Properties of Definite Integrals
Properties of Definite Integrals
Properties of Definite Integrals
Properties of Definite Integrals
Note that Property 2 of Theorem 4.7 can be
extended to cover any finite number of functions.
For example,
Properties of Def
Mathematics 63 TWHFV
Exercises (Rates of Change, Rectilinear Motion)
I. Determine if the following statements are true or false.
1. A particle moving in the positive direction of a line has positive acceleration.
2. If s = f (t) is the position function o
Exercises (Rolles Theorem and the Mean-Value Theorem)
Mathematics 63 TWHFV
1. Verify that the hypotheses of Rolles Theorem are satised on the given interval, and nd all values of
c in that interval that satisfy the conclusion of the theorem.
(a) f (x) = (
Mathematics 63 TWHFV
Exercises (Related Rates)
Solve the following problems completely.
1. If two resistors with resistances R1 and R2 are connected in parallel, the total resistance R, measured in ohms, is
1
1
1
given by =
+ . If R1 and R2 are increasing
Mathematics 63 TWHFV
Exercises (Limits involving infinity)
I. Homework. Write your complete solutions on one whole sheet of yellow pad paper. Due: 06
February 2015, Friday. Papers submitted after 11:45 am will be checked, but the score will
not be recorde
Mathematics 63 TWHFV Exercises
(Intermediate value theorem, Squeeze theorem, Limits involving trigonometric functions)
sin x ,
6x
1. Determine the interval of continuity of f (x) =
1
,
2x 1
jump or innite.
x
1
2
x>
1
2
. Classify each discontinuity as
Mathematics 63 (TWHFV)
Exercises on Definite Integrals and the Fundamental Theorem of Calculus
1. Use the interpretation of the denite integral of a continuous function on an interval as a net
signed area to evaluate the following denite integrals.
|2x +
Mathematics 63 TWHFV
Exercises on Applications of the Definite Integral
(Area, Volume, Arclength, Surface Area)
I. Use denite integrals to nd the area the following regions.
(a) the region bounded by the curves C1 : y = x2 2x + 1 and C2 : y = 7 x
(b) the
Mathematics 63
5th Long Exam (Solution Key)
2nd Sem AY 2014-15
16 May 2015
dy
(1 + csch x)log10 x x + cosh1 x
, where all factors
I. Using logarithmic dierentiation, nd
, if y =
dx
( + cos1 x)
are assumed to be positive.
(6 pts)
4
Solution:
1
ln(x + cosh1
Mathematics 63 TWHFV Exercises
(Tangent lines and the Derivative, Dierentiation formulas, Chain Rule)
1. Determine whether the statement is true or false.
(a) If f and g are functions such that f (x) = g(x), then f (x) = g (x).
(b) If the graph of f does
Mathematics 63 TWHFV
Exercises (Local linear approximation and differentials)
I. Find dy.
1. y =
2. y =
1
(2x3
x2
4. y = tan2 x sec 2x
4
+ 3)
5. y = x2 sin
2x + 3
1
x
1 x2
6. y =
cos x2
2 + cos x
3. y =
2 sin x
3
x cos x
II. 1. Use a local linear approxim
Mathematics 63 TWHFV
I. Find
Exercises on Hyperbolic Functions
dy
.
dx
1. y = sinh(2x ) cosh
2. csch y =
xsinh 1
log3 x
x2
coth x
3. y = (sech x)ln x +
1
+ tanh(3x + 2y 6)
II. Find the following antiderivatives.
1.
sinh z
cosh z dz
4.
2.
csch(cot1 x)
dx
1
Mathematics 63 TWHFV
Exercises (Exponential and Logarithmic Functions)
dy
I. Use logarithmic dierentiation to nd
. Assume all factors are nonzero, radicands are
dx
positive for even indices and nonzero for odd indices.
x2 cot4 x
1. y = x5 (x2 1)4 4 sin x
Demand, Revenue, Cost, &
Profit
Demand Function D(q)
p =D(q)
In this function the input is q and output p
q-independent variable/p-dependent variable
[Recall y=f(x)]
p =D(q) the price at which q units of the good can be sold
Unit price-p
Most demand
1
Pablo Echavarria #200385648
University of Regina
ENGL 100 005/015
Assignment #1
SECTION A: Kathiann Kowalski
It Is an alarming fact that college students spend more hours using their cellphones than sleeping.
In order to prevent an addiction to cellphon
1
Pablo Echavarria #200385648
University of Regina
Professor Sherry Klein
ENGL 100 - 005/015
02/16/2017
Assignment #2
SECTION ONE:
Why is John McCall the Art of Writing an F paper an ironic text?
John McCall text is an ironic text due to the tone in which
1
Pablo Echavarria #200385648
Professor Sherry Klein
Assignment #2
02/16/2017
Section One:
Why is John McCall the Art of Writing an F paper an ironic text?
John McCall text is an ironic text due to the tone in which he expresses the directions to
help stu
Pablo Echavarria #200385648
ENIN 370 Assignment 2
Page 3 of 4
02/08/2017
2. Plot the non-linearity error in terms of the percentage of the full-range reading versus the
wipers displacement, x for a rotary potentiometer whose resistance is R p = 500 (), th
1
Pablo Echavarria 200385648
Risk Assessment
Emmanuel Quaye
2/1/2017
Assignment 1
Question 1
The process that I am choosing is the process of installing a factory with assembly lines, it can be
subject to a risk analysis due to the way a factory is built,
Pablo Echavarria #200385648
Professor Emmanuel Quaye
ENIN 433 Risk Assessment
2/10/2017
Assignment #2
1. The process as Industrial Engineer I chose the creation of an industry or factory, which is
subject to risk analysis, in many ways along the process.
1
Pablo Echavarria 200385648
Professor Sherry Klein
ENGL 100-005/015
1 January 2017
Works cited page
1. Taylor, Michael. Shakespearean Heroines. The health Anthology of Shakespearean
Drama. Edited by J.K. kipling, Oxford, 1999, pp. 77-78.
2. Howard, Don.
Mathematics 63 TWHFV
I. Find
Exercises on Inverse Circular Functions
dy
.
dx
1. y = 5cos
1
sec1 (2x ) e
3. y =
log(3x) tan1 x
x
4. y = sin1 x2
2. y = sec(ln(csc1 x)
tan1
x
II. Find the following antiderivatives.
1
dx
x 25x2 4
1
1.
2.
(1 + x2 ) 16 (tan1 x)
Mathematics 63 TWHFV Exercises
(Increasing and decreasing functions, relative extrema and the rst derivative test, concavity and the second derivative
test, sketching graphs of functions)
I. Determine whether the statement is true or false.
1. If f (x) <
Mathematics 63 TWHFV
Additional Exercises Set A (Second Long Examination Coverage)
I. Find the indicated derivative. There is no need to simplify your answer.
3
1. Dx [sin( x5 ) (7x + 2)4 csc(x)]
3x
d2 y
, where y = 9y 3 + tan
dx2
2
d
3.
(cos 4x) (Use the