Calculus Answers

Find the linearization of the function z=x√y at the point (-9, 25).
L(x,y)=

Use the chain rule to find dzdt, where
z=x^2y+xy^2,x=−4+t^2,y=5−t^7

First the pieces:

∂z/∂x=

∂z/∂y=

dx/dt=

dy/dt=


End result (in terms of just t):
dz/dt=

Suppose z=x^2siny, x=−4s^2+t^2, y=−2st.

A. Use the chain rule to find ∂z/∂s and ∂z/∂t as functions of x, y, s and t.
∂z/∂s=

∂z/∂t=

B. Find the numerical values of ∂z/∂s and ∂z/∂t when (s,t)=(−5,−5).
∂z/∂s(−5,−5)=

∂z/∂t(−5,−5)=

Show that the ellipsoid 3x^2+2y^2+z^2= 9 and the sphere x^2+y^2+z^2-8x-6y-8z+24 = 0 are tangent to each other at the point (1,1,2), i.e., they have the same tangent plane there.

Find the linearizationL(x,y,z) of the function f(x,y,z) =sin(xy)/z at the point (pi/2,1,1).

Find the equation for the tangent plane to the surface x-z= 4 arctan(yz) at the point (1 +pi,1,1).

(Assume that 9 is the symbol for a partial derivative) For the equation (1/x)+(1/y)+(1/z)=1, evaluate 9z/9x and 9z/9y at the point (2,3,6)

A norman window with a perimeter of 20 units has the shape of a rectangle surrmounted by a semi circle. If its base is x units long and the height of rectangle is y units long, write a mathematical model for its area in term of x

Derive the reduction formula integration (x2+a2)n/2 dx = x(x2+a2)/n+1 + na2 /n+1 integration ( x2+a2)n/2-1 dx
Use the formula to integrate [(x2 +a2)n/2-1