Question

1.express dw/dt as a function of t, both by using the Chain Rule and by expressing w in terms of t and differentiating directly with respect to t. Then evaluate dw/dt at the given value of t.

a)w= -6x^2-10x^2 , x=cos t,y=sint, t=pi/4

b)w=4x^2y-4y^2x, x=cost y=sint, --> express n terms of t

2.Find the linearization L(x,y) of the function (x,y)=e^x cos(9y) at points (0,0) and (0,pi/2)

Answer #1

Use the Chain Rule to find dw/dt.
w = ln
x2 + y2 + z2
, x = 9 sin(t), y
= 4 cos(t), z = 5 tan(t)
dw
dt
=

Let w(x,y,z) = x^2+y^2+z^2 where x=sin(8t), y=cos(8t) , z=
e^t
Calculate dw/dt by first finding dx/dt, dy/dt, and dz/dt and using
the chain rule
dx/dt =
dy/dt=
dz/dt=
now using the chain rule calculate
dw/dt 0=

Let w = (x 2 -z)/ y4 ,
x = t3+7,
y = cos(2t),
z = 4t.
Use the Chain Rule to express dw/ dt in terms of t. Then
evaluate dw/ dt at t = π/ 2

Find dw/dt using the appropriate chain rule for w=x^2+y^2+z^2
where x=8tsin(s), y=8tcos(s) and z=5st^2

Use the Chain Rule to find dz/dt. (Enter your
answer only in terms of t.)
z=sqrt(1+x^2+y^2), x=ln(t), y=cos(t)
dz/dt=

(x+1)y'=y-1 2) dx+(x/y+(e)?)dy=0 3)ty'+2y=sint 4) y"-4y=-3x²e3x
5) y"-y-2y=1/sinx 6)2x2y"+xy'-2y=0 ea)y=x' b) x=0
2dx/dt-2dy/dt-3x=t; 2

Use the Chain Rule to find dz/dt. (Enter your answer only in
terms of t.) z = sin(x + 7y), x = 8t^2, y = 3/t

1.
a) Use the Chain Rule to calculate the partial derivatives.
Express the answer in terms of the independent variables.
∂f
∂r
∂f
∂t
; f(x, y, z) = xy +
z2, x = r + s −
2t, y = 6rt, z =
s2
∂f
∂r
=
∂f
∂t
=
b) Use the Chain Rule to calculate the partial derivative.
Express the answer in terms of the independent variables.
∂F
∂y
; F(u, v) =
eu+v, u =
x5, v = 2xy
∂F
∂y
=
c)...

Let h be the function defined by H(x)= integral pi/4 to x
(sin^2(t))dt. Which of the following is an equation for the line
tangent to the graph of h at the point where x= pi/4.
The function is given by H(x)= integral 1.1 to x (2+ 2ln( ln(t) ) -
( ln(t) )^2)dt for (1.1 < or = x < or = 7). On what
intervals, if any, is h increasing?
What is a left Riemann sum approximation of integral...

1. Solve the following differential equations.
(a) dy/dt +(1/t)y = cos(t) +(sin(t)/t) , y(2pie) = 1
(b)dy/dx = (2x + xy) / (y^2 + 1)
(c) dy/dx=(2xy^2 +1) / (2x^3y)
(d) dy/dx = y-x-1+(xiy+2) ^(-1)
2. A hollow sphere has a diameter of 8 ft. and is filled half way
with water. A circular hole (with a radius of 0.5 in.) is opened at
the bottom of the sphere. How long will it take for the sphere to
become empty?...

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