Student Handbook
for MATH 263 C and D
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”I’m back! I’m back in the saddle again!” - Aerosmith
Department of Mathematics
Ohio University
August 2011
Contents:
1. Material You Should Know – p. 2
2. Syllabus for 263C – p. 5
3. Suggested Problems for MATH 263C – p. 5
4. Syllabus for 263D – p. 6
5. Suggested Problems for MATH 263D – p. 6
Course Web Site: www.math.ohiou.edu/courses/math263/
O.U. Matlab Web Site: www.math.ohiou.edu/courses/matlab/
Calculus 263 Coordinator: Todd Young, math263coord@math.ohiou.edu, 593-1277
Math Department Office: Morton 321, mathematics@ohiou.edu, 593-1254
1
Material You Should Know Before MATH 263 C or D
Math is cumulative. To help you succeed in C and D these pages summarize the material
from A and B. Your Math 263C and 263D instructor will take for granted that you know
and can use the following material. You should review it before starting the course and
before each test.
If you are new at Ohio U. or did not take MATH 263 A and B here you should obtain a
copy of the Student Handbook for MATH 263 A & B from the course web site and read it
carefully. If you already have a copy of the handbook for MATH 263 A and B then you
should review it now.
Pre-Calculus Material
As a general rule you should understand and be able to use all the material in reference
pages 1 - 4 in the inside front and back covers of the textbook. You should memorize most
of the formulas on those pages. Page 3 of the Student Handbook for MATH 263 A & B
contains more detailed information.
Conic Sections:
Standard form of conic sections. Know how to graph them:
(x − h)2 (y − k)2
(x − h)2 (y − k)2
+
= 1,
Hyperbola:
−
= 1,
Ellipse:
2
2
a
b
a2
b2
Parabola: 4p(y − k) = (x − h)2
Limits:
Meaning of Limits
Limits Laws
One-sided Limits
Infinite Limits
sin x
=1
Basic Trig. Limit: lim
x→0 x
Continuity:
Continuity means: limx→a f (x) = f (a). Polynomials, Rational, Algebraic, Exponential,
Logarithmic, Trigonometric Functions all are continuous where defined.
Definition of derivative:
Wherever it exists: f (x) ≡ lim
h→0
Differentiation rules:
Linear Combinations:
f (x + h) − f (x)
.
h
d
af (x) + bg(x) = af (x) + bg (x).
dx
h(x) = f (g(x)), =⇒ h (x) = f (g(x))g (x)
Compositions (Chain rule):
d
f (x)g(x) = f (x)g(x) + f (x)g (x).
Products:
dx
f (x)g(x) − f (x)g (x)
d f (x)
=
.
Quotients:
dx g(x)
g(x)2
Derivatives to memorize: Reference Page 5 formulas: 1-9, 11, 13-15, 17, 19, 21, 25-27
d
... = ... .
Implicit Differentiation: Means: Differentiate the whole equation:
dx
Assume y is a function of x, so when you differentiate y you get dy/dx.
Graphs:
Vertical Line Test
Know the graphs of functions in the reference pages.
Graphing by hand:
- Find zeros f (x) = 0.
- Find vertical asymptotes at places where f (x) is undefined.
- Check for horizontal and slant asymptotes.
Divide quotients p(x)/q(x) if degree(p) ≥ degree(q).
- Find f (x) and check for critical points.
Either f (x) = 0 or f (x) does not exist.
- Find f (x) and check for possible inflection points.
Either f (x) = 0 or f (x) does not exist.
- List critical and inflection points.
- Graph.
- Clearly label all features.
Linear approximations: For x ≈ x0 : f (x) ≈ L(x) = f (x0 ) + f (x0 )(x − x0 ).
Max-Min:
- Global vs. Local extrema
- Find critical points: f (x) = 0 or f (x) does not exist.
- On a closed interval [a, b], must also check f (a) and f (b).
- First Derivative Test: Check sign of f (x) on each side of the critical point.
- Second Derivative Test: Check sign of f (x) at the critical point.
Intermediate Value Theorem If f (x) is continuous on [a, b] and d is any real number
between f (a) and f (b), then there exists a point c, a < c < b, such that f (c) = d.
Max/Min Theorem If f is continuous on the interval [a, b], then f has a maximum value
and a minimum value on [a, b].
Mean Value Theorem If f (x) is continuous on [a, b] and differentiable on (a, b), then
there exists a point c, a < c < b, such that f (c) = (f (b) − f (a))/(b − a).
Definitions related to Integration:
F (x) is an Antiderivative of f (x) means: F (x) = f (x)
Riemann Sum - Rn , Ln and Mn are examples.
Definite Integral - The limit as n → ∞ of any Riemann sum.
b
1
f (x) dx
Average of a Function: favg =
b−a a
Integration Theorems:
b
If f (x) is continuous on [a, b] then a f (x) dx exists, however, it might not be expressible
in terms of elementary (usual) functions.
Fundamental Theorem of Calculus: If f is continuous, and F is an antiderivative of f , then
x
b
d
f (s)ds = f (x). Part 2:
f (x) dx = F (b) − F (a).
Part 1:
dx a
a
Riemann sums and numerical integration:
Let (x0 , x1 , x2 , . . . , xn ) be evenly spaced, a = x0 , b = xn , ∆x = xi − xi−1 = (b − a)/n
Let (y0 , y1 , y2 , . . . , yn ) be values of f (x), i.e. yi = f (xi )
n−1
yi = ∆x(y0 + y1 + . . . + yn−1 )
left sum - Ln = ∆x
i=0
n
right sum - Rn = ∆x
yi = ∆x(y1 + y2 + . . . + yn )
i=1
trapezoid rule - Tn = ∆x/2 (y0 + 2y1 + 2y2 + . . . + 2yn−1 + yn )
Simpson’s rule - Sn = ∆x/3 (y0 + 4y1 + 2y2 + 4y3 + . . . + 2yn−2 + 4yn−1 + yn )
midpoint sum/rule - Mn = ∆x (f (x̄1 ) + f (x̄2 ) + . . . + f (x̄n ))
where x̄i = (xi−1 + xi )/2, i.e. the mid-points of the intervals: [xi−1 , xi ].
L’Hopital’s rule:
(x)
(x)
limx→a fg(x)
= limx→a fg (x)
if the first limit is 00 or ∞
∞ and the 2nd limit exists.
0
∞
0
Change ∞ · 0 and ∞ − ∞ to 0 or ∞ . Use ln for 0 , ∞0 , 1∞
Integrals to memorize:
un+1
+ C, n = −1.
un du =
n+1
1
du = ln |u| + C
u
eu du = eu + C
cos u du = sin u + C
sin u du = − cos u + C
1
√
du = sin−1 u + C
1 − u2
1
du = tan−1 u + C
2
1+u
sec2 u du = tan u + C
sec u tan u du = sec u + C
sinh u du = cosh u + C
cosh u du = sinh u + C
f (x) + g(x) dx = f (x) dx + g(x) dx
Three major integration
techniques:
Substitution:
Recognize
f (g(x)) g (x) dx, set u = g(x), and get: f (u) du.
By parts:
u dv = uv −
v du, first identify dv that can be integrated.
Partial fractions: First reduce by dividing
A2
A1
An
···
+
−→
+ ... +
Real roots:
n
2
(ax + b)
ax + b (ax + b)
(ax + b)n
Ax + B
···
−→ 2
Complex roots:
ax2 + bx + c
ax + bx + c
Differential Equation of exp. growth/decay:
Arc Length:
Parametric: x = f (t), y = g(t), α ≤ t ≤ β: L =
β
Polar: if r = r(θ), L = α r 2 + (dr/dθ)2 dθ
β
α
dy
= ay and y(0) = y0 , ⇒ y(t) = y0 eat .
dt
f (t)2 + g (t)2 dt
2
4.6
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
Syllabus for 263C
–
–
–
–
–
–
–
–
–
Newton’s Method
Sequences
Series
The Integral and Comparison Tests
Other Convergence Tests
Power Series
Representations as Power Series
Taylor and Maclaurin Series
Applications of Taylor Polynomials
Brief Review of Conic Sections: See review
section of this handbook.
3
10.1 – 3-D Coordinate Systems
10.2 – Vectors
10.3 – The Dot Product
10.4 – The Cross Product
10.5 – Equations of Lines and Planes
10.6 – Cylinders and Quadric Surfaces
10.7 – Vectors Functions and Space Curves
10.8 – Arc Length and Curvature**
10.9 – Motion in Space
** – Skip Normal and Binormal Vectors.
Homework Problems for MATH 263C
Section
4.6
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
Conics
10.1
10.2
10.3
10.4
10.5
10.6
10.7
10.8
10.9
Problems
1, 3, 5, 6, 9, 21, 22: ML1 Newton’s Method
3-27 odd
3-27 odd
3-27 odd, 30-32
2, 3-17 odd, 18, 19-29 odd, 35, 37
3-19 odd, 25, ML2 Summation of Series
1-17 odd, 23-29 odd, 35
1-17 odd, 27-35 odd, 43-46, 47-53 odd, 59, 61, 63
9-19 odd, 21-22, ML3 Taylor Series
ML4 Plane Curves
1-10, 11-15 odd, 21-33 odd, ML5 Space Curves
1-3, 5-19 odd, 22, 23
1-3, 5, 7, 11, 13, 15, 16, 17-33 odd, 37
1-9 odd, 13, 15, 23-27, 29, 31
1-15 odd, 16, 17-39 odd
1-31 odd
1-15 odd, 17-22, 23, 33-51 odd, 57-61 odd
1-5, 11-19 odd
1-11 odd, 15-25 odd
4
Syllabus for 263D
11.1
11.2
11.3
11.4
11.5
11.6
11.7
11.8
–
–
–
–
–
–
–
–
Functions of Several Variables
Limits and Continuity
Partial Derivatives
Tangent Planes and Approximation
The Chain Rule
Directional Derivatives and Gradient
Maximum and Minimum Values
Lagrange Multipliers
12.2
12.3
12.4
12.5
12.6
12.7
–
–
–
–
–
–
Double Integrals On General Regions
Double Integrals in Polar Coord.
Applications of Double Integrals
Triple Integrals
Triple Ints. in Cylindrical Coords.
Triple Ints. in Spherical Coords.
13.1 – Vector Fields
13.2 – Line Integrals
12.1 – Double Integrals Over Rectangles
5
Homework Problems for MATH 263D
Section
11.1
11.2
11.3
11.4
11.5
11.6
11.7
11.8
12.1
12.2
12.3
12.4
12.5
12.6
12.7
13.1
13.2
Problems
1-21 odd, 25-35 odd, 41-46, 51; ML1 Functions
3-13 odd, 23, 25, 27, 29, 30; ML2 Contour Plots
3, 4, 7-23 odd, 37-40, 43-51 odd; ML3 Partial Deriv.
1, 3, 5, 11-14, 17, 25, 27, 31
1-7 odd, 17, 22, 23, 25, 27, 29
1-17, 22, 26. 31-34, 37, 38, 44; ML4 Gradients
1-13 odd, 23-27 odd, 31-43 odd
1-15 odd, 16, 17, 38, 39; ML5 Lagrange Multipliers
1, 3, 4, 7-31 odd
1-13 odd, 17-25 odd, 31-41 odd
1-6, 7-27 odd; ML6 Double Integrals
1-13
3-39 odd, 43, 45; ML7 Approximate Integrals
1-13, 15-27 odd
1-10, 11-27 odd, 35, 36, 40
1, 3, 11-18, 21-26
1-19 odd, 16, 27, 28, 33, 35, 37