Midterm 1
Let u = <-3, 1, 1> and v = < 2, 1 ,-2>. Comput u • v.
-7
Find the Domain of f(x,y)=√(x+y).
x+y≥0
What is the second derivative test and its conditions.
fxx(a,b)fyy(a,b)−[fxy(a,b)]2
what is dA in the integral?
dxdy
Let u = <-3, x, 1> and v = < 2, x ,-x>. Find all values x of such that u and v are orthogonal.
x = 3, -2
Find fx for f(x,y,z)=sin(x2y−z)+cos(x2−yz).
fx(x,y,z)=2xycos(x2y−z)-2xsin(x2−yz)
How does one find the absolute maximum or minimum?
Check all critical points and boundary points.
What si the graphical/physical meaning of a double integral
Volume under the graph of z=f(x,y)
Let v = <1, 1, 0> and w = <1, 0, -1>. Find v • <w × v>.
0
Find the equation of the tangent plane to the surface defined by the function f(x,y)=sin(2x)cos(3y) at the point (π/3,π/4).
z=√(2)/2 x −3√(6)/4 y−6√4−π√(2)/6+3π√(6)/16.
Find and classify all the critical points of f(x,y)=4+x3+y3−3xy
(1,1): minimum
(0,0): saddle point
Compute the double integral f(x,y)=1 over the region R=[0,1]× [0,1]
1
Find the acceleration vector of r(t)=⟨cos(2t),−sin(2t),4t⟩.
⟨−4cos(2t),4sin(2t),0⟩
Let z=cos(yx2), x=t4−2t, y=1−t6. Find dz/dt
−2(t4−2t)(1−t6)(4t3−2)sin((1−t6)(t4−2t)2)+6t5(t4−2t)2sin((1−t6)(t4−2t)2)
Find the absolute minimum and absolute maximum of f(x,y)=x2+4y2−2x2y+4 on the rectangle given by −1≤x≤1 and −1≤y≤1.
min: 4
max: 11
Compute the average value of f (x, y) = x cos(xy) over the rectangle R = [0, π] × [0, 1].
2/π
Does the line <2, 5, 4> + t<1, 1, 1> lie on the plane 2x+3y-5z+1=0.
Yes
f(x, y) = x2y5. Find the gradient of f at (-2, 1).
<-4, 20>
Find the maximum and minimum of f(x,y)=5x−3y subject to the constraint x2+y2=136. Use lagrange multipliers
min: -68
max: 68
∬yey^2−4xdA
With R=[0,2]×[0,√8]
(e8+e−8−2)/8