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MTH 251 Lab 5: Sketching Functions with Given Properties, Assignments of Calculus

A lab assignment for a mathematics course (mth 251) where students are required to sketch functions based on given properties such as continuity, derivative signs, and vertical asymptotes. The assignment includes five activities with figures and instructions.

Typology: Assignments

Pre 2010

Uploaded on 08/18/2009

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Name______________________ MTH 251/Lab 5 Graded Homework
Page 1 of 2
Activity 3
Sketch onto Figure 3 a function that satisfies each of
the following properties.
The function is everywhere continuous.
()
0>
xf on 3, 2
(
)
.
()
0<
xf on −∞,3
(
)
, 2,4
(
)
, and 4,
(
)
.
() ()
043 =
=
ff
()
0>
xf on −∞,2
(
)
and 2,4
(
)
.
()
0<
xf on 4,
(
)
.
()
20 =f
Activity 4
Sketch onto Figure 4 a function that is everywhere continuous and whose first derivative
function is always positive and always decreasing; draw the function so that
()
(
)
100
=
=ff .
Sketch onto Figure 5 a function that has vertical asymptotes at 2
=
x and 3=x (but no
other discontinuities) and whose first derivative value continually increases on each of the
intervals −∞,2
(
)
, 2,3
(
)
, and 3,
(
)
.
1 2 3 4 5 6-1-2-3-4-5-6
1
2
3
4
5
6
-1
-2
-3
-4
-5
-6
x
y
Figure 3:
()
xfy
=
1 2 3 4 5 6-1-2-3-4-5-6
1
2
3
4
5
6
-1
-2
-3
-4
-5
-6
x
y
Figure 4:
()
xfy =
1 2 3 4 5 6-1-2-3-4-5-6
1
2
3
4
5
6
-1
-2
-3
-4
-5
-6
x
y
Figure 5:
(
)
xfy
=
pf2

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Name______________________ MTH 251/Lab 5 Graded Homework

Page 1 of 2

Activity 3

Sketch onto Figure 3 a function that satisfies each of the following properties.

  • The function is everywhere continuous.

• f ′( )^ x > 0 on (− 3,2).

• f ′( )^ x < 0 on (−∞ , − 3 ), (2,4 ), and (4, ∞).

• f ′(^ − 3 ) = f ′( ) 4 = 0

• f ′′^ ( ) x > 0 on (−∞ ,2) and (2,4 ).

• f ′′^ ( ) x < 0 on (4, ∞).

• f ( ) 0 = 2

Activity 4

  • Sketch onto Figure 4 a function that is everywhere continuous and whose first derivative

function is always positive and always decreasing; draw the function so that f ( ) 0 = f ′( ) 0 = 1.

• Sketch onto Figure 5 a function that has vertical asymptotes at x =− 2 and x = 3 (but no

other discontinuities) and whose first derivative value continually increases on each of the

intervals (−∞ , − 2 ), (− 2,3), and (3, ∞).

x

y

Figure 3: y = f ( ) x

x

y

Figure 4: y = f ( ) x

x

y

Figure 5: y = f ( x )

Page 2 of 2

Activity 8

Consider the function w shown in Figure 8. Sketch onto Figure 9 the continuous antiderivative of

w that passes through the point ( −1, 0 ).

Activity 6

A function, y = f ( ) x , passes through the points shown in Table 3. The same function's first

derivative is shown in Figure 7. Answer questions 6.2 and 6.3 in reference to these functions.

6.2 Which is greater, f ( ) 6 or f ( ) 7? How do you know?

6.3 Where over the interval ( 0 , 9 )is the graph of y = f ( x )concave down? How do you know?

Figure 9

An antiderivative of w

Figure 8

y = w x ( )

Figure 7: y = f ′( x )

Table 3: y = f ( ) x

x y