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Functions Polynomial Functions: y=a0+a1x+a2x2+…+anxn Linear function: n=1 y= a0+a1x; a0: intercept. It is the y value when x=0. a1: slope, slope indicates the rate of change in y as x changes It is one of the most commonly used functions, including environmental modeling. For example, if assuming you are 100 miles on your way away from home, you are driving at 60 miles/hr, how far away you are from home 3 hours later. a0=100 a1=60 y= a0+a1x=100+60*3=280. Quadratic functions: n=2 y=a0+a1x+a2x2 Solution of a quadratic equation: ax2+bx+c=0 if (b2-4ac)>0, then the equation has a two different real solutions, x1 b b 2 4ac ; 2a x2 b b 2 4ac 2a if (b2-4ac)=0, then the equation has a two identical real solutions, x1 x 2 b 2a if (b2-4ac)<0, then the equation has no real solutions. Examples: x2-5x-6=0; -3x2+2x-5=0; x2-4x+4=0; Exponential Functions: y=ax y=ex=exp(x), where e=2.71828, the base of natural log. x a x 1 1 x a a ax+y =ax×ay a x y ax ay x ax a x b b (ax)y=ax×y (ab)x=axbx (a+b)2=a2+2ab+b2 (a-b)2=a2-2ab+b2 a2 – b2=(a-b)(a+b) Exponentiation: a n a a ... a (note: a a ... a ?) n n 0 a =1 1 a-1= a a1 a a 0.5 1 1 1 1 a a 1 2 a a Logarithmic Functions: y=loga(x), x=ay 100=102, thus, 2=log10100 (in Matlab: log10(100)) a=e, y=ln(x) log(xy)=log(x) + log(y) log(x/y)=log(x) – log(y) log(xc)=c*log(x) logaa=1 loga1=0 logay = log b y , e.g. log1001000000=log1000000/log100=6/2=3 log b a Trigonometric Functions: sin(x) cos(x) tan(x) ctan(x) A*sin(ωx+φ): A is amplitude, p=2π/ω, φ=phase. δ= 23.5×sin(2π*(284+Jday)/365), which as period of 365 day and phase of 2π*284/365. sin2(x)+cos2(x)=1 tan(x)=1/ctan(x) tan(x)=sin(x)/cos(x) sin( x y ) sin x cos y cos x sin y cos( x y ) cos x cos y sin x sin y sin 2x 2 sin x cos x Degrees and radiances: a circle is 360 in degrees and 2π in radiances, thus 1 degree=π/180 radiances 1 radiance=180/π ≈57.3 degrees Numerical Exercises 1. plot y=ax+b with x varies from -10 to 10: (1) where a=-2, -1, 0, 1, 2 and b=5 all in a single plot. (2) where a=1, b= -2, -1, 0, 1, 2 all in a single plot. (3) based on the two plots, describe how “a” and “b” control the line. 2. plot y=exp(ax): with x varies from 0 to 10: (1) a=0.1 (2) a=-0.1 (3) based on the two plots, describe how the sign of “a” control the line. 3. Solve x from the following quadratic equations: (1) x2+2x +1=0 (2) 5x2-3x+6=0 (3) 5x2-xx-6=0 (4) ax2+3mx+3tx2+2nx+10+c=0 2 (5) e 2 x x 10 4. Evaluate the following expressions without using a calculator unless asked. (1) Log(100) (2) log(200)-log2 (3) log100(1000000) (4) log5(5) (5) ln(e) (6) ln(10) (7) log(-1) (8) log(1.0) (9) log(4)+log(25) (10) a5+a3 (11) a5*a3 (12) a5/a3 (13) (1)-1 (14) (-1)2n , where n is a positive integer. (15) (-1)2n+1, where n is a positive integer. (16) sin(75) (17) sin(120) (18) Given sin(x)=0.3, what is cos(x)? Please use a calculator to figure out what the angle of x is in degrees? 2 (19) 2 3 2 (20) 2 3