MATH 1325 Lab 1 – Limits, Continuity, Rates of Change, Derivatives
Question # 00561949
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Updated on: 07/18/2017 12:20 AM Due on: 07/18/2017
1 of 3
?12 ?11 ?10 ?9 ?8 ?7 ?6 ?5 ?4 ?3 ?2 ?1 1 2 3 4 5 6
?9
?8
?7
?6
?5
?4
?3
?2
?1
1
2
3
4
5
6
7
8
x
yInstructions: Please show all work, and write your solutions to the problems neatly on this handout. Draw box
around your final answer. If you need help, please feel free to consult with me or go to the Math Lab and ask for
assistance.
1. For the graphed function, answer the following.
a. )( lim
0
xf
x? ?
=
b. )( lim
0
xf
x? +
=
c. )( lim
0
xf
x?
=
d. f (0) =
e. )( lim
2
xf
x? ?
=
f. )( lim
2
xf
x? +
=
g. )( lim
2
xf
x?
=
h. f (2) =
i. )( lim
5
xf
x
?
??
=
j. ( ) limx
f x
???
=
k. )( lim xf
x ??
=
l. List the values of x where the function does not have a derivative.
2 of 3
2. Use your calculator to complete the table for
3
1
( )
1
x
f x
x
+
=
+
, and then use it to predict
1
lim ( )
x
f x
??
.
x ?1.1 ?1.01 ?1.001 ?1 ?.999 ?.99 ?.9
f x( ) DNE3
1
1
limx 1
x
?? x
+
=
+
__________
3. Use the properties of limits to find the following limits, if the limit exists.
(a)
2
6
4 12 limx 6
x x
? x
? ?
?
(b)
25
5
limx 25
x
? x
?
?
(c)
2
2
2 1 limx 4 3
x x
??? x
? ?
+
(d)
2
3
5 9 limx 3 2
x
?? x
?
?
4. If ( )
3
lim 8
x
f x
?
= and
3
lim ( ) 16
x
g x
?
= , then find ( )
3
2
lim
( ) ( ) x
f x
? f x g x ?
.
5. Find all values x a = where the function is discontinuous. 2
1
( ) x
f x
x x
+
=
+
3 of 3
6. Find the average rate of change of the function 3 2 y = 3x + 2x 4x + 1 ? ? between x = 5 and x = 8.
7. Use the definition of derivative to find f x ?( ) for the function f x x ( ) 3 = ?
8. Using the result from 7 to find the equation of the tangent line to the curve f x x ( ) 3 = ? at x = 4
9. The cost (in cents) of producing x tacos is given by 2
C(x) = 1000 + 0.24x .
(a) Find the marginal cost function.
(b) Calculate the marginal cost at a production level of 200 tacos. Interpret this result using a complete
sentence.
?12 ?11 ?10 ?9 ?8 ?7 ?6 ?5 ?4 ?3 ?2 ?1 1 2 3 4 5 6
?9
?8
?7
?6
?5
?4
?3
?2
?1
1
2
3
4
5
6
7
8
x
yInstructions: Please show all work, and write your solutions to the problems neatly on this handout. Draw box
around your final answer. If you need help, please feel free to consult with me or go to the Math Lab and ask for
assistance.
1. For the graphed function, answer the following.
a. )( lim
0
xf
x? ?
=
b. )( lim
0
xf
x? +
=
c. )( lim
0
xf
x?
=
d. f (0) =
e. )( lim
2
xf
x? ?
=
f. )( lim
2
xf
x? +
=
g. )( lim
2
xf
x?
=
h. f (2) =
i. )( lim
5
xf
x
?
??
=
j. ( ) limx
f x
???
=
k. )( lim xf
x ??
=
l. List the values of x where the function does not have a derivative.
2 of 3
2. Use your calculator to complete the table for
3
1
( )
1
x
f x
x
+
=
+
, and then use it to predict
1
lim ( )
x
f x
??
.
x ?1.1 ?1.01 ?1.001 ?1 ?.999 ?.99 ?.9
f x( ) DNE3
1
1
limx 1
x
?? x
+
=
+
__________
3. Use the properties of limits to find the following limits, if the limit exists.
(a)
2
6
4 12 limx 6
x x
? x
? ?
?
(b)
25
5
limx 25
x
? x
?
?
(c)
2
2
2 1 limx 4 3
x x
??? x
? ?
+
(d)
2
3
5 9 limx 3 2
x
?? x
?
?
4. If ( )
3
lim 8
x
f x
?
= and
3
lim ( ) 16
x
g x
?
= , then find ( )
3
2
lim
( ) ( ) x
f x
? f x g x ?
.
5. Find all values x a = where the function is discontinuous. 2
1
( ) x
f x
x x
+
=
+
3 of 3
6. Find the average rate of change of the function 3 2 y = 3x + 2x 4x + 1 ? ? between x = 5 and x = 8.
7. Use the definition of derivative to find f x ?( ) for the function f x x ( ) 3 = ?
8. Using the result from 7 to find the equation of the tangent line to the curve f x x ( ) 3 = ? at x = 4
9. The cost (in cents) of producing x tacos is given by 2
C(x) = 1000 + 0.24x .
(a) Find the marginal cost function.
(b) Calculate the marginal cost at a production level of 200 tacos. Interpret this result using a complete
sentence.

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Solution: MATH 1325 Lab 1 – Limits, Continuity, Rates of Change, Derivatives