lines/velocity deriv. 1 deriv. 2 deriv. 3 r.o.c/dy, ybrigh042/docs/math 1371/practice...
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Powerpoint Templates Page 1
!Lines/Velocity Deriv. 1 Deriv. 2 Deriv. 3 R.o.C/dy,
100
200
300
400
500
100
200
300
400
500
100
200
300
400
500
100
200
300
400
500
100
200
300
400
500
y∆
Powerpoint Templates Page 2
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)− −
Powerpoint Templates Page 3
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)− −
Powerpoint Templates Page 4
Lines/Velocity 200
Find the limits:2
23
4
22 1(1 )lim(
lim
)6 18
12 27
x
x
xx
xx x
xx→
→
++ ++
− +
−
Powerpoint Templates Page 5
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→
→
→ →
+ − +=
+ + + +−
−=
=
=− +
−≠ = = = −
− − − −
Powerpoint Templates Page 6
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 −= +
Powerpoint Templates Page 7
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 − −= = = −
− −− − −
= = =
−
−−
= = =
Powerpoint Templates Page 8
Lines/Velocity 400
Find the instantaneous velocity at t=1 using limits if
2 5( ) 64s t t t+ −=
Powerpoint Templates Page 9
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+ −=
Powerpoint Templates Page 10
Lines/Velocity 500
Find the secant slope and equation of the tangent line at t=3 using limits if
2( 5) 21t tf + −=
Powerpoint Templates Page 11
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
→ →
=
=
+ − ⇒ =
− + − − − + − − − −= = =
− − − − − −− − − +
= = ≠− − − − − − − −
− +=
− − −
− + −= = =
−− − −
− = −
Powerpoint Templates Page 12
Derivatives 1 100
Differentiate3 3( ) 52f x x x+ +=
Powerpoint Templates Page 13
Derivatives 1 100
3
2
3 52
'( 3
( )
) 3
x
f
f x
x
x
x
= + +
= +
Powerpoint Templates Page 14
Derivatives 1 200
Differentiate and find the tangent line at the given point.
( 4) 2 at xf xx
x += =
Powerpoint Templates Page 15
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
+ =
=
= − + = − + =
− = −
=
Powerpoint Templates Page 16
Derivatives 1 300
Differentiate using the definition of a derivative:
2 4 5x xy = + +
Powerpoint Templates Page 17
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
→ →
→ →
= + +
− + − − + − + −= =
− −− + + −
= ≠ = + + = +−
Powerpoint Templates Page 18
Derivatives 1 400
Differentiate
2( 1) )2
( x xxh xx
e + +−
=
Powerpoint Templates Page 19
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
+ +−
− −= + + + +
−−
=−−
+
=
Powerpoint Templates Page 20
Derivatives 1 500
Differentiate3
2
( 4) )(( )
xe xfx c x
x +−
=
Powerpoint Templates Page 21
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
+−
− + + + − +
−
−=
=
Powerpoint Templates Page 22
Derivatives 2 100
Differentiate
( ) tan( )sec( )f x x x=
Powerpoint Templates Page 23
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
=
= +
= +
Powerpoint Templates Page 24
Derivatives 2 200
Differentiate
(sincot
)( )
xx eyx
=
Powerpoint Templates Page 25
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+
=
=
− −
Powerpoint Templates Page 26
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= =
Powerpoint Templates Page 27
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
=
= = =
=
=
=
− = −
=
Powerpoint Templates Page 28
Derivatives 2 400
Differentiate
2
2
2 1)( sin(cos(
)2 1)zhz
zz
z − +=
+ +
Powerpoint Templates Page 29
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
− +=
+ +
+ + − + − − − + − + +=
+ ++
Powerpoint Templates Page 30
Differentiate
Derivatives 2 500
23 2
1
ln[54 ] 2sin( )( )sinh(cos ( ))
yy yyy
eg −
+=
Powerpoint Templates Page 31
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
− −
− −
−
+=
− + + − + −
+ + +
=
=
+ +
Powerpoint Templates Page 32
Differentiate
Derivatives 3 100
2 2( ) ln(cos 8h )(3 )xf x +=
Powerpoint Templates Page 33
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
= =
=
+ +
+ = ++
Powerpoint Templates Page 34
Find y’
Derivatives 3 200
( )46tanh ln 81y x x+ −=
Powerpoint Templates Page 35
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
+ − + −
− = + − + −
=
=
Powerpoint Templates Page 36
Derivatives 3 300
Find y’2 ( 1)45 2 1
cax bxe xy y+− + + = +
Powerpoint Templates Page 37
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
+
+
−
−
− −
− −
− −
+ + = +
+ − = −
+ = − −
+ + = −
+ +∴ = =
−
Powerpoint Templates Page 38
Derivatives 3 400Find y’
( ) ( ) ( )3 2tan sin cosyx x e x y y= − −+
Powerpoint Templates Page 39
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
− −
+
+ =
+ = −
= − −
= = − −
+ − − −
+ + − − −
+ +− − −
Powerpoint Templates Page 40
Derivatives 3 500
Find y’
( )( )2 3 3cot( ) ln csc( 2)cotx x y x y− + =
Powerpoint Templates Page 41
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
+
− +
−
=+
=
Powerpoint Templates Page 42
R.o.C & Differentials 100
Find dy with the given information
2 3 at( ) 4 4, 0.1x x x dxf x x= + − + = =
Powerpoint Templates Page 43
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
+ − + = = =
= + −
= + − =
= =
=
=
= =
Powerpoint Templates Page 44
R.o.C & Differentials 200
Find with the given information
2 22 6 at 2,( ) 5 0.210
x x xf x x − − = ∆ = ==
y∆
Powerpoint Templates Page 45
R.o.C & Differentials 200
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
− − = ∆ = =
∆ = + ∆ − = −= − =
=
Powerpoint Templates Page 46
R.o.C & Differentials 300
Find both dy and with the given information
2 5 2 at 2, 0) 3 .( 1x x d xf x x x− + = = == ∆
y∆
Powerpoint Templates Page 47
R.o.C & Differentials 300
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 − + = = ∆ =
= − == = =
∆ = + ∆ − = − = −
=
=
Powerpoint Templates Page 48
R.o.C & Differentials 400
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= + +
Powerpoint Templates Page 49
R.o.C & Differentials 400
2( ) 1000 24 0.31'( ) 24 0.62'(50) 55
C x x xC x xC
= + += +=
Powerpoint Templates Page 50
R.o.C & Differentials 500
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∆
Powerpoint Templates Page 51
R.o.C & Differentials 500
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
=
=
∆ = = + ∆ −
= ∆ =
= ⇒ = ⇒ =
= =
−
−