Calculus hw - 3 hrs due
vzan0222
03f4534b2658927baf2db2634a97e977.docSAMPLE TEST NAME_________________ (200 points Total) Math 5A Professor O. LePoint
Instructions:
Print out . Complete all the questions correctly for a max of 40 extra credit points.
_____________________________________________________
QUESTION 1: Define the following
Definition of limit
Definition of Continuity
Definition of derivative
Mean Value Theorem (and draw a diagram).
Fundamental Theorem of Calculus
QUESTION 2: (10 points)
=
-
®
)
(
lim
5
x
f
x
________
=
-
®
)
(
lim
2
x
f
x
_______
=
+
®
)
(
lim
2
x
f
x
________
=
-
®
)
(
lim
3
x
f
x
_________Is
f
continuous at x=2 Mathematically explain why or why not below:QUESTION 3:
a) Evaluate:
14
10
16
lim
2
2
4
+
-
-
®
x
x
x
x
b) Evaluate:
2
3
3
5
8
3
4
lim
x
x
x
x
x
+
-
-
¥
®
QUESTION 4:
Use the definition of derivative to find the derivative of
F(x) = 3x2 -5x
QUESTION 5:
Consider the following graphs of the functions f and g below . Beside each, sketch the graphs of f’(x) and g’(x) respectively. Consider finding the derivative at 3-5 distinctive (x,y) points.
INCLUDEPICTURE "http://cnx.org/content/m19113/latest/lines6b.png" \* MERGEFORMATINET
QUESTION 6:
Differentiate the following.
a)
)
4
(
sin
5
)
(
2
3
x
x
f
=
b)
6
2
)
(
3
2
+
-
+
=
x
x
x
x
f
QUESTION 7:
Find y’
)
2
cos(
3
2
y
x
=
QUESTION 8:
Find the equation of a passing through x = 10 if
)
3
3
(
)
(
2
-
=
x
x
f
QUESTION 9:
Find the x = c that satisfies the Mean Value Theorem for the function f(x) = x3 with endpoints x = 0 and x = 2.
QUESTION 10:
Suppose that you wanted to find the
5
by using Newtons’ Method.a) Define a function
)
(
x
f
for Newton’s Method.b) Show the method to find
2
x
such what when Newton’s method is applied,QUESTION 11.
2
3
16
4
)
(
x
x
x
f
-
=
(More space on next page)
a) Find the critical points.
b) Find the intervals where
)
(
x
f
increases.c) Find the local minimum and maximum values on [-3,5].
d) Identify all inflection points.
e) Find the intervals of upward concavity.
f) Graph the function and label the critical and inflection points.
QUESTION 12:
Find two nonnegative numbers whose sum is 9 and so that the product of one number and the square of the other number is a maximum.
QUESTION 13:
a) Find the exact value of the integral:
ò
-
-
3
1
)
2
3
(
dx
x
B
ò
-
+
1
0
1
dx
e
e
x
x
b) Find each anti-derivative:
q
q
q
d
2
5
sec
)
tan
1
(
ò
+
dx
x
x
x
ò
+
-
-
2
2
)
6
2
(
3
3
dx
x
x
ò
-
2
4
1
QUESTION 14: (10 points)
ò
-
p
p
2
)
cos(
2
dx
x
QUESTION 15: (10 points)
Draw the region R enclosed by the curves y =x and y =x2 which is rotated about the x-axis. Find the volume of the solid of the resulting solid.
QUESTION 16: (10 points)
In your own words, what is the main point of the study of Calculus?
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