M-1000
Topics and techniques covered by the Final Exam
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0. Precalculus techniques. Please pay attention to:
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Brackets, parentheses
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Addition (subtraction) of fractions.
Wrong:
" (a+b)/(c+d)=
a/c + c/d "
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Algebraic identities, such as
x3-y3=(x-y)
(x2+xy+y2)
See back of the Textbook, Special Factors
and Binomial Theorem.
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The Quadratic Formula
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Trigonometric identities (See back of the Textbook)
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Simplification of expressions: similar terms, cancelations.
Do not wrongly oversimplify. Avoid meaningless
"simplifications" like this
Wrong: " (sin x)/ x= sin
"
(Now think: sin of what?
-- there is no way to compute it, therefore we got something that doesn't
make sense!)
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Do not jeopardize on additivity:
it is not generally true that
"
sin (x+y)=sin(x)+sin(y) ",
etc. Yet many students tend to use such "rules" in a rush.
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Identities with powers and exponents (See back of the Textbook)
Remember:
roots and inverse powers are particular cases of the
power function xn ,
here n
is a constant.
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Identities involving logarithms. (See handout on Logarithmic Identities)
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Concepts of a constant, an independent variable, a parameter.
The same notation can stand for either of these in different
contexts.
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Concept of a function. The main thing to remeber is that a function
f(x) cannot take more than one value at given x.
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Geometry: area of a triangle (not necessarily a right triangle),
Pithagorean Theorem, areas of a trapezoid, a circle;
Volumes of a cube, a cone, a cylinder, a sphere.
(See Textbook, inside front page).
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1. Limits (including infinite limits)
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Substitution (should be attempted first in most cases, cf. Text, Sect.1.3,
pp. 75-77)
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Divide-out (Text, p.79)
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Rationalizing technique (p.80)
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Special limits: as x->0 ,
(sin x / x) ->
1
(exp(x)-1) / x -> 1
(cos x-1) / x -> 0
(Th.1.9, p.81)
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Infinite limits and vertical asymptotes (Sect.1.5)
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2. Find Derivative by Definition (Sect.2.1, p.115 and on)
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3. Continuity (Sect.1.4)
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Removable/non-removable discontinuities
Possible sources of discontinuity
denominator = 0 (in rational fractions as well as in trigonometric
functions like
tan x = sin x / cos x ,
sec x = 1 / cos x , csc x = 1 / sin x
piecewise-defined function
log |x|
at x=0
and, in general,
logarithm of a non-negative function f(x)
at points where
f(x)=0 .
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4. Differentiation using rules
&nvsp; (see summary in Sect.2.4, p.155; Handout)
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Sum, Difference, Product, Quotient.
(Sect.2.2, Th.2.5 and Sect.2.3, Ths.2.8, 2.9)
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Very basic rules:
const'=0 , x'=1.
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Power rule (derivative of
xn), including roots
and negative powers (Th.2.3, p.124, and Th.2.12, p.148)
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Chain rule (Sect.2.4)
[f(g(x)) ]' = f'(y) g'(x) where
y=g(x) .
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Derivatives of the functions:
ex, sin x, cos x, tan x, cot x,
sec x, csc x,
ln x, ln|x|
(See front page, also pp.128-129)
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Logarithmic Differentiation (pp.153 and 166; Handout)
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5. Equation of tangent line at given point to the graph of
y=f(x) (pp.113-114)
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(y-y0) / (x-x0)= m
(slope),
= f'(x0)
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6. Implicit Differentiation
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(Sect.2.5, p.161; algorithm on top of p.162)
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7. Related rates
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(Sect.2.7, p.175; algorithm on p.176)
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8. Accelerated motion (p.142, also Excercises 103-105 on p.145)
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9. Optimization (Sect.3.7, algorithm on p.251)
A notion of target function (notes).
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13. Elements of curve sketching (Sect.3.6; also review
midterm problems 6,7, worksheet on graphing, and similar problems
in earlier final exams.)
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Behaviour at infinity. (Positive or negative growth, or existence
of limits(s) as
x-> infinity; horizontal asymptotes.)
(Sect.3.5, esp. p.231 - Hor.As.; p.232 - Rational functions;
p.236 - infinite limits at infinity).
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Intercepts, Critical values, Inflection Points.
(Cf. tables in examples of Sect.3.6)
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Intervals of monotonicity and intervals of concavity.
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Fit parameters of a curve of a given type to satisfy certain conditions.
(Review examples in notes: Cubic curve (Tue Apr.8), Parabola (Wed Apr.9))
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10. Curve sketching (Sect.3.6)
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a) sketch a graph of a given function
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b) graph of a function satisfying given condition &npsp;&npsp;
(points, limits at infinities, asymptotes, critical value / horizontal
tangent, intervals of monotonicity and/or concavity).
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12. Integration
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Indefinite integrals (= antiderivatives) (Sect.4.1)
Don't forget +C
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Definite integrals (Th. 4.9, p.309)
Constant C isn't needed
(it cancels anyway).
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13. Area
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To find area between two given curves
y=f(x) and
y=f(x) ,
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1) Solve the equation f(x)=g(x)
to find two solutions x=a
and x=b.
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2) Evaluate the definite integral
from a to
b
of f(x)-g(x) .
M-1000-5 Homepage