Unit 3 Pretest (No Calculator)
Determine whether Rolle's THM can be applied.
If it can, find all values of c in the interval such
that f '  c   0.
f  x    x  3 x  1
2
[1,3]
THM 3.3 Rolle's THM
Let f be continuous on [a, b] and differentiable on (a, b).
f (a)  f (b)   at least one number c in ( a, b) such that f '(c )  0.
Unit 3 Pretest (No Calculator)
Determine whether Rolle's THM can be applied.
If it can, find all values of c in the interval such
5
 c  , 1
3
[1,3]
that f '  c   0.
f  x    x  3 x  1
2
f '  x    x  1   x  3  2  x  1
f  1  0  x 2  2 x  1   x  3 2 x  2 
2
f  3  0
 x  2x 1 2x  4x  6
2
2
 3 x  2 x  5   3 x  5  x  1
2
Unit 3 Pretest (No Calculator)
Find all values of c in the interval  a, b  such that
f 'c 
f  x  x  2
f b  f  a 
ba
.
[2,6]
THM 3.4 The Mean Value THM
f is continuous on [a, b] and differentiable on (a, b) 
 a number c in (a, b) such that
f (b)  f (a)
f '(c) 
ba
Unit 3 Pretest (No Calculator)
Find all values of c in the interval  a, b  such that
f 'c 
f  x  x  2
f b  f  a 
ba
.
[2, 6]
f  2  0
y 1
f  6  2 

x 2
1
1
1
1/ 2
f '  x    x  2

  x  2 1 x  2 1
2
2 x2 2
x3
Unit 3 Pretest (No Calculator)
Find all values of c in the interval  a, b  such that
f 'c 
f b  f  a 
f  x   sin x
ba
.
[0,  ]
f  0  0
y
f    0 
0
x
f '  x   cos x  0 @ x 

2
Unit 3 Pretest (No Calculator)
4. Use the first derivative test to investigate
1 4
f  x   x  8 x for relative extrema.
4
3
f '  x   x  8  x  2 is a critical number.
 , 2
 2,  
Interval
3
0
Test Value


Derivative
Conclusion Decreasing Increasing
f has a relative min of (2, 12)
Unit 3 Pretest (No Calculator)
5. Use the first derivative test to investigate
f  x   ( x  1) for relative extrema.
3
f '  x   3  x  1
2
 x  1 is a critical number.
 , 1
Interval
2
Test Value

Derivative
Conclusion Increasing
 1, 
0

Increasing
f does not have relative extrema
Unit 3 Pretest (No Calculator)
6. Use the second derivative test to investigate
f  x   x  cos x, 0  x  2 , for concavity.
(List any inflection points.)
f '  x   1  sin x
f ''  x    cos x  0  x 
 3
     3 3 
 , , , 
2 2  2 2 
,
2 2
are the possible locations of inflection points
 0,  / 2
 / 2,3 / 2 3 / 2, 2 
Interval

 /4
7 / 4
Test Value



2nd Deriv.
Conclusion Conc. Dwn Conc. Up
Conc. Dwn
Unit 3 Pretest (No Calculator)
7. Use the second derivative test to determine
which critical numbers, if any, give a relative max.
f ''  x   6 x  10 and f  x  has critical numbers at
1 & 7/3.
f '' 1  0
f ''  7 / 3  0
f  x   x3  5 x 2  7 x
1
3
 x  1 is the location of a relative max.
Unit 3 Pretest (No Calculator)
2
2x
8. Find all horizontal asymptotes for f  x   2
.
3x  5
2
y  is a horizontal asymptote
3
Unit 3 Pretest (No Calculator)
2x
9. Given f  x   2
, find lim f  x  .
x 
3x  5
lim f  x   0
x 
Unit 3 Pretest (No Calculator)
10. Find all horizontal asymptotes for f  x  
x0
f  x 
3x
x2  4

3x
x

x2  4
3
4
1 2
x
x2
x  0 3x
3x
3
x
f  x 


4
x2  4
x2  4
 1 2
x
 x2
y  3 are horizontal asymptotes
3x
x2  4
.
Unit 3 Pretest (No Calculator)
11. Find the dimensions of a rectangle of maximum area,
with sides parallel to the coordinate axes, that can be
  x, y 
 x, y 
inscribed in the ellipse given by
x2 y 2

1
144 16
a) Write Area in terms of x and y.   x,  y 
y
x
A  2 x  2 y  4 xy
x
y
 x,  y 
Unit 3 Pretest (No Calculator)
11. Find the dimensions of a rectangle of maximum area,
with sides parallel to the coordinate axes, that can be
  x, y 
 x, y 
inscribed in the ellipse given by
x2 y 2

1
144 16
a) Write Area in terms of x and y.   x,  y 
 x,  y 
b) Write the ellipse formula in terms of x.
2
2
2
y
x
x
2
 1
 y  16 
16
144
9
2
144  x
1
2
2
y 
 y   144  x
9
3
Unit 3 Pretest (No Calculator)
11. Find the dimensions of a rectangle of maximum area,
with sides parallel to the coordinate axes, that can be
  x, y 
 x, y 
inscribed in the ellipse given by
x2 y 2

1
144 16
a) Write Area in terms of x and y.   x,  y 
 x,  y 
b) Write the ellipse formula in terms of x.
c) Write the area function in terms of x.
A  2 x  2 y  4 xy
4
2
A  x   x 144  x
3
1
y   144  x 2
3
Area will be positive
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
x2 y 2

1
144 16
1
2
y
144  x
3
  x, y 
 x,  y 
d) Use the first derivative test to find the 
x and y that give maximum area.
4
A  x   x 144  x 2
3
dA 4 
1

2
2 1/ 2
  144  x  x  144  x   2 x   
dx 3 
2

 4  144  2 x 2 
4
x2
2
  144  x 
 
0
2
2
3
144  x  3  144  x 
 x  72  6 2
 x, y 
12,0
 x,  y 
0  x  12
1
y
144  72  2 2
3
Unit 3 Pretest (Calculator)
Determine the absolute extrema of the function and
the x-value in the closed interval where it occurs.
f  x   x  12x
3
f '  x   3x  12  0
2
 3x  12  x  2
2
 2,16 ;  4,16 ;  2, 16
x2
0, 4
Unit 3 Pretest (Calculator)
Use a calculator to graph the function.
Determine the absolute extrema of the function and
the x-value in the closed interval where it occurs.
2
f  x 
2 x
[0, 2)
 0,1
Unit 3 Pretest (Calculator)
14. The cost C of producing x units per day is
1 2
C  x  62 x  125
4
and the price per unit is
2
d P
7


2
dx x  11 6
1
p  75  x.
3
What daily output produces maximum profit?
1 2 1 2
P  xp  C  75 x  x  x  62 x  125
3
4
7 2
P   x  13x  125
12
dP
7
  x  13  0
dx
6
7
x  13  x  11
6
The 2nd Derivative test will confirm that this is the location of a max.
Unit 3 Pretest (Calculator)
15. The cost C of producing x units per day is
1 2
C  x  62 x  125
4
and the price per unit is
2
d C
250

2
3
dx x  22 22
1
p  75  x.
3
What daily output produces minimum average cost?
1
125
C  x  62 
4
x
dC 1 125
  2 0
dx 4 x
1 125
 2
4 x
2
x  500  x  22
The 2nd Derivative test will confirm that this is the location of a min.