Question 1 Find the (indicated) first partial derivatives. a) f(x,t)=xInt b) f(x,y)=x" c) f(x, y) = [cos(t)dt y d) u = xysin '(yz) c) u= x f) u= = x + x + ... + x g) f(x,y) = ln(x+x + y); , (3,4) h) f(x,y,z)= y x+y+z f,(2,1,-1) i) f(x, y, z)=sin x+sin y+sin z; f.(0,0,/4) Question 2 A) Find az/ax and oz/oy: a) xz=arctan(y) b) yz=ln(x+2) c) x + y+z = 3xyz B) Find all the second partial derivatives (i) w=u+v (i) ==arctan x+y 1-xy (iii) f(x,y)=x'y' +2x+y Question 3 Show that each of the following is a solution of the wave equation u = au... (i) u=sin(kx) sin(akt) (ii) u t (iii) u=(x-at)+(x+at)" Question 4 The kinetic energy of a body of mass m and velocity v is K = Question 5 Show that =K. Om Ov 60 The temperature at a point (x,y) on a flat metal plate is given by 7(x,y) = 1 + x + y where T is measured in Cand x,y in metres. Find the rate of change of temperature with respect to distance at the point (2,1) in (a) the x-direction and (b) the y-direction.