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Calc 3 11.5

 

<== Calculus 3
Number Question Answer
See the work
1. Use the chain rule to find rac{dz}{dt}, where

z = x^2 y + x y^2,\quad  x = -5 - t^4,\quad y = -4 + t^4

First the pieces:

rac{\partial z}{\partial x} =
rac{\partial z}{\partial y} =
rac{dx}{dt} =
rac{dy}{dt} =
rac{dz}{dt} =
2xy + y^2

x^2 + 2xy

-4t^3

4t^3

36t^3+72t^7
The work
2. Suppose \displaystyle w = rac{x}{y} + rac{y}{z}, where
x = e^{5t},\ y = 2 + \sin \left( 5t 
ight), and z = 2 + \cos \left( 6t 
ight).

A ) Use the chain rule to find rac{dw}{dt} as a function of x, y, z, and t.
rac{dw}{dt} =
B ) Use part A to evaluate rac{dw}{dt} when t = 0.

A)

B) 35/12
The work
3. Suppose z = x^{2} \sin y,

A. Use the chain rule to find rac{\partial z}{\partial s} and rac{\partial z}{\partial t} as functions of x, y, s and t.
rac{\partial z}{\partial s} = =
rac{\partial z}{\partial t} = =

B. Find the numerical values of rac{\partial z}{\partial s} and rac{\partial z}{\partial t} when (s , t) = ( 4 , 4 ).
rac{\partial z}{\partial s} \left( 4 , 4 
ight) =
rac{\partial z}{\partial t} \left( 4 , 4 
ight) =

A)
=
=

B)
-17070.39
-14952.91
The work
4. Let

w = -2xy  - yz  - 3xz, x = st, y = e^{st}, z = t^2
Compute
\displaystyle rac{\partial w}{\partial s}(4, -5) = =
\displaystyle rac{\partial w}{\partial t}(4, -5) = =
= 375
= -900
The work
5. Let W(s,t) = F(u(s,t), v(s,t)) where

u(1,0) = 8, u_s(1,0) = 2, u_t(1,0) = -4
v(1,0) = -8, v_s(1,0) = 5, v_t(1,0) = -3
F_u(8, -8) = 4, F_v(8, -8) = -1

W_s(1,0) = W_t(1,0) =
3
-13
The work
6. Consider the curve x^{2} + 6 xy + y^{5} = 8
The equation of the tangent line to the curve at the point (1,1) has the form y = mx + b where
m = and b =
m = -8/11
b = 19/11
The work
7. Find the slope of the tangent line to the curve
\displaystyle \sqrt{2x +4y}  + \sqrt{2xy} = 
    \sqrt{22} + \sqrt{24} at the point ( 3,4 ).
The slope is .
m = -0.991 The work
8. Consider the surface F(x,y,z) = x^{3}z^{7} + \sin(y^{8}z^{7})  - 5 = 0.
Find the following partial derivatives
∂z/∂x =
∂z/∂y =
∂z/∂x =
∂z/∂y =
The work
9. Consider the surface F(x,y,z) = x^{8}z^{3} + \sin(y^{7}z^{3}) + 5 = 0.
Describe the set of points on the surface for which it is not possible to define the surface as the graph of a differentiable function z = f(x,y).
Your answer should be in the form g(x,y,z) = 0
= 0.
= 0 The work
10. Suppose that z = f(x,y) is defined implicitly by an equation of the form F (x,y,z) = 0. Find formulas for the partial derivatives ∂f/∂x and ∂f/∂y in terms of F1, F2, F3.

∂f/∂x =
∂f/∂y =
∂f/∂x = -F1 / F3
∂f/∂y = -F2 / F3
The work
11. The radius of a right circular cone is increasing at a rate of 3 inches per second and its height is decreasing at a rate of 5 inches per second. At what rate is the volume of the cone changing when the radius is 30 inches and the height is 50 inches?

cubic inches per second

4712.39 The work
12. In a simple electric circuit, Ohm's law states that V = IR, where V is the voltage in volts, I is the current in amperes, and R is the resistance in ohms. Assume that, as the battery wears out, the voltage decreases at 0.03 volts per second and, as the resistor heats up, the resistance is increasing at 0.02 ohms per second. When the resistance is 300 ohms and the current is 0.02 amperes, at what rate is the current changing?

amperes per second

-0.0001013 The work