Change in Entropy and Enthalpy found over a temperature range and asked to compute the change in Gibb's Energy I was given the following question by my tutor:

"The variation of enthalpy ($\Delta H$) and entropy ($\Delta S$) in a reaction carried out at
constant pressure in the temperature range between 300 ºC and 350 ºC, are 1700 kcal and
15kcal/K respectively. Calculate the range of temperatures in which this reaction is
spontaneous."

But my book on thermodynamics says that the change in Gibb's energy for a process can only be calculated if we start and end at the same temperature (the temperature can vary during the process as heat is added or lost from the system to the surroundings, but the end points must have constant temperature and pressure.) In this question the changes in entropy and enthalpy are found over a range of temperatures where the final temperature is not the same as the initial temperature, but the pressure is always constant.  

Is my tutor's question invalid, is my book wrong in saying that the temperature has to be the same at the initial and final values, or am I thinking about this wrong?
This should address your confusion about constant T:

$$G = H - TS \Rightarrow dG = dH -TdS - SdT$$

$$ \Rightarrow \text{(at constant T) } dG = dH -TdS \Rightarrow \Delta G = \Delta H - T \Delta S$$

So, yes, you need $\Delta T = 0$ for $\Delta G = \Delta H - T \Delta S$ to hold.

However, this holds at any T.  

E.g., suppose you do the reaction at $300 K$. $\Delta T = 0$, so $\Delta G = \Delta H - T \Delta S$ holds.

But suppose you repeat reaction at $350 K$. It's still the case that $\Delta T = 0$, so $\Delta G = \Delta H - T \Delta S$ still holds.

As far as the question itself goes, I agree with Chet Miller that it appears to be badly worded.  It seems what your tutor meant to say was:

"Between 300 ºC and 350 ºC, assume that $\Delta H$ and $\Delta S$ are constant, with values of 1700 kcal and 15kcal/K, respectively. Calculate the range of temperatures over which $\Delta G &lt;=0$, i.e., the range the textbook labels as 'spontaneous'".