restart: This Maple Worksheet provides an example of the solution to the Maple Computer Project #2 for Math 152 Spring 2005 using a different energy gain function from the one in the project. You will have to modify the below to define the functions f, fgain, fsolve and averagegain, and use the value of c calculated based upon your birthdate. It is assumed that for the below, the birthdate is halfway through the year, so that c=10*.5 = 5 First set up the function to take the derviative of (equation 1) f:=t->K*(1-exp(-c*t^2))/(t+m); NiM+SSJmRzYiZio2I0kidEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKihJIktHRiUiIiIsJkYuRi4tSSRleHBHNiRJKnByb3RlY3RlZEdGM0koX3N5c2xpYkdGJTYjLCQqJkkiY0dGJUYuOSQiIiMhIiJGO0YuLCZGOUYuSSJtR0YlRi5GO0YlRiVGJQ== Next, take the derivative of this and simplify it diff(f(t),t); NiMsJiosSSJLRzYiIiIiSSJjR0YmRidJInRHRiZGJy1JJGV4cEc2JEkqcHJvdGVjdGVkR0YtSShfc3lzbGliR0YmNiMsJComRihGJ0YpIiIjISIiRicsJkYpRidJIm1HRiZGJ0YzRjIqKEYlRicsJkYnRidGKkYzRidGNCEiI0Yz simplify(%); NiMqKEkiS0c2IiIiIiwqKihJImNHRiVGJkkidEdGJSIiIy1JJGV4cEc2JEkqcHJvdGVjdGVkR0YvSShfc3lzbGliR0YlNiMsJComRilGJkYqRishIiJGJkYrKipGKUYmRipGJkYsRiZJIm1HRiVGJkYrRjRGJkYsRiZGJiwmRipGJkY2RiYhIiM= Next, since we wish to find the critical points to find the maximum of equation (1), and the denomenator cannot =0, take the numerator numer(%); NiMqJkkiS0c2IiIiIiwqKihJImNHRiVGJkkidEdGJSIiIy1JJGV4cEc2JEkqcHJvdGVjdGVkR0YvSShfc3lzbGliR0YlNiMsJComRilGJkYqRishIiJGJkYrKipGKUYmRipGJkYsRiZJIm1HRiVGJkYrRjRGJkYsRiZGJg== simplify(%=0); NiMvKiZJIktHNiIiIiIsKiooSSJjR0YmRidJInRHRiYiIiMtSSRleHBHNiRJKnByb3RlY3RlZEdGMEkoX3N5c2xpYkdGJjYjLCQqJkYqRidGK0YsISIiRidGLCoqRipGJ0YrRidGLUYnSSJtR0YmRidGLEY1RidGLUYnRiciIiE= and this is the answer to part a of the project Next, define a procedure which will find the critical points by setting this =0 and solving numerically (since this is a trancendental equation) for the optimal t*. Note that since K is a constant it does not need to be included in the below and is thus ignored from now on. Simply copy and paste the expression above into the below where appropriate, skipping the K mysolve:=proc(c,m) fsolve(2*c*t^2*exp(-c*t^2)+2*c*t*exp(-c*t^2)*m-1+exp(-c*t^2)=0,t,0.01..20); end proc; NiM+SShteXNvbHZlRzYiZio2JEkiY0dGJUkibUdGJUYlRiVGJS1JJ2Zzb2x2ZUc2JEkqcHJvdGVjdGVkR0YtSShfc3lzbGliR0YlNiUvLCoqKDkkIiIiSSJ0R0YlIiIjLUkkZXhwR0YsNiMsJComRjNGNEY1RjYhIiJGNEY2KipGM0Y0RjVGNEY3RjQ5JUY0RjZGPEY0RjdGNCIiIUY1OyRGNCEiIyIjP0YlRiVGJQ== Note that the above assumes the optimum t* will occur before time 20. Now to plot the graph for part b, define a new function for this using the value of c for the appropriate birthdate partb:=m->mysolve(5,m); NiM+SSZwYXJ0Ykc2ImYqNiNJIm1HRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJS1JKG15c29sdmVHRiU2JCIiJjkkRiVGJUYl and graph this over the given fange of m values plot(partb,1..20); 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For Part c, do a 3-d plot using a range of c and m values, and then rotate it so that you can see the graph in a manner that best visualizes it (so you can see the scales on all the axes) plot3d(mysolve,0..10,1..20,axes=BOXED); 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For part d, we first need to define a function for the instantaneous rate of gain of food in a patch and be able to evaluate it at the optimum time to leave a patch t*, and then compare this with the long-term avaerage rate of gain over time, which is equation 1 evaluated at t*. So first the food gained up to time t in a patch is fgain:=(K,c,t)->K*(1-exp(-c*t^2)); NiM+SSZmZ2Fpbkc2ImYqNiVJIktHRiVJImNHRiVJInRHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSomOSQiIiIsJkYwRjAtSSRleHBHNiRJKnByb3RlY3RlZEdGNUkoX3N5c2xpYkdGJTYjLCQqJjklRjA5JiIiIyEiIkY9RjBGJUYlRiU= with marginal gain (instantaneous rate of gain) the derviative of this with respect to t (the D[3] in the below holds all other values in fgain fixed and takes the derivative with respect to t) being fgainprime: =D[3](fgain);NiM+SStmZ2FpbnByaW1lRzYiZio2JUkiS0dGJUkiY0dGJUkidEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCQqKjkkIiIiOSVGMTkmRjEtSSRleHBHNiRJKnByb3RlY3RlZEdGN0koX3N5c2xpYkdGJTYjLCQqJkYyRjFGMyIiIyEiIkYxRjxGJUYlRiU= This is then evaluated at the optimum time t* and plotted as a function of m (holding K=1) and using the value of c chosen marginalgain:=m->fgainprime(1,5,mysolve(5,m)); NiM+SS1tYXJnaW5hbGdhaW5HNiJmKjYjSSJtR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUtSStmZ2FpbnByaW1lR0YlNiUiIiIiIiYtSShteXNvbHZlR0YlNiRGMDkkRiVGJUYl plot(marginalgain,1..20); 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 This is then compared to the average gain over a patch from equation 1 evaluated at t* averagegain:=(K,c,m,t)->K*(1-exp(-c*t^2))/(t+m); NiM+SSxhdmVyYWdlZ2Fpbkc2ImYqNiZJIktHRiVJImNHRiVJIm1HRiVJInRHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSooOSQiIiIsJkYxRjEtSSRleHBHNiRJKnByb3RlY3RlZEdGNkkoX3N5c2xpYkdGJTYjLCQqJjklRjE5JyIiIyEiIkY+RjEsJkY8RjE5JkYxRj5GJUYlRiU= patchaveragegain:=m->averagegain(1,5,m,mysolve(5,m)); NiM+STFwYXRjaGF2ZXJhZ2VnYWluRzYiZio2I0kibUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUksYXZlcmFnZWdhaW5HRiU2JiIiIiIiJjkkLUkobXlzb2x2ZUdGJTYkRjBGMUYlRiVGJQ== plot(patchaveragegain,1..20); 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 and we see that the plots are identical by plotting the difference difference:=m-> patchaveragegain(m)-marginalgain(m);NiM+SStkaWZmZXJlbmNlRzYiZio2I0kibUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYtSTFwYXRjaGF2ZXJhZ2VnYWluR0YlNiM5JCIiIi1JLW1hcmdpbmFsZ2FpbkdGJUYvISIiRiVGJUYl plot(difference,1..20); 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these differences are very small - due to numerical round-off differences in the two - so the marginal value theorem is seen to hold for this case.