PHY183-Lecture02 - S c h e d u le fo r t h is w e e k … !...

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Unformatted text preview: S c h e d u le fo r t h is w e e k … ! Today: • Math Primer (algebra, trigonometry, geometry) • Read appendix at the end of the book Physics for Scientists & Engineers 1 Fall Semester 2006 Lecture 2 - Math Primer ! Read Chapter 1 of the book by Tomorrow ! Wednesday, Thursday: • • • • 1 Vectors Metric and British units, unit conversion Orders of magnitude in nature How to solve common physics problems Physics for Scientists&Engineers 1 2 August 29, 2006 Physics for Scientists&Engineers 1 August 29, 2006 F e w P o in t s fr o m M o n d a y ! LON-CAPA OK? ! Calculators for exams. ! Homework? Learning Center ! DOCS? ! Read Chapter 2 up to page 32 by Thursday B a s ic A lg e b r a ! Products (commutative property) ! Common factors (distributive property) xy = yx ax + bx + cx = (a + b + c) x ! Squares (a + b )2 = a 2 + 2 ab + b 2 (a ! b )2 = a 2 ! 2 ab + b 2 (a + b )(a ! b ) = a 2 ! b 2 You should know these relations by heart August 29, 2006 Physics for Scientists&Engineers 1 3 August 29, 2006 Physics for Scientists&Engineers 1 4 Q u a d r a t ic E q u a t io n ! An equation of the form E x p o n e n ts ! If a is a real number and n an integer, then an means multiplying a with itself n times: ax 2 + bx + c = 0 ! has the two solutions a n = a ! a"a ! .."a !"!#". ! $ ! The exponent does not have to be an integer, but can be any real number x ! Rules: No real solution otherwise n factors x= !b + b ! 4 ac 2a 2 ! The roots are real, if 2 !b ! b 2 ! 4 ac x= 2a b ! 4 ac a0 = 1 a =a 1 5 August 29, 2006 a! x = x y 1 ax x+y a=a 1 2 a =na 1 n a a =a ax = ax! y y a (a ) xy = a xy 6 August 29, 2006 Physics for Scientists&Engineers 1 Physics for Scientists&Engineers 1 L o g a r it h m ! Logarithm is inverse function to exponentiation In t e g r a t io n ! P o ly n o m i a ls : ! T r ig f u n ct ion s: ! E x p o n e n t i a l, lo g : ! P r o d u c t r u le : ! C h a i n r u le : 7 August 29, 2006 y = a x ! x = log a y (x = logarithm of y with respect to basis a). ! Two most common bases: 10 and e=2.7182… ! Notation: loge=ln ! Rules (valid for any base): dn x = nx n !1 dx d sin ( ax ) = a cos ( ax ) dx d ax e = aeax dx d 1 ln ( ax ) = dx x log 1 = 0 log(ab ) = log a + log b ! a$ log # & = log a ' log b " b% August 29, 2006 d df ( x ) $ ! dg( x ) $ ( f ( x )g( x )) = ! # & g( x ) + f ( x ) # " dx % " dx & % dx y (u ( x)) ! dy dy du = dx du dx 8 log(a x ) = x log a Physics for Scientists&Engineers 1 Physics for Scientists&Engineers 1 G e o m e try A = area, V = volume, C = circumference T r ig o n o m e t r y ! (Right) triangles appear in many places in introductory physics problems • It pays to remind ourselves of some basic trigonometry sin ! = a opposite side = c hypotenuse cos ! = tan ! = August 29, 2006 Physics for Scientists&Engineers 1 9 August 29, 2006 b adjacent side = c hypotenuse sin ! a = cos ! b cot ! = Physics for Scientists&Engineers 1 cos ! 1 b = = sin ! tan ! a 10 T r ig o n o m e t r ic F u n c t io n s 2! P y th a g o ra s ! Right triangles: • Pythagoras • Same, using trig functions ! Small angles (! ! 1): sin ! = tan ! = ! cos ! = 1 a2 + b2 = c2 Trig functions are periodic: sin 2 ! + cos 2 ! = 1 ! General triangles • Law of cosines 2! ! sin (! + 2" ) = sin ! cos (! + 2" ) = cos ! tan (! + " ) = tan ! cot (! + " ) = cot ! 11 c 2 = a 2 + b 2 ! 2 ab cos " • In general, angle sum is 1800 ! +" +# =$ August 29, 2006 Physics for Scientists&Engineers 1 12 August 29, 2006 Physics for Scientists&Engineers 1 ...
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