the derivative with respect to x
< Assuming - y is a differentiable function of x >
32y
4y
2 3x y
=
=
2 ๐๐ฅ๐๐ฅ +3 ๐๐ฆ๐๐ฅ=2+3 ๐๐ฆ๐๐ฅ
4 ๐ฆ 3(2๐ฅ+2)
6 ๐ฆ2(2 ๐ฅ+2)
Implicit Differentiation
Find the derivative with respect to x
< Assuming - y is a differentiable function of x >
๐ฅ๐ฆ=ยฟ ๐ฅ ๐๐ฆ๐๐ฅ +๐ฆ ๐๐ฅ๐๐ฅ ยฟ ๐ฅ ๐๐ฆ๐๐ฅ + ๐ฆ
๐ฅ2+ ๐ฆ2=ยฟ2 ๐ฅ ๐๐ฅ๐๐ฅ +2 ๐ฆ ๐๐ฆ๐๐ฅ ยฟ2 ๐ฅ+2 ๐ฆ ๐๐ฆ๐๐ฅ
sin (๐ฅ๐ฆ) ยฟcos (๐ฅ๐ฆ )โ(๐ฅ ๐๐ฆ๐๐ฅ +๐ฆ ๐๐ฅ๐๐ฅ )ยฟ ๐ฅcos (๐ฅ๐ฆ ) ๐๐ฆ๐๐ฅ + ๐ฆ cos (๐ฅ๐ฆ)
Implicit Differentiation
(D) Product Rule
2xy ยฟ ๐ฅ (2 ๐ฆ ) ๐๐ฆ๐๐ฅ + ๐ฆ2 ๐๐ฅ๐๐ฅ
ยฟ2 ๐ฅ๐ฆ ๐๐ฆ๐๐ฅ +๐ฆ2
ยฟ ๐ฅ ๐๐ฆ๐๐ฅ+ ๐ฆโ ๐๐ฅ๐๐ฅ
ยฟ ๐ฅ ๐๐ฆ๐๐ฅ +๐ฆโ
๐ฅ๐ฆ
Implicit Differentiation
(E) Chain Rule 3( )xy
Product inside a chain
3 (๐ฅ๐ฆ )2(๐ฅ ๐๐ฆ๐๐ฅ +๐ฆ ๐๐ฅ๐๐ฅ )
3 ๐ฅ3 ๐ฆ2 ๐๐ฆ๐๐ฅ+3 ๐ฅ2 ๐ฆ3
ยฟ3 ๐ฅ2 ๐ฆ2(๐ฅ ๐๐ฆ๐๐ฅ + ๐ฆ ๐๐ฅ๐๐ฅ )
sin (๐ฅ๐ฆ)
cos (๐ฅ๐ฆ )โ(๐ฅ ๐๐ฆ๐๐ฅ + ๐ฆ ๐๐ฅ๐๐ฅ )
๐ฅcos (๐ฅ๐ฆ ) ๐๐ฆ๐๐ฅ + ๐ฆ cos(๐ฅ๐ฆ )
Implicit Differentiation
(E) Chain Rule
Product inside a chain
๐ข=๐ฅ๐ฆ ๐๐ข=๐ฅ ๐๐ฆ๐๐ฅ +๐ฆ ๐๐ฅ๐๐ฅ
sin๐ข (๐๐ข)
cos๐ข(๐ฅ ๐๐ฆ๐๐ฅ +๐ฆ ๐๐ฅ๐๐ฅ )
cos ๐ฅ๐ฆ (๐ฅ ๐๐ฆ๐๐ฅ )+cos ๐ฅ๐ฆ (๐ฆ ๐๐ฅ๐๐ฅ )
Implicit Differentiation
To find implicitly.
EX: Diff Both Sides of equation with respect to x
Solve for
dydx
2 2 9x y dydx2 ๐ฅ ๐๐ฅ๐๐ฅ+2 ๐ฆ ๐๐ฆ๐๐ฅ=0
2 ๐ฅ+2 ๐ฆ ๐๐ฆ๐๐ฅ=0
๐๐ฆ๐๐ฅ=
โ2๐ฅ2 ๐ฆ =
โ๐ฅ๐ฆ
29y x
29y x
Need both x and y to find the slope.
C CONVERSATION: Voice level 0. No talking!
HHELP: Raise your hand and wait to be called on.
AACTIVITY: Whole class instruction; students in seats.
M MOVEMENT: Remain in seat during instruction.
P PARTICIPATION: Look at teacher or materials being discussed. Raise hand to contribute; respond to questions, write or perform other actions as directed.NO SLEEPING OR PUTTING HEAD DOWN, TEXTING, DOING OTHER WORK.
S
Activity: Teacher-Directed Instruction
Objective
โข C: The swbat differentiate implicitly equations in more than one variable.
โข L: the sw explain to others how to find derivatives of multiple types of problems verbally and demonstratively
EX 1:3 2 25 4y y y x
(a) Find the derivative at the point ( 5, 3 ) , at ( -1,-3 )
(b) Find where the curve has a horizontal tangent.
(c) Find where the curve has vertical tangents.
3 ๐ฆ2 ๐ ๐ฆ๐๐ฅ +2 ๐ฆ ๐๐ฆ๐๐ฅ โ5 ๐๐ฆ๐๐ฅ โ2๐ฅ ๐๐ฅ๐๐ฅ=0
(3 ๐ฆยฟยฟ2+2 ๐ฆโ5)๐๐ฆ๐๐ฅ=2 ๐ฅยฟ
๐๐ฆ๐๐ฅ =
2๐ฅ3 ๐ฆ2+2 ๐ฆโ5
๐๐ฆ๐๐ฅ |ยฟ (5,3)=10
28๐๐ฆ๐๐ฅ |ยฟ (โ1 ,โ3)=โ2
16
EX 1:3 2 25 4y y y x
(b) Find where the curve has a horizontal tangent. Horizontal tangent has a 0 slope
๐๐=0โด๐=0
2 ๐ฅ=0๐ฅ=0
EX 1:3 2 25 4y y y x
(c) Find where the curve has vertical tangents. Vertical tangent has an undefined slope๐
๐๐ข๐๐๐๐ ๐=0
3 ๐ฆ2+2 ๐ฆโ5=0(3 ๐ฆ+5)(๐ฆโ1)
3 ๐ฆ+5=0๐ฆ=
โ53
๐ฆโ1=0๐ฆ=1
Ex 2:
3 3 2x y xy
< Folium of Descartes >
3 ๐ฅ2 ๐๐ฅ๐๐ฅ +3 ๐ฆ2 ๐๐ฆ
๐๐ฅ=2(๐ฅ ๐๐ฆ๐๐ฅ +๐ฆ ๐๐ฅ๐๐ฅ )3 ๐ฅ2+3 ๐ฆ2 ๐๐ฆ
๐๐ฅ=2๐ฅ ๐๐ฆ๐๐ฅ +2 ๐ฆ
3 ๐ฆ2 ๐๐ฆ๐๐ฅ โ2 ๐ฅ ๐๐ฆ๐๐ฅ=2 ๐ฆโ3 ๐ฅ2
๐๐ฆ๐๐ฅ ( (3 ๐ฆ
2โ2 ๐ฅ)(3 ๐ฆ 2โ2 ๐ฅ))= 2 ๐ฆโ3๐ฅ2
(3 ๐ฆ2โ2๐ฅ )
๐๐ฆ๐๐ฅ =
2 ๐ฆโ3๐ฅ2
3 ๐ฆ2โ2๐ฅ
3 ๐ฅ2โ2 ๐ฆ=2 ๐ฅ ๐๐ฆ๐๐ฅ โ3 ๐ฆ2 ๐๐ฆ๐๐ฅ
3 ๐ฅ2โ2 ๐ฆ=(2 ๐ฅโ3 ๐ฆ2)๐๐ฆ๐๐ฅ
3 ๐ฅ2โ2 ๐ฆ(2 ๐ฅโ3 ๐ฆ2)
=๐๐ฆ๐๐ฅ
Why Implicit?
3 3 2x y xy
< Folium of Descartes > Explicit Form:
3 6 3 3 6 33 31
1 1 1 18 82 4 2 4
y x x x x x x
3 6 3 3 6 33 32 1
1 1 1 1 13 8 82 2 4 2 4
y y x x x x x x
3 6 3 3 6 33 33 1
1 1 1 1 13 8 82 2 4 2 4
y y x x x x x x
2nd Derivatives
NOTICE:The second derivative is in terms of x , y , AND dy /dx.
The final step will be to substitute back the value of dy / dx into the second derivative.
EX: Our friendly circle. Find the 2nd Derivative.2 2 9x y
2 ๐ฅ ๐๐ฅ๐๐ฅ +2 ๐ฆ ๐๐ฆ๐๐ฅ=0
2 ๐ฅ+2 ๐ฆ ๐๐ฆ๐๐ฅ=0
2 ๐ฆ ๐๐ฆ๐๐ฅ =โ2 ๐ฅ
๐๐ฆ๐๐ฅ=
โ2๐ฅ2 ๐ฆ
๐๐ฆ๐๐ฅ=
โ๐ฅ๐ฆ
๐2 ๐ฆ๐๐ฅ2 =ยฟ
๐ฆ (โ1 ๐๐ฅ๐๐ฅ )โ(โ ๐ฅ) ๐๐ฆ๐๐ฅ๐ฆ2
๐2 ๐ฆ๐๐ฅ2 =
โ๐ฆ+๐ฅ ๐๐ฆ๐๐ฅ๐ฆ 2
๐2 ๐ฆ๐๐ฅ2 =
โ๐ฆ+๐ฅ (โ๐ฅ๐ฆ )๐ฆ2 ( ๐ฆ๐ฆ )
โ๐ฆ 2โ๐ฅ2
๐ฆ3๐2 ๐ฆ๐๐ฅ2 =ยฟ
2nd DerivativesEX: Find the 2nd Derivative.
23 5xy
0โ(๐ฅ 2 ๐ฆ ๐๐ฆ๐๐ฅ + ๐ฆ2 ๐๐ฅ๐๐ฅ )=0
โ2 ๐ฅ๐ฆ ๐๐ฆ๐๐ฅ โ ๐ฆ2=0
โ2๐ฅ ๐๐ฆ๐๐ฅ โ ๐ฆ (โ2 ๐๐ฅ๐๐ฅ )(โ2๐ฅ)2
๐2 ๐ฆ๐๐ฅ2 =ยฟ
๐2 ๐ฆ๐๐ฅ2 =ยฟ
๐2 ๐ฆ๐๐ฅ2 =ยฟ
โ2๐ฅ ๐๐ฆ๐๐ฅ +2 ๐ฆ
4 ๐ฅ2
โ2๐ฅ ( ๐ฆโ2๐ฅ )+2 ๐ฆ
4 ๐ฅ2๐๐ฆ๐๐ฅ =
๐ฆ 2
โ2๐ฅ๐ฆ=๐ฆโ2๐ฅ
๐2 ๐ฆ๐๐ฅ2 =ยฟ
๐ฆ+2 ๐ฆ4 ๐ฅ2
3 ๐ฆ4 ๐ฅ2
๐2 ๐ฆ๐๐ฅ2 =ยฟ
Higher DerivativesEX: Find the Third Derivative.
sin( )y x
cos (๐ฆ ) ๐๐ฆ๐๐ฅ=๐๐ฅ๐๐ฅ
๐๐ฆ๐๐ฅ =
1cos(๐ฆ )
๐๐ฆ๐๐ฅ =sec (๐ฆ )
๐2 ๐ฆ๐๐ฅ2 =ยฟ
๐2 ๐ฆ๐๐ฅ2 =ยฟ
๐2 ๐ฆ๐๐ฅ2 =ยฟ
sec (๐ฆ ) tan (๐ฆ )๐๐ฆ๐๐ฅ
sec (๐ฆ ) tan ( ๐ฆ ) sec (๐ฆ )
๐ ๐๐2 (๐ฆ ) tan (๐ฆ )
Last update
โข 10/19/10
p. 162 1 โ 29 odd