# Download Functions of Two Variables by Seán Dineen (auth.) PDF

By Seán Dineen (auth.)

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**Example text**

A further interesting point is how naturally the function of two variables g(x, Llx) entered into our discussion of one variable theory. We only considered the continuity properties of g(x, Llx) when we kept the variable x fixed and let the variable Llx tend to O. From our experience of partial derivatives we might, say that 9 is partially continuous (in the second variable). igate what happens if 9 is a continuous function of both variables. This is another fruitful line of investigation but again we will not pursue it here.

O h = °ofx (x, y) . VI + °ofy (x, y). O lim K(x, y, hVl, hV2) 11 vII of = of ox(x,y) ·Vl + Oy(x,y) ·V2· Hence of ov of of = VI oX + V2 oy of · . · and smce ox and of oy are b oth contmuous we see t h at of. ov IS alSo contmuous. We have proved the following result. Proposition 36. If f is a continuous function defined on an open subset ofR2 and ~~ and ~~ both exist and are continuous then for any vector v in R 2 the directional derivative then ~~ exists and is continuous. 1 Let f (x) denote a company's profit expressed as a function of the capital invested x.

Now substituting in this value for A we have f(x + Llx) = f(x) + B· Llx ± h(x, Llx) . Llx and f(x + Llx) Llx f(x) = B· Llx ± h(x, Llx) . Llx = B ± h(x, Llx). Llx Hence f'(x) = lim f(x+Llx)-f(x)=B± lim h(x,Llx) ~z~o Llx ~z~o =B. From this we conclude that f(x) + f'(x) ·Llx = A+B ·Llx, that g(x, Llx) = ±h(x, Llx) and the tangent line is the closest line to the graph of f. What happens if we use more derivatives? The answer is interesting and you might like to try experimenting to see what happens.