Implicit Differentiation. If 3x 2 +16x-y-17 = xy 5 find y’ @ (1,-1) 6x + 16 - y’ = y’ =

Implicit Implicit DifferentiationDifferentiation

If 3xIf 3x22 +16x-y-17 = xy +16x-y-17 = xy55 find y’ @ (1,-1) find y’ @ (1,-1)

6x + 6x +

16 -16 -

y’ =y’ =

[xy[xy55 ]’ = ]’ =

A.A. 5xy5xy44

B.B. 5xy5xy44 y’ y’

C.C. x5yx5y44 y’+y y’+y55

D.D. x5yx5y44 y’+y y’+y55 y’ y’

If 3xIf 3x22 +16x-y-21 = xy +16x-y-21 = xy55 find y’ @ (1,-1) find y’ @ (1,-1)

6x + 16 – y’ = x5y6x + 16 – y’ = x5y44 y’+y y’+y55

6x + 16 – y’ = x5y6x + 16 – y’ = x5y44 y’+yy’+y55

A.A. 6x + 16 - y6x + 16 - y55 = y’ + 5 x y = y’ + 5 x y44 y’ y’

B.B. 6x + 16 - y6x + 16 - y55 = - y’ - 5 x y = - y’ - 5 x y44 y’ y’

C.C. 6x + 16 + y6x + 16 + y55 = - y’ + 5 x y = - y’ + 5 x y44 y’ y’

6x + 16 – y’ = x5y6x + 16 – y’ = x5y44 y’+yy’+y55

A.A. 6x + 16 - y6x + 16 - y55 = y’ + 5 x y = y’ + 5 x y44 y’ y’

B.B. 6x + 16 - y6x + 16 - y55 = y’ - 5 x y = y’ - 5 x y44 y’ y’

C.C. 6x + 16 + y6x + 16 + y55 = - y’ + 5 x y = - y’ + 5 x y44 y’ y’

6x + 16 - y6x + 16 - y55 = y’ + 5 x y = y’ + 5 x y44 y’y’

A.A. 6x + 16 - y6x + 16 - y55 = y’ (1 + 5 x y = y’ (1 + 5 x y44 ) )

B.B. 6x + 16 - y6x + 16 - y55 = y’ (y’ + 5 x y = y’ (y’ + 5 x y44 ) )

C.C. 6x + 16 + y6x + 16 + y55 = y’ ( 5 x y = y’ ( 5 x y44 ) )

6x + 16 - y6x + 16 - y55 = y’ (1 + 5 x = y’ (1 + 5 x yy44 ) )y’ = y’ =

A.A. (6x + 16 - y(6x + 16 - y55) / (1 + 5 x y) / (1 + 5 x y44 ) )

B.B. 6x + 16 - y6x + 16 - y55 = /(-1 – 5 x y = /(-1 – 5 x y44 ) )

y’=(6x+16-yy’=(6x+16-y55) /(1+5xy) /(1+5xy44))@ (1,-1)@ (1,-1)

A.A. -23/6-23/6

B.B. 23/623/6

C.C. 21/621/6

D.D. -21/6-21/6

s = 4 s = 4 r r 22

Relate ds/dt to dr/dt.Relate ds/dt to dr/dt.

s = 4 s = 4 r r 22 ds/dt = 8 ds/dt = 8 r dr/dt r dr/dt

A.A. TrueTrue

B.B. FalseFalse

ds/dt = 8 ds/dt = 8 r dr/dt If r dr/dt If r=1r=1& ds/dt = 8 then dr/dt = & ds/dt = 8 then dr/dt =

A.A. TrueTrue

B.B. FalseFalse

V = I RV = I RR = R =

#5 Find dR/dt when V is #5 Find dR/dt when V is increasing at 1 volt per increasing at 1 volt per second and I is decreasing at second and I is decreasing at 1/3 amp per second, V = 12 1/3 amp per second, V = 12 and I = 2.and I = 2.

Find R when V = 12 and I = 2Find R when V = 12 and I = 2

V = I RV = I RR = 6 ohmsR = 6 ohms

#5 Find dR/dt when V is #5 Find dR/dt when V is increasing at 1 volt per increasing at 1 volt per second and I is decreasing at second and I is decreasing at 1/3 amp per second, V = 12 1/3 amp per second, V = 12 and I = 2.and I = 2.

Do the calculus, V’ = Do the calculus, V’ =

V = I RV = I RV’ = I R’ + I’ RV’ = I R’ + I’ R

#5 Enter answer on next #5 Enter answer on next slide.slide.

d) Find dR/dt when V is d) Find dR/dt when V is increasing at 1 volt per increasing at 1 volt per second and I is decreasing at second and I is decreasing at 1/3 amp per second, V = 12 1/3 amp per second, V = 12 and I = 2.and I = 2.

V’ = I R’ + I’ R when V’ = I R’ + I’ R when V=1 V=1 I=-1/3, V=12, I=2 R’ I=-1/3, V=12, I=2 R’ = ?= ?1.51.5

11

22. The rope is hauled in22. The rope is hauled in

At 2 feet/sec and y=6’At 2 feet/sec and y=6’

is constant.is constant.

a)a)How fast is the boat approaching How fast is the boat approaching the dock when 10’ of rope is out?the dock when 10’ of rope is out?

rope is hauled inrope is hauled inat 2 feet/sec and y=6’at 2 feet/sec and y=6’is constant.is constant.How fast is the boat How fast is the boat approaching the dock approaching the dock when 10’ of rope is when 10’ of rope is out?out?

-2.5-2.5

0.050.05

How fast is the area of the How fast is the area of the triangle changing then?triangle changing then?

s’ = -2s’ = -2

x’ = -2.5x’ = -2.5

A = 0.5 x 6 = 3 xA = 0.5 x 6 = 3 x

How fast is the area of the How fast is the area of the triangle changing then?triangle changing then?

s’ = -2s’ = -2

x’ = -2.5x’ = -2.5

A = 0.5 x 6 = 3 xA = 0.5 x 6 = 3 x

A’ = 3 x’ sq. ft. / secA’ = 3 x’ sq. ft. / sec

A’ = -7.5 sq. ft. / secA’ = -7.5 sq. ft. / sec

Related RatesRelated RatesSuppose a radar gun on first base Suppose a radar gun on first base catches a baseball 30 feet away catches a baseball 30 feet away from the pitcher and registers 50 from the pitcher and registers 50 feet per second. How fast is the feet per second. How fast is the ball really traveling?ball really traveling?

a baseball 30 feet away a baseball 30 feet away from the pitcher and from the pitcher and registers 50 registers 50 . feet per . feet per second.second.

A.A. x=30 y’=50x=30 y’=50

B.B. x=30 y =50x=30 y =50

C.C. x’=30 y’=50x’=30 y’=50

D.D. x’=30 y=50x’=30 y=50

Related RatesRelated Rates

The calculus.The calculus.

x = 30 y’ = 50 y = ?x = 30 y’ = 50 y = ?

The algebra.The algebra.

2 2 22 45 x y

2 ' 2 'xx yy

2 2 22 45 30 y

4950 y

DifferentiateDifferentiateimplicitlyimplicitly

A.A. 2x = 2y y’2x = 2y y’

B.B. 2x = y y’2x = y y’

C.C. 2x x’ = y y’2x x’ = y y’

D.D. 2x x’ = 2y y’2x x’ = 2y y’

2 2 22 45 x y

Related RatesRelated Ratesx = 30 y’ = 50 y = ?x = 30 y’ = 50 y = ?

Back to the calculus.Back to the calculus.

2 ' 2 'xx yy

4950 y

2(30) 'x

a baseball 30 feet away a baseball 30 feet away from the pitcher and from the pitcher and registers 50 registers 50 . feet per . feet per second.second.A.A. x’ = 0.6 root(4950)x’ = 0.6 root(4950)

B.B. x’ = 50 root(4950)x’ = 50 root(4950)

C.C. x’ = 30 root(4950)x’ = 30 root(4950)

D.D. x’ = 5/3 root(4950)x’ = 5/3 root(4950)

2 ' 2 4950 'xx y

Related RatesRelated RatesBack to the calculus.Back to the calculus.

x = 30 y’ = 50 y = ?x = 30 y’ = 50 y = ?

2 ' 2 'xx yy4950 y

2(30) ' 2 4950 'x y

Related RatesRelated RatesBack to the calculus.Back to the calculus.

x = 30 y’ = 50 y = ?x = 30 y’ = 50 y = ?

2 ' 2 'xx yy4950 y

2(30) ' 2 4950(50)x

Related RatesRelated Ratesx’ = 117.260 feet/sec.x’ = 117.260 feet/sec.

X = 30 y’ = 50 y = ?X = 30 y’ = 50 y = ?

2 ' 2 'xx yy 4950 y

4950(50)'

30x

2(30) ' 2 4950(50)x

Related RatesRelated RatesRead the problem, drawing a pictureRead the problem, drawing a picture

No non-constants on the pictureNo non-constants on the picture

Write an equationWrite an equation

Differentiate implicitlyDifferentiate implicitly

Enter non-constants and solveEnter non-constants and solve

Implicit Differentiation. If 3x 2 +16x-y-17 = xy 5 find y’ @ (1,-1) 6x + 16 - y’ = y’ =

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Transcript of Implicit Differentiation. If 3x 2 +16x-y-17 = xy 5 find y’ @ (1,-1) 6x + 16 - y’ = y’ =