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Honours Calculus at the University of Guelph, Winter 2007 to the present

Memories of Math 1200 & 1210

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plotx2

5K 3, x2 K 5, 0.5 sin t K 1, 0.75 cos t C 1, t =Kp ..p , 0.5 sin t C 1,K0.75 cos t

C 1, t =Kp ..p , x =K1.58 ..1.58, color = blue, axes = none

Interactive ClassesConnect the algebra and the picture

Solve 7$x = xK 4.Method 1: Cases

Method 1: CasesCase 1: 7$xR 0 Case 2: 7$x! 0

Case 1: 7$xR 0 Case 2: 7$x! 0

rxR 0 and 7$x = 7$x rx! 0 and 7$x =K7$x

Solve 7$x = xK 4 Solve K7$x = xK. After

After work work work, we get x =K23

work work work, we get x = 12

No solution, since xR 0. No solution, since x! 0.

Let's see what is really going on!

P1 d plot 7 x , xK 4, , x =K4 ..6, color = black, red , thickness = 2 :P2 d plot K7 x, x = 0 ..2, color = black, thickness = 2, linestyle = dash :P3 d plot 7 x, x =K2 ..0, color = black, thickness = 2, linestyle = dash :P d plots display P1, P2, P3 , labels = "", "" :P

Method 2: Square both sides (and check!)

7$x = xK 4r 49$x

2= x

2K 8$xC 16

Solving eventually gives x =K23

, x = 12

. Checking in the original equation

shows in both cases, LS≠RS r No solution. Again, let's SEE.plot 49$x

2, xK 4

2, x =K2.1 ..4.1, y =K1 ..30, color = black, red ,

thickness = 2

Learning by intimidation

x4$sin x

5K 3 dx

x5$sin x

5K 3 dx

Effective use of animation and a bonus bit of funcycloid:n. the curve described by a point on (or rigidly connected to) the circumference of a circle as it rolls without slipping on a straight line.

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with plots :

animate s$tK sin s$t , 1Kcos s$t , t = 0 ..1 , sC cos l , 1C sin l , l

= 0 ..2$p , sK q$sin s , 1K q$cos s , q = 0 ..1 , sK sin s K 3C 6

$ w, 1K cos s K3$ sin s

1K cos sC w$ 6$ sin s

1K cos s, w = 0 ..1 , s

= 0 ..9$p, frames = 150, thickness = 2, scaling = constrained, tickmarks

= 2, 2 , view = K3 ..9$p,K3.5 ..3.5 , color = blue

Bringing Negative Radius in Polar Coords to Life With a Special Maplet!

#Title: PolarAnimate #Author: Gord Clement, August 2010 #Description: For a given function and interval, this animates the function in polar coordinates. # The radial arm is animated as the animation moves through the interval.

# When the radius is negative, the function is plotted with dots as if the radius were positive

#Usage: #Call : PolarAnimate function, interval #function: polar function to be used for the animation #interval: interval to be used in the animation, entered

in standard Maple notation, ie [a,b] would be entered q = a ..b # Note, the variable used in the interval will be interpretted as q in the animation PolarAnimate d proc expr, range

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#local declarations local signOfFunction, var, minX, maxX, function, maxR; #function that is -1 is the given value is negative, 0 otherwise signOfFunction d x/piecewise x ! 0,K1 ; #extract variable and endpoints var d op 1, range ; minX d evalf op 1, op 2, range ; maxX d evalf op 2, op 2, range ; #calculate maximum radius length for length of radial armmaxR d maximize abs expr , range ; function d var/expr; #create animationplots animate plot, expr$cos var , expr$sin var , var = minX ..s , expr

$signOfFunction expr $ cos var , expr$signOfFunction expr $sin var , var = minX ..s , maxR$ t$cos s , maxR$ t$sin s , t = 0 ..1 , color = red, black, black , thickness = 2,4, 1 , linestyle = solid, dot, dash , s = minX ..maxX, scaling = constrained, frames= 100

end proc:PolarAnimate 3$sin 2$q , q = 0 ..2$PiPolarAnimate 1K2$ cos q , q = 0 ..2$PiPolarAnimate 1C 2$ sin q , q = 0 ..2$PiPolarAnimate sin 0.4$q , q = 0 ..10$Pi

PolarAnimate cos e$q , q = 0 ..20$Pi

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Resources for you!http://maplesoft.com/contact/webforms/CalculusKit.aspx

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Mobification!Go to Mobius