lines/velocity deriv. 1 deriv. 2 deriv. 3 r.o.c/dy, ybrigh042/docs/math 1371/practice...

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!Lines/Velocity Deriv. 1 Deriv. 2 Deriv. 3 R.o.C/dy,

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y∆

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Lines/Velocity 100

Find the equation of the line in both slope-intercept form and point-slopeform that goes through the points

( 2,10) and (6, 2)− −



Lines/Velocity 100

point-slope: 1

2 10 12 46 ( 2) 8 3

4 ( 2)3

4slope-intercept

0

: 3

223

x

y

m

x

y

− − −= = −

− −

− +

= − +

=

− =

( 2,10) and (6, 2)− −



Lines/Velocity 200

Find the limits:2

23

4

22 1(1 )lim(

lim

)6 18

12 27

x

x

xx

xx x

xx→

→

++ ++

− +

−



Lines/Velocity 2002 4

2

3

3 3

2

2

1 4 16 131) 4 2 1 7

6 18 012 2

(

7 06( 3) 6 6when 3, 1

( 3)

1 )lim(

lim

lim lim( 9) ( 9) 6

x

x

x x

xx

xx x

x xx x x

xx→

→

→ →

+ − +=

+ + + +−

−=

=

=− +

−≠ = = = −

− − − −



If

Find the slope of the secant line between:

** x=1 and x=4

** x=4 and x=9

Lines/Velocity 300

( 9 1) 4xg xx −= +



Lines/Velocity 300( 9 1) 4xg xx −= +

(4) (1) 3 6 14 1 4 1

(

(1) 6, (4) 3,

9) (4) 8 3

(9) 8

119 4 5 5

s

s

g g

gm

g g

g

g

m − −= = = −

− −− − −

= = =

−

−−

= = =



Lines/Velocity 400

Find the instantaneous velocity at t=1 using limits if

2 5( ) 64s t t t+ −=



Lines/Velocity 400

2

3 3

( ) (1) 4 5 9 (4 9)( 1) when 11 1 ( 1)

(1) 3

4 9

lim lim(4 9) 21

avg

t

avg

inst avg t

s t s t t t t t

s

t t tv

v tv v t

→ →

=

=

= +

− + − + −= = ≠

− − −

= = + =

2 5( ) 64s t t t+ −=



Lines/Velocity 500

Find the secant slope and equation of the tangent line at t=3 using limits if

2( 5) 21t tf + −=



Lines/Velocity 5002( 5) 21t tf + −=

( ) ( )

( )

( )

2

2 2 2

2

2

2 2

2

3 3 2

25 (3) 5

( ) (3) 1 25 5 4 25 4 253 3 3 4 25

9 ( 3)( 3) when 3,( 3) 4 25 ( 3) 4 25

( 3)

4 25

( 3) 6lim lim8 44 25

5 (

( ) 1

3

34

3)

s

s

s

T st t

t f

f t f t t tmt t t t

t t tm tt t t t

tmt

t

f

m mt

y t

t

→ →

=

=

+ − ⇒ =

− + − − − + − − − −= = =

− − − − − −− − − +

= = ≠− − − − − − − −

− +=

− − −

− + −= = =

−− − −

− = −



Derivatives 1 100

Differentiate3 3( ) 52f x x x+ +=



Derivatives 1 100

3

2

3 52

'( 3

( )

) 3

x

f

f x

x

x

x

= + +

= +



Derivatives 1 200

Differentiate and find the tangent line at the given point.

( 4) 2 at xf xx

x += =



Derivatives 1 200

2

2 at 4

5(4) 2.5 or 2

2 1 1 1 1'( ) ; '(4)8 4 82

12.5 ( 4

(

)8

) x xx

f

f x f

f

y

x

x x

x

+ =

=

= − + = − + =

− = −

=



Derivatives 1 300

Differentiate using the definition of a derivative:

2 4 5x xy = + +



Derivatives 1 300

2

2 2 2 2

4 5 take x=b as "given point"

( 4 5) ( 4 5) ( ) 4( )' lim lim

( )( ) 4( )lim when , lim( 4) 2 4

x b x b

x b x b

x x

x x b b x b x byx b x b

x b x b x b x b x b xx b

y

→ →

→ →

= + +

− + − − + − + −= =

− −− + + −

= ≠ = + + = +−



Derivatives 1 400

Differentiate

2( 1) )2

( x xxh xx

e + +−

=



Derivatives 1 400

2

2 2

2 2 2 2

( 1)2

( 2) (2 )'( ) ( 1)

( )

( 2)( 2) )

2( 2

x

x x x

xxx

x x xh x e x e e

h

xx x

x e

x

+ +−

− −= + + + +

−−

=−−

+

=



Derivatives 1 500

Differentiate3

2

( 4) )(( )

xe xfx c x

x +−

=



Derivatives 1 500

( )( )

( )

3

2

2 2 3 3 2

22

( 4)( )

1( ) (3 ) ( 4) ( 4) ( ) (2 )2'( )

( )

** can simplify slightly if desi d

(

re

)x

x x x

e xx c x

x c x x x e e x x c x

f

xxf x

x c x

x

e

+−

− + + + − +

−

−=

=



Derivatives 2 100

Differentiate

( ) tan( )sec( )f x x x=



Derivatives 2 100

2

2 3

( ) tan( )sec( )'( ) tan( )(sec( ) tan( )) sec( )(sec ( ))tan ( )sec( ) sec ( )

f x x xf x x x x x x

x x x

=

= +

= +



Derivatives 2 200

Differentiate

(sincot

)( )

xx eyx

=



Derivatives 2 200

x x

100-x

100-x

2

2

sincot( )cot( )(s

( )

( ))' in( ) cos( )) sin( ) ( co )

s(

cc t

x

x x x

x e

e

yxx x e x xy

xe x+

=

=

− −



Derivatives 2 300

Differentiate and find the tangent line at the given point

2 3 2( ) tan ( )csc ( ) at / 3sin( )

cos( ) x x xx

f xx π= =



Derivatives 2 3002 3 2

32

3 2

( ) tan ( )csc ( ) at / 3sin( )sin ( ) 1cos ( )

1cos ( ) sin ( )( ) sec( )sin( )

23

'( ) sec( ) tan( )

' 2

cos(

33

2

)

cos

2 3

( )

3

x x x xxxxx xf x x

x

f

f x x

f

x

y x

x

x

f

π

π

π

π

=

= = =

=

=

=

− = −

=



Derivatives 2 400

Differentiate

2

2

2 1)( sin(cos(

)2 1)zhz

zz

z − +=

+ +



Derivatives 2 400

2

2

2 2 2 2

2 2

sin(cos(cos( cos(

2 1)( )2 1)2 1) 2 1)(2 2) 2 1)( sin 2 1)(2 2))'( )

2 1sin(

)(

cos (

zh zzz z z z z

zzz z z zh z z

zz

− +=

+ +

+ + − + − − − + − + +=

+ ++



Differentiate

Derivatives 2 500

23 2

1

ln[54 ] 2sin( )( )sinh(cos ( ))

yy yyy

eg −

+=



Derivatives 2 500

( )

23 2 2 2

1 1

1 2 2 2 1

2

2 1

ln[54 ] 2sin( ) ln 54 3ln( )sinh(cos sinh(cos

2sin( )( )) ( ))

3 1( )) 2 2cos(sinh(cos ln 54 3ln cos

sinh (

)(2 ) 2sin( ) cosh

cos

( ( ))1

'( )( ))

y yy y

y y y y y yy

y e

y

y y yg y

y

y

g

y

y

− −

− −

−

+=

− + + − + −

+ + +

=

=

+ +



Differentiate

Derivatives 3 100

2 2( ) ln(cos 8h )(3 )xf x +=



Derivatives 3 100

2 2 2

2 22

( ) ln(cosh ) 2 ln(cosh(3 )2'

(3 8) 8)

8)(6 ) 12 tanh(3 8)8

( ) sinh(3cosh( )3

f x x

f x

x

xx xx

x

= =

=

+ +

+ = ++



Find y’

Derivatives 3 200

( )46tanh ln 81y x x+ −=



Derivatives 3 200

( ) ( )

( ) ( )( )

4 4

32 4

4

6 81 6 81

81 4' sec 6 81

1tanh ln tanh ln2

1 l6 81

n2 2

x x x x

xy h x

x

y

xx

+ − + −

− = + − + −

=

=



Derivatives 3 300

Find y’2 ( 1)45 2 1

cax bxe xy y+− + + = +



Derivatives 3 300

( )

( )

2 ( 1)

2 ( 1)

2 2

2 2

2 2

5

5

5 5

5 5

5

4

4

3 4

4 3

4

3

5

2 1

2 1

(2 ) (4 )

(2 ) 1 4

(2 )'1 4

ca

ca

x b

x b

x x

x x

x x

xe xy yd xe y xydx

dy dyxe x e x y ydx dx

dyxe x e y xydx

dy xe x e yydx xy

+

+

−

−

− −

− −

− −

+ + = +

+ − = −

+ = − −

+ + = −

+ +∴ = =

−



Derivatives 3 400Find y’

( ) ( ) ( )3 2tan sin cosyx x e x y y= − −+



Derivatives 3 400

( ) ( ) ( )( )

( )( )

( )

3

2 3 2

2 3 2

2

3 2

tan sin cos

sec cos sin( ) sin (3 )

sec cos sin (3 ) sin( )

sec cos'si

2

( ) tan( ) 2 1

( ) tan( ) 2 1

( ) tan( )2 1n (3 ) sin( )

y

y y

y y

y

y

x x e x y

dy dyx x x e y ydx dx

dyx x y y x edx

dy x xydx y y x

y

x x e y

x e y

x xe

x

ey

− −

+

+ =

+ = −

= − −

= = − −

+ − − −

+ + − − −

+ +− − −



Derivatives 3 500

Find y’

( )( )2 3 3cot( ) ln csc( 2)cotx x y x y− + =



Derivatives 3 500

( )( )2 3 3

2 3 3

3 3 2 22 2 2 2

3

22 2 2 2 2

cot( ) ln csc( 2)cot

1cot( ) ln(csc( 2)) ln(cot( ))2csc( 2)cot( 2)(3 ) ( )( )) cot( )

csc( 2)

( )) (

( csc )( csc 32cot( )

csccot( csc 32cot )

) 3(

c

x x y x y

x x y x y

y y y dy dyx ydx x

xx xy

x

dx

dyx x yx dx

x y

− + =

− + − =

+ ++

− −− − =−

+

− + −

= 3

22 2 2

2 2 3

csccot( csc2cot( )'

3

ot( 2)

( )) ( )

3 cot( 2)

y

xx

y

x xdy xydx y y

+

− +

−

=+

=



R.o.C & Differentials 100

Find dy with the given information

2 3 at( ) 4 4, 0.1x x x dxf x x= + − + = =



R.o.C & Differentials 1002 13 at 4, 0.1

10

1'( ) 821 143'(4) 32 4 35.75

(

4 4143 1 143'( ) 3.575

4 0 40

)

1

4 x x x dx

f x x xx

f

d

f x x

y f x dx

+ − + = = =

= + −

= + − =

= =

=

=

= =




Find with the given information

2 22 6 at 2,( ) 5 0.210

x x xf x x − − = ∆ = ==

y∆




2 22 6 at 2, 0.210

( ) ( ) (2.2) (2) 13.8 10 3.8

( ) 5 x x x

y f x

f x x

x f x f f

− − = ∆ = =

∆ = + ∆ − = −= − =

=




Find both dy and with the given information

2 5 2 at 2, 0) 3 .( 1x x d xf x x x− + = = == ∆

y∆




2 5 2 at 2, 0.1

'( ) 6 5, '(2) 7'( ) (7)(.1) 0.7

( ) ( ) (2.1) (2) 4

(

.73

3

0.73

)

4

x x dx x

f x x fdy f x dx

y f x x f x f f

f x x − + = = ∆ =

= − == = =

∆ = + ∆ − = − = −

=

=




The price of producing a commodity is given by

Find the rate of change when 50 units are being produced

2( ) 1000 24 0.31C x x x= + +




2( ) 1000 24 0.31'( ) 24 0.62'(50) 55

C x x xC x xC

= + += +=




A spherical balloon is being painted. The original volume of the balloon is and the new volume is .

How much paint do the painters estimate they use?

How much is ?What is the value of dV if dr= ?

2563π

5003π

r∆r∆




3

3 2

256 2443 3

43

256 ( ) ( )3

4,

50

1

4( ) '( ) 4 '

03

5003

(4) 643'( ) 64

V r

V f r r f r

r r

f r r f r r f

dV f r dr

π π π

π

π π

π π π

π

=

=

∆ = = + ∆ −

= ∆ =

= ⇒ = ⇒ =

= =

−

−


lines/velocity deriv. 1 deriv. 2 deriv. 3 r.o.c/dy, ybrigh042/docs/math 1371/practice...

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