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(4) Let if (z, y)メ(0,0) if (x, y) (0,0) f(z, y) / 0 a) Show that f is a continuous function b) Show that f has partial derivatives at (0,0) and find (0,0) as well as c) Is f differentiable at (0, 0)? d) Are the partial derivatives r tinuous at (0,0)? (0,0) (5) Let A E M(mx n,R) and f : RnRm be the linear map f(x)-Ax. Show that f is a differentiable function and find the derivative of f. (6) The pressure P (in kilopascals), volume V (in liters), and temperature T (in kelvins) of a mol of an ideal gas are related by the equation P. V-8.31 . T. a) Find the rate at which the pressure is changing when the temperature is 300K and increasing at a rate of 0.1K/s and the volume is 100L and increasing at a rate of 0.2L/s b) Find the rate at which the volume is changing when the temperature is 320K and increasing at a rate of 0.15K/s and the pressure is 20kPa and increasing at a rate of 0.05kPa/s (7) Let g(t) = (at, bt) for a,be R and if(x, y) if (x, y) (0,0) (0,0) f(x, y) 0 a) Show that f has partial derivatives at (0,0) and find 쓿(0,0) as well as (0,0) Use these to show that ▽f(0,0) . g(0) b) Show that f og is differentiable and 0 c) Explain why the results of a) and b) do not contradict the Chain Rule

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