Determine whether Rolle's THM can be applied.

If it can, find all values of in the interval such

that ' 0.

c

f c

23 1 [ 1,3]f x x x

Unit 3 Pretest (No Calculator)

THM 3.3 Rolle's THMLet be continuous on [ , ] and differentiable on ( , ).

( ) ( ) at least one number in ( , ) such that '( ) 0.

f a b a b

f a f b c a b f c

Determine whether Rolle's THM can be applied.

If it can, find all values of in the interval such

that ' 0.

c

f c

Unit 3 Pretest (No Calculator)

2

2

2

2 2

2

3 1 [ 1,3]

' 1 3 2 1

2 1 3 2 2

2 1 2 4 6

3 2 5 3 5 1

f x x x

f x x x x

x x x x

x x x x

x x x x

5, 1

3c

1 0

3 0

f

f

Find all values of in the interval , such that

' .

c a b

f b f af c

b a

2 [2,6]f x x

Unit 3 Pretest (No Calculator)

THM 3.4 The Mean Value THM is continuous on [ , ] and differentiable on ( , )

a number in ( , ) such that

f a b a b

c a b

( ) ( )

'( )f b f a

f cb a

Find all values of in the interval , such that

' .

c a b

f b f af c

b a

1/ 2

2 [2,6]

2 0

16 2

21 1 1

' 2 2 1 2 12 22

32

f x x

f

yf

x

f x x x xx

x

Unit 3 Pretest (No Calculator)

Find all values of in the interval , such that

' .

c a b

f b f af c

b a

sin [0, ]

0 0

0 0

c 02

' os @

f x x

f

yf

x

f x x x

Unit 3 Pretest (No Calculator)

4

4. Use the first derivative test to investigate

1 8 for relative extrema.

4f x x x

3' 8 2 is a critical number.f x x x

Interval

Test Value

Derivative

Conclusion Decreasing Increasing

, 2 2,

0 3

has a relative min of (2, 12)f

Unit 3 Pretest (No Calculator)

Unit 3 Pretest (No Calculator)

3

5. Use the first derivative test to investigate

( 1) for relative extrema.f x x

2' 3 1

1 is a critical number.

f x x

x

does not have relative extremaf

Interval

Test Value

Derivative

Conclusion Increasing Increasing

, 1 1,

2 0

' 1 sin

3'' cos 0 ,

2 2are the possible locations of inflection points

f x x

f x x x

Unit 3 Pretest (No Calculator)

6. Use the second derivative test to investigate

cos , 0 2 , for concavity.

(List any inflection points.)

f x x x x

3 3, , ,

2 2 2 2

Interval

Test Value

2nd Deriv.

Conclusion Conc. Dwn Conc. Up

0, / 2 / 2,3 / 2

/ 4

Conc. Dwn

3 / 2,2

7 / 4

Unit 3 Pretest (No Calculator)

7. Use the second derivative test to determine

which critical numbers, if any, give a relative max.

'' 6 10 and has critical numbers at

1 & 7/3.

f x x f x

1 is the location of a relative max.x

'' 1 0

'' 7 / 3 0

f

f

1

3

3 25 7f x x x x

Unit 3 Pretest (No Calculator)

2

2

28. Find all horizontal asymptotes for .

3 5

xf x

x

2 is a horizontal asymptote

3y

Unit 3 Pretest (No Calculator)

2

29. Given , find lim .

3 5 x

xf x f x

x

lim 0x

f x

Unit 3 Pretest (No Calculator)

2

310. Find all horizontal asymptotes for .

4

xf x

x

2 2

22

33 3

44 4 1

xx xf x

x xxx

3 are horizontal asymptotesy

2 2

22

33 3

44 4 1

xx xf x

x xxx

0x

0x

Unit 3 Pretest (No Calculator)

2 2

11. Find the dimensions of a rectangle of maximum area,

with sides parallel to the coordinate axes, that can be

inscribed in the ellipse given by

1144 16

a) Write Area in terms of

x y

x

and .y

,x y ,x y

,x y ,x yx x

y

y

2 2 4A x y xy

2 2

2

22

22 1

144

1 1616 144

1

3

9

44

9

y x x

y x

y

xy

Unit 3 Pretest (No Calculator)

2 2

11. Find the dimensions of a rectangle of maximum area,

with sides parallel to the coordinate axes, that can be

inscribed in the ellipse given by

1144 16

a) Write Area in terms of

x y

x

and .

b) Write the ellipse formula in terms of .

y

x

,x y ,x y

,x y ,x y

Unit 3 Pretest (No Calculator)

2 2

11. Find the dimensions of a rectangle of maximum area,

with sides parallel to the coordinate axes, that can be

inscribed in the ellipse given by

1144 16

a) Write Area in terms of

x y

x

and .

b) Write the ellipse formula in terms of .

c) Write the area function in terms of .

y

x

x

,x y ,x y

,x y ,x y

24144

3A x x x

2 2 4A x y xy 21

1443

y x Area will be positive

2 2

11. Find the dimensions of a rectangle of maximum area,

with sides parallel to the coordinate axes, that can be

inscribed in the ellipse given by

1144 16

d) Use the first derivativ

x y

e test to find the

and that give maximum area.x y

,x y ,x y

,x y ,x y

2

1/ 22 2

2 22

2 2

4144

3

4 1144 144 2

3 2

4 4 144 2144 0

3 3144 4

2

1 4

72 6

A x x x

dAx x x x

dx

x xx

x x

x

1

144 7 22 23

y

12,0

0 12x

21144

3y x

Determine the absolute extrema of the function and

the -value in the closed interval where it occurs.x

3 12 0, 4f x x x

2

2

' 3 12 0

3 12 2

f x x

x x

2,16 ; 4,16 ; 2, 16

2x

Unit 3 Pretest (Calculator)

Use a calculator to graph the function.

Determine the absolute extrema of the function and

the -value in the closed interval where it occurs.x

2 [0, 2)

2f x

x

0,1

Unit 3 Pretest (Calculator)

Unit 3 Pretest (Calculator)

2

14. The cost of producing units per day is

1 62 125

4 and the price per unit is

1 75 .

3 What daily output produces maximum profit?

C x

C x x

p x

2

2

21 175 62 125

3 47

13 12512

P

P xp C x x

x

x x

x

713 0

1

6

13 17

6

dPx

dx

x x

The 2nd Derivative test will confirm that this is the location of a max.

2

2

7

6

d P

dx

11x

Unit 3 Pretest (Calculator)

2

15. The cost of producing units per day is

1 62 125

4 and the price per unit is

1 75 .

3 What daily output produces minimum average cost?

C x

C x x

p x

2

1 12562

4

1 1250

4

C xx

dC

dx x

2

2

1

2

125

4

00 25x x

x

The 2nd Derivative test will confirm that this is the location of a min.

2

2 3

250

22

d C

dx22x