Calculus Answers

limx→−2x3+32x3−4x+1
= …….

A Dam is overflowing at the rate of
V ′(t) = 3t/(t^2+1)^(2/3) thousand m3 per day, V ′(t) = 6t(2t + 1)−2 thousand m3 per day,
where t is the time in days after the overflow begins. Find the amount of water that overflows from the dam during the second day of the overflow. Give the answer exactly first and then to 4 significant figures.

Find the total mass of a plate in the shape of the region bounded by y=x^(-1) and y=0 for 1<x<4 and the mass density is f(x,y)=y/x

2
limx→−2x3+32x3−4x+1
= …….
57
−57
11−15
1115

help me answer questions
1)
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2)
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3)
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1)Calculate the average value of the function f(x,y)=e^(x+y) on the square [0,1]x[0,1]
2)Calculate the average height above the x-axis of a point in the region 0<=x<= x^(2)

1)Find the directional derivative of f(x,y,z)=xy^(2)-z^(2)in the direction of v=(1,-2,2) at (2,1,3)
2)Suppose f(1,0,0)=-3,fx(1,0,0)=-2,fy(1,0,0)=4 and fz(1,0,0)=2 . Use linear approximation to estimate f(1.02,0.01,-0.03).
3)Find an equation of the tangent plane at the point(4,1) of the surface z=x^(2)+y^(-2)

a) Find the directional derivative of
f (x, y)=
( )
 

 


=
¹
+
0 , ( , ) (0,0)
, , (0,0)
2 4
2 2
x y
x y
x y
xy
at (0, 0) in the direction
3
q =p .

b) Compute the Jacobian matrices using the chain rule for z = u2 + v2 where (3)
u = 2x + 7, v = 3x + y + 7 .
c) If f (x, y) =
( )
 

 


= =
¹
+
0 , 0 ,
, , 0
2 2
4 2 2
x y
x y
x y
x y
(5)
then show that fxy (0,0) = f yx(0,0)

Find
dt
dw
at
2
p
t = , where
w = x2 + y2 + 2x + 3y, x =cos t, y =sin t