Please help with this calculus limit problem?

Question

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^3+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 plus or minus 1 degree Celsius, what range of wattage is allowed for the input power?

c) in terms of the epsilon, delta definition of lim, as x approaches a, 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 1

a) T'(w) = 0.3w² + 2.155 > 0 then the function T is ascendant.
If we want T(w) >= 200, then we must have w >= 11.575 watts (found with a calculator)

b) If I understand (!), the temperature is between 199 and 201 degrees.
If it's true, then the wattage must be betwwen 11.551 and 11.598 watts

c) x is the variable
f(x) is its image by the function f
a is a known number
L is the limit (known or to find)

the value of epsilon is as you want (very small) : epsilon is greater than the difference between f(x) and L

and delta is greater than the difference between x and a. It is very small too.

Smaller epsilon is, smaller delta is. delta depends of epsilon.

For example, I want to prove that the limit when x approaches 2 of f(x) = x² is 4 :

To simplify, I choose x greater than 2

I choose epsilon = 0.001

then I have f(x) - 4 < 0.001

I want to find delta to have x - 2 < delta, if it is possible !

f(x) - 4 = x² - 4 = (x - 2)(x + 2) < 0.001

then x - 2 < (0.001)/(x + 2)
and x + 2 > 2 + 2 = 4

then x - 2 < (0.001)/4

x - 2 < 0.00025

if I choose delta = 0.00025
then I have x - 2 < delta

In this example I can choose delta = epsilon divided by 4 (for any epsilon, as smaller as I want).

Answer 2

word that 4x - x^2 = x(4 - x) = x(2 + sqrt(x))(2 - sqrt(x)) now sub in to furnish: lim (2 - sqrt(x)) / [x(2 + sqrt(x))(2 - sqrt(x)) (2 - sqrt(x)) / (2 - sqrt(x)) cancels to a million as long as x =/= 4, and is a detachable discontinuity you are able to substitute in 4 for x in what's left to locate the shrink a million / [x(2 + sqrt(x)] as x ==> 4 is a million / [4(2 + 2)] = a million/sixteen shrink as x ==> 4 = a million/sixteen once you initially substitue 4 for x, you get 0 / 0, it particularly is indeterminate, telling you which you will seek for components of 0 to cancel..