1.
Co-operated
Fulfillment
Humble
Belief
Benefit
Promise
Equal
Supportive
Trustful
Redeem
2.
The purpose of listing the examples of the authors service to the British Empire is so that he has
leverage and some credibility when he starts to talk negatively
Calculus 2
Test 2
June 19, 1996
A. For each of items 1, 2, 3, set up the integral (or integrals) whose value gives
the indicated quantity; do not solve.
1
1
1. (5 pts.) Find the average value of f (x) =
over the interval [0, ].
2
2
1x
2. (10 pts.) If x a
Calculus 2
Test 1
1. (15 pts.) The graph of y = f (t) is given below. Set
Z x
A(x) =
f (t) dt for 0 x 7 .
May 24, 1996
0
(a) True or False: A(x) is continuous on the interval (0, 7).
(b) Is A(x) differentiable on the interval (0, 7)? If yes, describe A0 (
Calculus 1 Exam 3 October 29, 1993 Name
Instructions: Write your answers on separate answer sheets, not on this paper.
(
x2 ; x < 1
(1) (20 pts) Consider the function f (x) =
.
2x ; x 1
(a) Sketch a graph of f . (b) For which x values does f 0 (x) exist?
Calculus 1 Exam 2 October 8, 1993
Instructions: The following 10 questions are worth 10 points each. Write your answers on
separate answer sheets, not on this paper.
(1) Sketch the graph of V = V0 (1 et ) where t is time, V0 is a positive constant and V
i
Calculus 1 Make-up Exam 1 September, 1993
Instructions: Do not write your answers on this paper, use separate answer
sheets.
(1) Generally, the more fertilizer that is used, the better the yield of the crop.
However, if too much fertilizer is applied, the
MAC2311 CALCULUS I, FALL SEMESTER 1990
MAC 2311
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Final Exam
December 17, 1990.
1
(10 pts.) f (x) =
x+1
(a) Find the slope of the line between (0, f (0) and a variable point (x, f (x) on the graph.
(b) Find f 0 (0).
(5 pts. each)
MAC2311 CALCULUS I, FALL SEMESTER 1990
MAC 2311
Test 4
December 13, 1990.
(1) (10 pts. each)
Z x
d
sec t dt =
(a)
dx
Z 2 0
(b)
1/x4 dx =
Z1
(c)
sec2 (2 + 2x) dx =
Z 1
1
(d)
dx =
1 + 4x
0
(2) (15 pts.) Set up and evaluate an integral that gives the area of
MAC2311 CALCULUS I, FALL SEMESTER 1990
MAC 2311
(1)
(2)
(3)
(4)
(5)
(6)
Test 3
November 14, 1990.
x
(a) (5 pts.) lim
=
x
9x2 + 1
x3
(b) (5 pts.) lim
=
x
1 (x 1)2
(20 pts.) As a right circular cylinder is being heated, its radius is increasing at a rate
MAC2311 CALCULUS I, FALL SEMESTER 1990
MAC 2311
Test 2
d
(sin x + tan x + x + 1/x) =
(1)
dx
x3 2x
d
(2)
=
dx 1 + 3x x2
2
d
2 + x5 sin(x2 2x + 4) =
(3)
dx
tan 2x
(4) lim
=
x
0 x
October 12, 1990. The following are each worth 10 points.
d
1
to find
f (2x)
MAC2311 CALCULUS I, FALL SEMESTER 1990
MAC 2311
Test 1
September 19, 1990.
p
(1) (15 pts.) (a) Sketch the graph of x = y + 1 = 0. (b) If this equation defines y as a function of x,
find the domain and range.
(2) (15 pts.) If |f (x) 4| < 3, find a bound fo
MAC 2312
Test 2 B Answer Key
2/23/17
1. Use the data in the following table and the graph to find solutions for parts (a)
and (b).
x
0 1 2 3 4 5 6 7 8
f (x) 14 13 12 11 9 7 5 3 0
y=f(x)
x
2.
3.
4.
5.
(a) (6pt) Use
R 8the Midpoint Rule and the given data t
MAC 2312
Test 2 A Answer Key
2/23/17
1. Use the data in the following table and the graph to find solutions for parts (a)
and (b).
x
0 3 6 9 12
f (x) 10 9 7 4 0
y=f(x)
x
2.
3.
4.
5.
(a) (6pt) Use the
R 12Trapezoidal Rule and the given data to estimate the
MAC 2312
Test 1 A Answer Key
2/2/17
1. (5pt) Find the FORM of the partial fraction decomposition (do not find the nu8x2
merical value of the constants).
(x + 2)2 (x2 + 4)
A
B
Cx + D
+
+ 2
2
x + 2 (x + 2)
x +4
Z
p
2. (5pt) Complete the statement. To evalua
MAC 2312
Test 2 C Answer Key
2/23/17
1. Use the data in the following table and the graph to find solutions for parts (a)
and (b).
x
0 1 2 3 4 5 6 7 8
f (x) 14 13 12 11 9 7 5 3 0
y=f(x)
x
2.
3.
4.
5.
(a) (6pt) Use
R 8the Midpoint Rule and the given data t
MAC 2312
Test 1 B Answer Key
2/2/17
1. (5pt) Find the FORM of the partial fraction decomposition (do not find the nu8x
merical value of the constants).
(x 2)2 (x2 + 9)
B
A
Cx + D
+
+
x 2 (x 2)2
x2 + 9
Z
p
2. (5pt) Complete the statement. To evaluate the i
MAC 2312
Test 1 C Answer Key
2/2/17
1. (5pt) Find the FORM of the partial fraction decomposition (do not find the nu8x3
merical value of the constants).
(x + 4)2 (x2 + 9)
A
B
Cx + D
+
+ 2
2
x + 4 (x + 4)
x +9
Z
p
2. (5pt) Complete the statement. To evalua
Name: _
Class:
Date: _
(Subsequent Pages)
1.
Use the given graph of f to find the the open interval(s) on which f is concave upward.
1. 4.3 Increasing and Decreasing
Definitions 1.1. Let f be a continuous function on the interval (a, b).
(1) We say f is i
1. 4.1 Definitions of Minimum and Maximum Values
Definitions 1.1. In the definitions below, assume y = f (x) is a function with domain
D and c is a number in the domain of f .
(1) f has an absolute maximum or global maximum at x = c if f (c) f (x)
for all
1. 4.6 Graphing with Technology
Example 1.1. Graph f (x) = ex 0.186x4 . Where (approximately) is the function
increasing, decreasing, concave up concave down?
Using WolframAlpha website:
Now use some calculus to refine:
f 0 (x) = ex .744x3 and f 00 (x) =
1. 4.5 Summary of Graphing Tools
(1) Find from y = f (x):
(a) Domain: where is f defined?
(b) x: set y = 0 and solve for x
(c) y: set x = 0 and solve for y
(d) Symmetry
(i) y = f (x) is symmetric to the y axis if f (x) = f (x).
(ii) y = f (x) is symmetric
1. 4.2 Rolles Theorem and the Mean Value Theorem
Theorem 1.1 (Rolles Theorem). Let f be a function satisfying the following properties:
(1) f is continuous on the interval [a, b]
(2) f is differentiable on the interval (a, b)
(3) f (a) = f (b)
Then there
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Class:
Date: _
(First Page)
Name: _
Class:
Date: _
(Subsequent Pages)
1. 4.3 Concavity
1.
Use the given graph of f to find the the open interval(s) on which f is concave upward.
Definitions 1.1. Let f be a con
1. 4.4 Indeterminant Forms
(1)
0
0
(2)
(3) 0
(4)
(5) 00
(6) 0
(7) 1
2. LHospitals Rule
f (x)
. If this limit has the indeterminant form
xa g(x)
f (x)
f 0 (x)
lim
= lim 0
.
xa g(x)
xa g (x)
Consider lim
0
0
or
, then
To use LHospitals Rule for any other
1. 4.7 Optimization - main steps
Step 1. Read problem and express all information from the problem mathematically.
Use variables to represent any quantity that changes. Numbers may be used
for quantities that remain constant.
Step 2. Find a function for t
1. Section 3.7 Recall
The instantaneous velocity of a particle with position at time t given by s(t)
at time t is.
The instantaneous acceleration of a particle with position at time t given by
s(t) at time t is.
The jerk of a particle with position at
1. 3.8 Exponential Growth and Decay
Definition 1.1. There are many applications where a function is proportional to its
first derivative. In other words,
dy
= ky.
dx
This differential equation is called the natural law of growth (k > 0) or decay
(k < 0).
1. 3.10 Linearization
Recall that the tangent line of y = f (x) at the point (a, f (a) is the line the
best approximates the graph of y = f (x) at that point. This leads to the following
definition.
Definition 1.1. The linearization of f at a is the linea
1. 3.11 Definitions of Hyperbolic Functions
sinh x =
ex ex
2
cosh x =
ex + ex
2
tanh x =
sinh x
cosh x
coth x =
1
tanh x
sech x =
1
cosh x
csch x =
1
sinh x
2. Derivatives
d
sinh x = cosh x
dx
d
cosh x = sinh x
dx
d
tanh x = sech2 x
dx
d
coth x = csch2 x