Math 2414 Exam 2 Review Answers
1.
2.
3.
1
1
1
x e dx = 4 x e 8 xe + 32 e + C
sec d = tan + ln cos + C
2 4x
2 4x
4x
4x
2
1
1
x sin x cos x dx = 4 x cos 2 x + 8 sin 2 x + C
3 3x
2
e cos 2 x + sin 2
Math 2414 Exam 2 Review
Evaluate the following integrals.
1. x 2 e 4 x dx
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
sec d
x sin x cos x dx
e cos 2 x dx
x ln x dx
sec x ln ( tan x + 2 ) dx
cos 20 x dx
sin
Formulas for test 2
The equation of the tangent plane at point P (x0 , y0 , z0 )
given expliit equation of surfae z = f (x, y)
z z0 = fx (x0 , y0 )(x x0 ) + fy (x0 , y0 )(y y0 ).
given impliit equat
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MATH 2415 Review for tat 4
10.
1.
2
3
4
5
6.
7
8
9
Sections: 14.1-14.8 FmM/cfw_m WI 0% a Law 83 W 'A 5
Concepts:
A
FYtl 5" View (a; C" J 04 he 1
vector elds, graphical representation, st
MATH
2415
Take-home quiz
Solutions
#5
f
f
x
1. Let f (x, y) = sin
. Calulate
and
.
1+y
x
y
x
1
fx (x, y) = cos
1 + y 1 + y
x
x
fy (x, y) = cos
.
1+y
(1 + y)2
2. Find the four seond partial derivatives
. Form A Test 2, Math 2415
Mrs
Name _ _. _ Seat
Show all work and use correct mathematical notation. l
2
my . . . _ . .
1. Let x, 3;) = m. Evaluate the 11m1t (3131(09) x, 3) on the paths.
(a) y = m
MATH
2415
Take-home quiz
Solution
#9
1. Find the volume of the solid bounded by the paraboloid z = x2 + y2 and the plane z = 9.
Solution. The intersetion urve between the paraboloid and the plane is t
MATH
2415
Take-home assignment
Solutions
#1
1. Let v = h2, 1i. Express v in terms of the basis unit
vetors i and j; then represent the vetors
y
v, v, 2v
2v
in the given system of oordinates in standar
'Pmewhsb
MATH 2415 Review for test 3
Sections: 13.113.7, 14,4-
Concepts:
1. double and triple integrals and their use in computing areas and volumes
2. changing order of integration in iterated integr
MATH 2415
Review for test 4
Setions: 14.1-14.8
Conepts:
1. vetor elds, graphial representation, streamlines
2. line integrals of salar funtions
3. line integrals of vetor elds and their interpretation
Numerical integration. Simpsons method
Using the right Riemann sum for f on the interval [a, b] with x =
!b
f (x) dx
y = f (x)
y = f (x)
y
xk
a
b
f (xk ) x
k=1
a
y
n
"
ba
we have
n
x
xk
a
y = f (x)
y
Evaluating areas using random numbers.
Basic probability principle: Given a sample space with equally likely outcomes (like heads and tails
when flipping a coin) the probability of an event (a set of
BriggsCochran_SGs
4/9/10
7:44 AM
Page 8
Briggs/Cochran
Calculus: Multivariable Study Card
Vectors and the Geometry of Space
Three-Dimensional Coordinate Systems
The distance between P1(x1, y1, z1) and
RENDER NOT INCLUDED IN QUOTE.
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ALL DIMENSIONS TO BE CONFIRMED ON SITE
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PREFABRICATING WORKS.
Student Study Session
Calculus with Graphing Calculators Solutions
Multiple Choice Solutions
(Note: When justifying answers, do not write calculator syntax on paper. Calculus notation
must be used on
Differentiation of Exponential and Logarithmic Functions
MODULE - V
Calculus
23
Notes
DIFFERENTIATION OF EXPONENTIAL
AND LOGARITHMIC FUNCTIONS
We are aware that population generally grows but in some
Maxima and Minima
MODULE - V
Calculus
25
Notes
MAXIMA AND MINIMA
You are aware that in any transaction the total amount paid increases with the number of items
purchased. Consider a function as f ( x
Form A. Test 3, Math 2415
SM'Ws
Show all work and use correct. mathematical notatiOn.
he: rm
1. Set up an iterated integral that gives'the volume of the: bounded by f (3:1 3;] = 2:5 + 3; over the
_
2:
MATH 2415
Take-home quiz 4 : Riemann sum approximations for line integrals
The work done by F along 0 is given by fc F - dr.
Assume C has parametrization given by r(t) with t in cfw_11,11. An approxim
Math 2414 Exam 1 Review Answers
dy
dx
dy
2.
dx
dy
3.
dx
dy
4.
dx
1.
= 2 cosh
x
3
= tanh x
= x
1
x3
= 3sech ( 3 x 4 ) tanh ( 3 x 4 )
ln 2
5.
1
17
tanh 2 x dx = 2 ln 8
0
ln 2
6.
4e
sinh d = ln 4
0
7.
Math 2414 Exam 1 Review
Find the derivative of the following functions. Simplify your answers where possible.
x
1. y = 6 sinh
3
2. y = ln cosh x
3. y = cosh ln x 2
4. y = sech ( 3 x 4 )
Evaluate the f
MATH 2415
Average distane
1. The irle r = 2 sin has radius 1 and is entered at (0, 1). Set up an integral to nd the average
distane of the points on the irle to the x-axis.
y
The irle with enter at (0
Math 2415
The xyz -oordinate system.
z
This system is reated by adding a new axis,
alled the
z -axis
We will be
to the
xy -oordinate
working with the
oordinate system:
system.
right-handed
if the nger
The roller oaster urve
Consider the paraboloid given by equation
f (x, y) = x2 + y 2
A problem we an solve is to nd the extrema of
f
given the onstraint that the points
(x, y)
are
on the ellipse
x2 +
Finding the distane from a point to a Line
The problem of nding the distane from a point to a line an be solved by one of the
following methods:
1.
2.
3.
4.
as an optimization problem;
by nding a veto
Distane from a point to a line
Reall the ross produt property
|u v| = |u|v| sin ,
0
u and v. This leads to the fat that the area of the
1
triangle formed by u and v is
|u v| sine the height h of the t
Change of variables:
!
S
f (x, y) dxdy =
!
D
f (T (u, v)
(x, y)
dudv.
(u, v)
The idea of linear approximation from one variable functions is at the basis of the formulas for change of variables.
We en
Center of the rst otant of solid sphere with uniform density
4a3
.
We know that the volume of a sphere of radius a is
3
Let E be the portion of the sphere that lies in the rst otant.
E = cfw_(x, y, z)
6. Consider the surfae given by z = f (x, y).
Use the following notation for the partial derivatives of f evaluated at (x0 , y0 )
p=
f
(x0 , y0 ) ,
x
q=
f
(x0 , y0 ) .
y
(a) Desribe a normal vetor to
MATH
2415
# 16
Worksheet
1. Let A(2, 3) and B(1, 2). Find AB and BA.
2. Give a geometri desription of the following sets of points:
x2 + y 2 + z 2 2x + 6y 6
3. Let a = 3i j + 4k and b = 5i + j, ompute