Question

The number of minutes taken for a chemical reaction if f(t,x). it depends on the temperature t degrees celcius, and the quantity, x grams, of a catalyst present. when the temperature is 30 degrees celcius and there are 5 grams of catalyst, the reaction takes 50 minutes. increasing the temperature by 3 degrees reduces the time taken by 5 minutes. increasing the amount of catalyst by 2 grams decreases the time taken by 3 minutes. use this information to find the partial derivatives fx(30,5) and ft(30,5). use the tangent plane approximation to find f(33,4).

Answer 1

Hello,

Just for the fun, it's been a long time for that.

$f(30,5)=50\\ f(t+3,x)=f(t,x)-5\ ==\textgreater \dfrac{f(t+3,x)-f(t,x)}{3}= -\frac{5}{3} \\ f(t,x+2)=f(t,x)-3\ ==\textgreater\dfrac{f(t,x+2)-f(t,x)}{2}= -\frac{3}{2} \\ f(t+\Delta t,\Delta x)=f(f,t)+ \dfrac{\partial{f(t,x)}}{\partial{t}} *\Delta t+ \dfrac{\partial{f(t,x)}}{\partial{x}} *\Delta x\\ f(33,4)\approx f(30,5)- \dfrac{5}{3} *3-\dfrac{3}{2}*(-1)= 50- 5+1.5=46.5$