e'y x+y A surface is defined by the following equation: z(x,y)=- a) Find the equation of...

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e'y x+y A surface is defined by the following equation: z(x,y)=- a) Find the equation of the tangent plane to the surface at the point P(0, -2). Present your answer in the exact form (don't use a calculator to convert your result to the floating- point format). b) Find the gradient of function z(x, y) at the same point P. c) Find the angle between the gradient and the x-axis. Present your answer in degrees up to one decimal place. Calculate the total charge Q of a thin plate with the charge density distribution given by the formula: p(x, y) = (1+x) mC/m. The plate shape is restricted by the lines: x=0, x= 1, y=0, and y=x, where x and y are measured in metres. a) Sketch the plate shape. b) Present the total charge through the double integral. c) Reduce the double integral to the repeated integrals and show limits of integration. d) Calculate the integral and present your answer with two significant figures. = 48ff. D Consider the double integral V=4 cos dA over the region D enclosed between the lines: 0rcos, 0 /4. a) Reduce the integral to the repeated integral and show limits of integration. b) Calculate the integral and present your answer up to two significant figures. Calculate the line integral of the vector-function F(x, y, z) = (y + z) i-yzj+xk along the path L: {x= 3 sint, y=t, z=2 cost} (0 The line mass density of a wire is given by the formula p(x, y) = (3x + 5y +422) g/cm. Calculate the total mass M of the wire if its shape is described by the equations: x (t) = 3 sint, y(t) = 3 cost, z(t)=2 sint, where 0t7/2 and length is measured in centimetres. Present you answer in the exact form (don't use a calculator), and then calculate the mass of the wire up to three significant figures. Show your answer with the physical dimension (g, cm, sec). e'y x+y A surface is defined by the following equation: z(x,y)=- a) Find the equation of the tangent plane to the surface at the point P(0, -2). Present your answer in the exact form (don't use a calculator to convert your result to the floating- point format). b) Find the gradient of function z(x, y) at the same point P. c) Find the angle between the gradient and the x-axis. Present your answer in degrees up to one decimal place. Calculate the total charge Q of a thin plate with the charge density distribution given by the formula: p(x, y) = (1+x) mC/m. The plate shape is restricted by the lines: x=0, x= 1, y=0, and y=x, where x and y are measured in metres. a) Sketch the plate shape. b) Present the total charge through the double integral. c) Reduce the double integral to the repeated integrals and show limits of integration. d) Calculate the integral and present your answer with two significant figures. = 48ff. D Consider the double integral V=4 cos dA over the region D enclosed between the lines: 0rcos, 0 /4. a) Reduce the integral to the repeated integral and show limits of integration. b) Calculate the integral and present your answer up to two significant figures. Calculate the line integral of the vector-function F(x, y, z) = (y + z) i-yzj+xk along the path L: {x= 3 sint, y=t, z=2 cost} (0 The line mass density of a wire is given by the formula p(x, y) = (3x + 5y +422) g/cm. Calculate the total mass M of the wire if its shape is described by the equations: x (t) = 3 sint, y(t) = 3 cost, z(t)=2 sint, where 0t7/2 and length is measured in centimetres. Present you answer in the exact form (don't use a calculator), and then calculate the mass of the wire up to three significant figures. Show your answer with the physical dimension (g, cm, sec).