Q.57. Rate of Change, Limits.

Question 57.

A crystal growth furnace is used in research to determine how best to manufacture crystals used in electronic components for the space shuttle. For proper growth of the crystal, the temperature must be controlled accurately by adjusting the input power. Suppose the relationship is given by:

T(w)= 0.1w^2 +2.155w +20

where T is the temperature in degrees Celsius and w is the power input in watts:

a) how much power is needed to maintain the temperature at 200 degrees Celsius?

b) if the temperature is allowed to vary from 200 degrees Celsius by up to ±1 degree Celsius, what range of wattage is allowed for the input power?

c)in terms of the epsilon, delta definition, as x approaches a, of lim f(x)=L, what is x? what is f(x)? what is a? what is L? What value of epsilon is given? what is the corresponding value of delta?

Answer 57.

a)

T(w)= 0.1w^2 +2.155w +20

200 = 0.1w^2 +2.155w +20

w^2 + 21.55w -1800 = 0

w = 33 watt

b)

δT = 0.2w*δw + 2.155δw

1 = [0.2(33) + 2.155]δw

δw = 1 / 8.755 = 0.114

wattage allowed for input power = 33 ± 0.114 watt

c)

x is a variable whose value lies in the interval (a - δ, a +δ). This is also called δ-neighbourhood of a.

f(x) is the value of the function for values of x lying in the above interval.

L is the limit of the function as x approaches a, i.e., as x takes up values closer and closer to a from values larger than a as well as from values smaller than a but always within the δ-neighbourhood of a.

Any value of ε greater than zero can be taken.

To find the corresponding value of δ, start with the inequality

l f(x) - L l < ε and find out from it l x - a l < ?. The value of '?' you find is the value of δ. While going from l f(x) - L l < ε, it should be ensured that one reaches l x - a l < δ following <=> all throughout so that it can be concluded that for every δ > 0,

lx - al < δ => l f(x) - L l < ε.

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