PHY183-Lecture2 - Physics for Scientists &...

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Unformatted text preview: Physics for Scientists & Engineers 1 Spring Semester 2006 Lecture 2 - Math Primer January 11, 2006 Physics for Scientists&Engineers 1 1 Few Points from Monday LON-CAPA/ Clicker Registration/ DOCS? You will need a calculator. Final- Thursday, May 4, 20:00-22:00 Honors Option Questions? January 11, 2006 Physics for Scientists&Engineers 1 2 Schedule of this week... Today: Math Primer (algebra, trigonometry, geometry) Read appendix at the end of the book Read Chapter 1 of the book by tomorrow Wednesday, Thursday: Vectors Metric and British units, unit conversion Orders of magnitude in nature How to solve common physics problems January 11, 2006 Physics for Scientists&Engineers 1 3 Basic Algebra Products (commutative property) xy = yx Common factors (distributive property) ax + bx + cx = (a + b + c)x Squares (a + b) 2 = a 2 + 2ab + b 2 (a - b) 2 = a 2 - 2ab + b 2 (a + b)(a - b) = a 2 - b 2 To be known by heart January 11, 2006 Physics for Scientists&Engineers 1 4 Quadratic Equation An equation of the form ax + bx + c = 0 has the two solutions -b + b 2 - 4ac x= 2a -b - b 2 - 4ac x= 2a 2 The roots are real, if January 11, 2006 No real solution b 4ac Physics for Scientists&Engineers 1 5 2 Exponents If a is a real number and n an integer, then an means multiplying a with itself n times: a n = a a a L a n factors The exponent does not have to be an integer, but can any real number x be Rules: a =1 a =a January 11, 2006 0 1 a = x a -x a 1/ 2 = a a 1/ n =n a 1 ax x-y x y x +y =a a a =a y a for Scientists&Engineers Physics 1 (a x y ) = a xy 6 Logarithm Logarithm is inverse function to exponentiation y = a x x = log a y (x = logarithm of y with respect to base a). Two most common bases: 10 and e=2.7182... Notation: loge=ln Rules (valid for any base): a^ log ~ = log a - logb b January 11, 2006 log1 = 0 log(ab) = log a + logb log(a x ) = x log a Physics for Scientists&Engineers 1 7 Example 1 Simplify ln a - ln(b 2 /a) + 2ln(ba) + 2ln( a-2 ) = ln a - ln(b 2 ) + ln a + 2(lnb + ln a) - 4 ln a = ln a - 2lnb + ln a + 2lnb + 2ln a - 4 ln a = (1+ 1+ 2 - 4)ln a + (-2 + 2)lnb =0 January 11, 2006 Physics for Scientists&Engineers 1 8 Geometry A = area, V = volume, C = circumference January 11, 2006 Physics for Scientists&Engineers 1 9 Example 2 The radius of the earth is R=6.37x106 m. What is the surface area of the earth? A = 4pR2 A = 4pR2 = 4p (6.37x106 m)2 = 5.099043638x1014 m2 Too many Significant figures! Enter in LON-CAPA as 5.10E14 m^2 or 5.10E14 m*m January 11, 2006 Physics for Scientists&Engineers 1 10 Trigonometry (Right) triangles appear in many places in introductory physics problems It pays to remind ourselves of some basic trigonometry a opposite side sina = = c hypotenuse b adjacent side cosa = = c hypotenuse sin a a tana = = cosa b January 11, 2006 cosa 1 b cot a = = = sina tan a a Physics for Scientists&Engineers 1 11 Trigonometric Functions p 2p Small angles (a << 1): sina = tan a = a cosa = 1 Trig functions are periodic: sin(a + 2p ) = sin a cos(a + 2p ) = cosa tan(a + p ) = tan a cot (a + p ) = cot a Physics for Scientists&Engineers 1 12 2p p January 11, 2006 Pythagoras Right triangles: Pythagoras a2 + b2 = c 2 Same, using trig functions sin2 a + cos 2 a = 1 General triangles Law of cosines c 2 = a 2 + b 2 - 2abcos g In general, angle sum is 1800 a+b+g =p January 11, 2006 Physics for Scientists&Engineers 1 13 Derivatives Polynomials: Trig functions: Exponential, log: Product rule: Chain rule: January 11, 2006 d n x = nx n -1 dx d sin ( ax ) = a cos ( ax ) dx d ax e = ae ax dx d 1 ln ( ax ) = dx x d df (x) ^ dg(x) ^ g(x) + f (x) ( f (x)g(x)) = dx ~ dx ~ dx dy dy du y (u ( x)) fi = dx du dx Physics for Scientists&Engineers 1 14 Example 3 Let y = ax + bx + c dy =0 dx 2 At what value of x is y minimum? dy = d (ax 2 ) + d (bx ) + d (c ) = 2ax + b = 0 dx dx dx dx -b -(-3) fix= = = 0.3 2a 2(5) d2y >0 How do we know it is a minimum? 2 dx 2 d y d = 2ax + b) = 2a = 2(5) = 10 > 0 dx 2 dx ( What is the value of y at minimum? the y = 5(.3) - 3(.3) + 2 = 1.55 > 0 15 2 January 11, 2006 Physics for Scientists&Engineers 1 ...
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