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Class Notes for Mathematics at University of Illinois

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MATH 221 Lecture Notes - Lecture 20: Function Composition
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MATH 221 - LECTURE #20 - DERIVATIVES The derivative of a function is a function. If the derivative f(a) of some function f exists for all a in the domain of f, then we have a new function: namely, for each nu...

Mathematics
MATH 221
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MATH 221 Lecture Notes - Lecture 15: Polynomial Long Division
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MATH 221 - LECTURE # 15 - REVISION FOR MIDTERM Note: The lecture was a revision for the Limits, Continuity , Asymptotes topics discussed previously. Glimpses: For the general limit-finding problems: ...

Mathematics
MATH 221
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MATH 221 Lecture Notes - Lecture 17: Product Rule
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MATH 221 - LECTURE #17 - DERIVATIVES PROOF OF PRODUCT RULE Let h(x) = f(x)g(x). To nd the derivative we must express the change of h in terms of the changes of f and g: h(x) h(a) = f(...

Mathematics
MATH 221
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MATH 221 Lecture 21: DERIVATIVES
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MATH 221 - LECTURE #21 - DERIVATIVES Dealing with equations of the form F1(x, y) = F2(x, y). If the implicit definition of the function is not of the form F (x, y) = 0 but rather of the form F1(x, y)...

Mathematics
MATH 221
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MATH 221 Lecture 19: DERIVATIVES
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MATH 221 - LECTURE #19 - DERIVATIVES f(x )0= lim h>0 f(0 + h) -0f(x ) / h (f(x) +- g(x)) = f(x) +- g(x) (fg) = fg + fg 2 (f/g) = f(g) - fg / g ...

Mathematics
MATH 221
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MATH 221 Lecture 18: DERIVATIVES
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MATH 221 - LECTURE #18 - DERIVATIVES REMEMBER THESE: 2 sinx cosx = sin 2x Cos x = (cos 2x + 1) / 2 2 Sin x = (cos2x - 1) / 2 PROBLEM 1: Let f(x) =...

Mathematics
MATH 221
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MATH 221 Lecture Notes - Lecture 16: Constant Function
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MATH 221 - LECTURE #16 - DERIVATIVES The derivative of any constant function is zero. Let k(x)= c be a constant function. Then we have k(x) = limh0 ( k(x+h) ...

Mathematics
MATH 221
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MATH 221 Lecture 14: REVISION OF LIMITS
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MATH 221 - LECTURE #14 - REVISION OF LIMITS NOTE: The lecture covered the revision of Limits for upcoming Midterm. HELP ON MIDTERM : (Source : Review Packet for Midterm 1 CoveringLimits and Continuity) NOTE: The fol...

Mathematics
MATH 221
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MATH 125 Lecture Notes - Lecture 36: Unit Vector
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MATH 125 Lecture 36 Orthogonal Projections Vectors: If vector u = <u, u, ...u>, then u is equal to the length of vector u 12n 22 2 u12..n+u+.+u A unit vector is a vector whose length is 1 Example: u=<1,0> 2 2

Mathematics
MATH 125
Lena Folwaczny
MATH 125 Lecture 35: Selecting a Basis from a Set of Spanning Vectors
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MATH 125 Lecture 35 Selecting a Basis from a Set of Spanning Vectors Basis for a Subspace V: It is a finite set of vectors {v, .., v} if: 1n The set spans V The set is linearly independent Example: R= <1,0> and R=<0,1> 12 ...

Mathematics
MATH 125
Lena Folwaczny
MATH 221 Lecture 27: Physics and Calculus
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MAT221Lecture 27 Physics and Calculus Work= Force x Distance What happens when force is not constant? We have to use inte

Mathematics
MATH 221
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MATH 125 Lecture Notes - Lecture 34: Linear Independence, Linear Combination, Augmented Matrix
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MATH 125 Lecture 34 Linear DependenceIndependence and Basis Selection Linear Dependence: Let S={v,v,....,v} 12n S is linearly dependent if there exist constants c,c,....,c not all zero such that 12n cv+cv+....+cv=0 1122nn ...

Mathematics
MATH 125
Lena Folwaczny
MATH 125 Lecture 33: Review for Exam
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MATH 125 Lecture 33 Review for Exam Example 1: Compare the corresponding exact stable vector This is not possible because the question does not make sense Replacing the bottom row with 1s is only for stochastic ones T is n...

Mathematics
MATH 125
Lena Folwaczny
MATH 221 Lecture 26: Shell Method of Volume Approximation
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MAT221 Lecture 26 Shell Method of Volume Approximation In this method, we add pi, t

Mathematics
MATH 221
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MATH 125 Lecture Notes - Lecture 32: Contraposition, Scalar Multiplication, Null Character
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MATH 125 Lecture 32 More on Subspaces Zero Vector: All subspaces must contain the zero vector If it is a subspace, it contains the zero vector If it doesnt contain the zero vector, it is not a subspace (p q) = (q p) (p q) ...

Mathematics
MATH 125
Lena Folwaczny
Statistics - Reference Guides

This Guide provides an easy to follow summary of all basic principles that are the foundation to Statistics and Probabilities. Using a combination of charts and sample equations, the key concepts are developed and the esse...

Econometrics
ETC2430
All Professors
Calculus Derivatives - Reference Guides

This advanced Calculus Guide provides assistance to the user through its articulation of key Derivatives principles. Helpful tips to solve exponential and logarithmic functions are featured.

BAD - Business Administration
BAD 200
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Permachart - Marketing Reference Guide: Becquerel, Hyperbolic Function, International Standard Book Number

l e a r n • r e f e r e n c e • r e v i e w permacharts TM CCalculus Integrals II INTEGRALS INVOLVING a+bu & p+qu ...

BAD - Business Administration
BAD 200
All Professors
College Mathematics - Reference Guides

This comprehensive Guide provides concrete reference materials in three math fundamentals – Algebra, Calculus, and Trigonometry. Helpful sample equations and rules are the backbone to this Guide

BAD - Business Administration
BAD 200
All Professors
Permachart - Marketing Reference Guide: Inflection Point, Inflection, Farad

l e a r n • r e f e r e n c e • r e v i e w permacharts TM Differential Calculus Differential Calculus ...

BAD - Business Administration
BAD 200
All Professors
Permachart - Marketing Reference Guide: Iterated Integral, Centroid, Multiple Integral

permacht asrTM integral Calculus theorems types of iNtegrals averaGe value Theorem ...

BAD - Business Administration
BAD 200
All Professors
Laplace Transforms - Reference Guides

This useful Math Guide is a specific reference for techniques to convert ordinary differential equations to algebraic equations. Helpful charts and summaries are used to convey information such as exponential, square root,...

BAD - Business Administration
BAD 200
All Professors
Permachart - Marketing Reference Guide: Charge Pump, Diode, Electrolytic Capacitor

l e a r n • r e f e r e n c e • r e v i e w permacharts TM CConductors & Semiconductors BASICS TERMS Conductor Valence and conduction bands for electr...

Electrical Engineering
4400:307
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Permachart - Marketing Reference Guide: Steradian, Luminous Intensity, Si Base Unit

l e a r n • r e f e r e n c e • r e v i e w permacharts TM Metric System ...

Basic Engineering
BE 1101
All Professors
Intermediate Algebra - Reference Guides

This more advanced Guide builds upon the reference material set out in the Introductory Algebra Guide. Linear equations, polynomial equations and complex numbers are readily explained; clear and easy to follow emphasis is ...

MATH
MATH 504
All Professors
Introductory Algebra - Reference Guides

This Guide provides a concise and very readable summary of all key Algebraic principles. Special attention is directed to terminology, laws, and operations, with practical examples developed as illustration of these princi...

MATH
MATH 504
All Professors
Permachart - Marketing Reference Guide: Circumscribed Circle, Trapezoidal Rule, Quadrilateral

permacharts TM Geometry Geometry QUADRILATERALS CIRCLES & CONICS...

MATH
MATH 501
All Professors
Permachart - Marketing Reference Guide: Cubic Yard, Nautical Mile, Fluid Ounce

l e a r n • r e f e r e n c e • r e v i e w permacharts TM Weights & Measures THE U.S./IMPERIAL SYSTEM OF MEASUREMENTS ...

Physics
PHYS 24300
All Professors
Vectors & Matrices - Reference Guides

Vectors are an important area of study that develops naturally from a basic understanding of physics and is subsets – statics, dynamics, and mechanics. This practical Guide provides assistance in an often complex disciplin...

Physics
PHYS 24300
All Professors
Permachart - Marketing Reference Guide: Luminous Flux, Magnetic Flux, Magnetic Flux Quantum

permacharts TM F u n d a m e n t a lP h y s ic a lC o n s t a n t s General Constants Quantity...

Physics
PHYS 24300
All Professors

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