Implicit Differentiation and Max/Min Problems
The critical point of f(x,y)=x^2y+y^2-x.
What is (-1,-1/2)?
The double integral int_0^2int_0^1k s t \ ds \ dt simplified.
What is k?
The integral set up to find the curved surface area of a cylinder.
What is int_0^(2pi) int_0^(h) R \ d z \ d theta ?
The partial derivative of cos(x^2+y) with respect to x.
What is -2xsin(x^2+y)?
A local extremum is at the critical point of f(x,y)=x^2y+y^2-x, which is of this type. (Local maxima, local minima, or saddle point.)
What is a saddle point?
The integral set up to find the area of a circle.
What is int_0^(2pi)int_0^(R)r \ dr \ d theta
The centroid location of the triangle formed by the lines y=1/2x, x=1, and y=0.?
What is (2/3,1/6)?
The electric field at the center of a uniform hoop of charge.
What is 0?
(dx)/dt given x+e^x=t .
What is 1/(1+e^x)?
The integral set up to find the volume of a sphere.
What is int_0^(2pi) int_0^(pi)int_0^(R)r^2 sin(theta ) \ d r \ d phi \ d theta
The arclength of y=2/3x^(3/2) over the interval [-1,0].
What is 2/3 ?
dV in spherical coordinates.
What is r^2sin(theta) \ d r \ d theta \ d phi ?
The slope of the tangent line to the curve x^3-3y^3+xy +21=0 at the point (1,2) .
What is 1/7?
The integral set up to find the volume of a cone.
What is int_0^h int_0^(2pi)int_0^(R/h z)r \ d r \ d theta \ d z
The moment of inertia about a disk's center with density sigma = k \ r?
What is 3/5MR^2?
The cartesian coordinates in polar terms.
What is x = r sin(theta) cos(phi)
y = r sin(theta) sin(phi)
z = r cos(theta) ?
The ratio of a cylinder's radius and height that maximizes volume.
Hint: Use Lagrange multipliers. What is the equation you are maximizing (f), and what is the constraint equation (g)?
h=2r
The integral over a 2-dimensional Gaussian distribution.
What is 1?
The integral set up to find the moment of inertia of a cylinder with wavy sides ( r = cos(z), h = 2pi ).
What is rho int_0^(2pi) int_0^(2pi)int_0^(cos(z))r^3 d r \ d theta \ d z
The shape that has the greatest moment of inertia.
What is a hoop?