Use the tangent plane approximation to find f(33,4).

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 the tangent plane approximation to find f(33,4).

ft(30,5) = -5/3
fx(30,5) = -3/2

f(33,4)≈?

 

 

Answer 1

The tangent plane is an extension to an additional dimension of the "point-slope" form we use in two dimensions.

That is usually written as  y - y₀ = m(x-x₀) , or if rewritten in function notation similar to your problem (and moving the y₀ = f(x₀) to the other side):

  f(x) = f(x₀) + f_x(x0)(x-x₀) 

where I also replaced the slope m with the equivalent : f_x(x₀).

 

For your more dimensions case we have equivalently:

  f(t,x) = f(t₀,x₀) + f_t(t₀,x₀)(t-t₀) + f_x(t₀,x₀)(x-x₀)

Putting in the things we know, like f(30,5) = 50 minutes, gives the tangent plane approximation at a new t,x point:

  f(t,x) =   50     +  -5/3 (t-t₀)     + -3/2 (x-x₀)    minutes

So for the point t = 33 , x = 4 we get:

  f(33,4) ≈ 50 + -5/3*(33-30) + -3/2 *(4-5)

                = 50 -5 + 3/2

                = 46.5 minutes