Summary

Summary Advanced Calculus Mathematics 214

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Course
Mathematics 214

Institution
Stellenbosch University (SUN)

Book
Calculus: Early Transcendentals

Digitally summarised advanced calculus for Mathematics 214 Stellenbosch University. These notes are detailed and comprehensive to assist you in your studies. Detailed theory and examples included. Textbook used: Calculus: Early Transcendentals (Daniel K. Clegg, James Stewart, and Saleem Watson) ...

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Summarized whole book? No
Which chapters are summarized? 13, 14, 15, 16
Uploaded on June 24, 2023
Number of pages 89
Written in 2022/2023
Type Summary

mathematics 214

Book Title:Calculus: Early Transcendentals

Author(s):James Stewart, Daniel K. Clegg, Saleem Watson

Edition:2020
ISBN:9780357631478
Edition:Unknown

Class notes
Antiderivatives
Summary
Early Transcendentals Appendix A, B, C
Summary
Chapter 1 Calculus: Early Transcendentals

Institution
Stellenbosch University (SUN)
Course
Mathematics 214

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advanced calculus
⑧
12.6. Cylinders and quadric surfaces

cylinders
A cylinder is a surface that consists of all lines (called rulings) that are parallel to a given line and pass
through a given plane curve.

Sketch the the surface x
e.g. graph of z =

does notinvolve
equation y
·

·

any vertical plane with
equation y = K intersects the

graph in the curve moves
->

parallel along xz-plane
·
surface name: parabolic cylinder
rulings of cylinder parallel to
y-axis
·
are

if variable the surface is
rulings parallel to * one
(/y(z) is
missing, a

y-axis and
pass cylinder
through curve

e.g.(2 y2 + 1
=

·
circle with radius 1(z
of =k)

·

parallel to
cy plane

2
5.9.y2 z
+
1
=

radius k) NB:both these
circle with of 1(x
equations representa cylinder,
·
*
=

parallel this cylinder is
to
yz-plane Not
a circle. The trace of
·

a circle with z 0.
=

, surfaces,
quadric have the same characteristics as conic sections

A quadric surface is the graph of a second-degree equation in three variables x, y, and z. The
most general such equation is:

* Ax2
By (z2 Dxy Eyz fxz 6x Hy Iz 5 0
+
+
+ + + + +
+ + =

where A-5 are constants, butby translation and rotation can be boughtinto
I standard forms:

I
Ax2 Byz (z 2 5 0 or where C
A, B, O
· + +
=
+
=

Ax2 By2 Iz 0
+
· + =

e.g. Use traces to sketch the
quadric surface:( + YE +

=

· substitute z 0 = trace
=> in cy plane is x+
Y 1)
=

Ellipse
Horizontal in z K is:
trace plane
·
=

x2 y) 1
k2; z
k) ellipse provided 1
k 0 -> 12 4
- -

<
+ =
=

9 ↳
-> -
2<k < 2

and ellipses:
·vertical traces
parallel to
yz xz
planes are also

y2 z
+
1
=
-

k2;x kif = -
1ckx )

9 H

x2 z2 k2 if 3xk< 3
1
i( k
-

t
-
=
=

4 4

Z vertical:yz Z vertical:xz
Y Horizontal:
M xy M M

3 2

2

>x
>x
-y
C C C
- I I -
3 3 -
I I

-
2
-
3
2

W W W

combine these form ellipsoid (all in IR5:
we can traces to an 3 traces are
ellipses)

s
each trace is symmetric to each coordinate plane,
because there are only even
powers of and Z
x, y

, surface:(another def.)
TRACE
ofa
quadric
curve obtained by taking an intersection a
of
plane parallel to a coordinate plane (cy/zy/yz
plane) and the surface.
quadric

e.g. Determine the shape of the surface

[(1,4,z) G(R3: y2
+

z)
=

solution set

solution

① Find the traces where z =

K, KEIR Horizontal trace:Ellipse

vertical trace:Parabola

consider?+Y=K in a
plane (2D) <
Elliptic parabaloid

obtain solutions. know itdoes the
If
so, we no
So, we cross
not
negative Z axis.

k
If 0,x y
=
=
0
=
is the
only solution. y
a

If k >0:x2 yz
+ k
=

cy-plane
16

x2 k gets larger, ellipse
Y
- + =I AS

16k grows
I

() x2 +
y2 = j >x
L
I
(4) (r)
2
0
-
L

~
I

How does the ellipse grow? N

2
xz
plane x
y2 z
+ =
-

M
k2 = 116
k1
plane Z =

yz
-

A
N A
1 Set K
a y
=

Let x
k,
= then if k 0
=

z x2
2
1
y2
+
z
= =
I0 =

16 16
R
k 'x
=
ak 0 =
L -
>x
R

ifk 0
=

..
2 W
z
y
=

if k 1
=

↑ ·

=z
i y
+

W

c

, revision ofconic sections (10.5)

intersection a
of cone with a plane

⑪se
x2 + y2 = I a vertices: I a

92 b2
and Ib

⑭erbola xand
(*and-y:
+
y:
-

3
x -
1
=
>vertices: I a

92
assymptotes: x x

⑳bola
=
x2 OR Y2:AKI OR c2 =

dy>
b2
a = Lip
focus

Use traces to sketch z 4x2 yz
e.g.
+
=

·
substituting =
0, we
get: y2 = z

the
Thus, a
parabola parallel to
ccy-plane
·
FOr x K: =

z
412 z slice the parallel the yz-plane,
y => if we
graph with
any plane
to
+

2
=

yZ
have direction.
we a
positive parabola opening up in the upward
FOr y K:

xz/z
=

=4x + k2
parabola
= that
opens upward (steeper)
·
FOr z K:
=

cy( y2 4=
+ k ellipse
=
iff k>0

e.g.

·
fOr
Sketch

x K:
=
z
yz
=
-

FOr
x2

z K!= FOr
y K:
=
-pr
z =
yz
-

k2 k yz
=
-
x2 z x2
=
-
x2

42
a A xy xZ
yperbolic paraboloid

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