Calculus: The Language of Change

Exercise Set 4.2
Instantaneous Rates of Change

by Kevra Tanke

Exercise 1

Computer Project.    Use  k=0.0223118

Exercise 2 : Graphical Euler's Method

You only need to know where C starts (C[0]  =  75) 
          and your equation for change 	(dC/dt)= -kC to sketch its 
            curve .   
              (This is not very accurate but gives the right 
                  shape . )

part a

Sketch the curve C = C[t] when k = (1/50), so (dc/dt)= -(C/50).  When C = 75, this makes the slope of the curve -(3/2).  Move a small distance along the line of slope -(3/2) from the point (0, 75).  If C=60, how much is the slope of the curve?  If C = 5, how much is the slope of the curve?  If C = 0, what is the  slope of the curve?

Solution of part a

Graph

\(Needs["Graphics`Colors`"]; 

Clear[F, G, X, T]; 

    F[X_, T_] := \(-\(T\/50.0\)\); 

    G[{T_, X_}] := {T + dT, X + F[T, X]\ dT}; 
T0 = 0.0; 
X0 = 75.0; 

    finalTime = 100.0; 
dT = 10.0; 
steps = Floor[finalTime\/dT]; 

    solnList = NestList[G, {T0, X0}, steps]; 

    RedLine = ListPlot[solnList, PlotJoined -> True, PlotStyle -> Red]; 
    \)

Slopes

If C=75, how much is the slope of the curve?

	(dc/dt) =  -(75/50.0)

N[ -(75/50.0)]

If C=60, how much is the slope of the curve?

	(dc/dt) =  -(60/50.0)

N[-(60/50.0)]

If C = 5, how much is the slope of the curve?

	(dc/dt) =  -(5/50.0)

N[-(5/50.0)]

If C = 0, what is the  slope of the curve?

	(dc/dt) =  -(0/50.0)

N[-(0/50.0)]

part b

Sketch the curve C = C[t] on the same axes when k = 0.01, so (dc/dt) = -(C/100).

Solution of part b

Graph when k=.01.

\(Clear[f, g, x, t]; 
f[x_, t_] := \(-\(t\/100\)\); 

    g[{t_, x_}] := {t + dt, x + f[t, x]\ dt}; 
t0 = 0.0; 
x0 = 75.0; 

    finalTime = 100.0; 
dt = 10.0; 
steps = Floor[finalTime\/dt]; 

    solnList = NestList[g, {t0, x0}, steps]; 

    BlueLine = 
      ListPlot[solnList, PlotJoined -> True, PlotStyle -> Blue]; \)

Graph of both curves.

Show[RedLine, BlueLine];

part c

Explain the idea in the computer's recursive "Euler" computation of the tempurature.  Write a few sentences covering the main ideas without giving a lot of technical details but rather  only giving a description and a few sample computations.  In other words, convince us that you understand the computer program and it's connection with sketching the curve C = C[t] from a starting value and differential  equation.

Solution of part c

The computer's recursive "Euler" computation calculates the value of the slope at a given point, goes to the next point and recalculates the slope at that point, and continues to connect these points with line segments.  To get a 'smoother' looking graph calculate more points and in turn have shorter line segments.