Calculus Answers

Let f be the R^2 - R function defined by f(x,y) = ln xy and let r be the R-R^2 function defined by r(t) = (e^t,t).

a) Determine the composite function f o r?

b) Determine grad f(x,y) and r' (t)?

c) Determine the derivative function (f o r)' by

i) Differentiating the expression obtained in (a)?

ii) Using the Chain rule?

let f be the R^2-R function defined by f(x,y) = (x - y)^3

a) Determine the rate of increase in f at the point (2,1) in the direction of the vector (1,-1)?

b) What is the rate of increase in f at (2,1) in the direction of the negative X-axis?

c)Determine the maximum rate of increase in f at (2,1).In which direction is the maximum rate of increase?

Fi‎nd the flu‎x of ->F (x, y, z) =〈4x, 3z + x^2, y^2/2〉

a‎cross the po‎sitively or‎iented su‎rface S giv‎en by

->R(u, v) = 〈2u, 4v, −u^2〉, 1 ≤ u^2 + v^2 ≤ 4

Are the following statements true or false? Give reasons for your answers.

 a) 2 is a limit point of the interval [1,4]

 b) Every bounded sequence is convergent. 

 c) The function, f :R → R defined by f(x)= |x −1| + |3 − x |is differentiable at x =4

 d) The function f(x) = [x] − x is not integrable in [0,3] where [x] denotes the greatest 

integer function. 

 e) The function f defined by  f(x)= (e^x₊ e^-x )/2 when x not equal to 0 and 1/2 when x =0 is continuous in the closed interval, [-1,1 ]