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(1) Determine whether the following functions are differentiable. Are they even of class C1? f(x, y) = ( 2xy +77 b) f(x,y) c) f(x, y)-cos (ye) - sin() d) (2) Why should the graphs of f(x, y)-y e* and g(x, y)-y +yr +y yr2 be called tangent at (0,0)? (3) If three resistors with resistances R1, R2, and Rg are connected in parallel, the total electrical resistance is determined by a) Find OR b) Suppose that the values of Ri, R2, and Rs are currently 100, 200, and 300 ohms respectively. How fast is R changing with respect to R1? c) The measurements of R1, R2, and Rs in part b) have a possible error of 0.5% in each case. Estimate the maximum error in the calculated value of R(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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