Question

In the temperature range between 0C and 700C the resistance R [in ohms] of a certain platinum resistance thermometer is given by R = 10 + 0.04124T − 1.779 × (10^−5)T^ 2 where T is the temperature in degrees Celsius. Where in the interval from 0C to 700C is the resistance of the thermometer most sensitive and least sensitive to temperature changes? [Hint: Consider the size of dR/dT in the interval 0 ≤ T ≤ 700.

Answer 1

The derivative of the function measures the rate at which function changes with respect to the change of the variable. So, to find where the given function is more/less sensitive to the changes of time we should find maximum/minimum of its derivation.

$R = 10 + 0.04124T − 1.779 × 10^{−5}T^{2}$

${\frac {dR} {dT}}=0.04124-3.558*10^{-5}T$

Now to find extremums of this function we must find it's derivative

${\frac {d^{2}R} {dT^{2}}}=-3.558*10^{-5}$

Since that function is independent from t, it means that the function ${\frac {dR} {dT}}$ is monotonic. We can easily determine from the form that it is actually decreasing on the interval from 0 to 700, which means t = 0 is the point with the greatest sensitivity and t = 700 is the point with the least sensitivity.

${\frac {dR} {dT}}(0)=0.04124-3.558*10^{-5}*0=0.04124$

${\frac {dR} {dT}}(700)=0.04124-3.558*10^{-5}*700=0.04124-0.024906=0.016334$