Suppose that the equation z = f x , y is expressed in the polar form z = g r , θ by making the substitution x = r cos θ and y = r sin θ . (a) View r and θ as functions of x and y and use implicit differentiation to show that ∂ r ∂ x = cos θ and ∂ θ ∂ x = − sin θ r (b) View r and θ as functions of x and y and use implicit differentiation to show that ∂ r ∂ y = sin θ and ∂ θ ∂ y = cos θ r (c) Use the results in parts (a) and (b) to show that ∂ z ∂ x = ∂ z ∂ r cos θ − 1 r ∂ z ∂ θ = sin θ ∂ z ∂ y = ∂ z ∂ r sin θ + 1 r ∂ z ∂ θ = cos θ (d) Use the result in part (c) to show that ∂ z ∂ x 2 + ∂ z ∂ y 2 = ∂ z ∂ r 2 + 1 r 2 ∂ z ∂ θ 2 (e) Use the result in part (c) to show that if z = f x , y satisfies Laplace’s equation ∂ 2 z ∂ x 2 + ∂ 2 z ∂ y 2 = 0 then z = g r , θ satisfies the equation ∂ 2 z ∂ r 2 + 1 r 2 ∂ 2 z ∂ θ 2 + 1 r ∂ z ∂ r = 0 and conversely. The latter equation is called the polar form of Laplace’s equation.
Suppose that the equation z = f x , y is expressed in the polar form z = g r , θ by making the substitution x = r cos θ and y = r sin θ . (a) View r and θ as functions of x and y and use implicit differentiation to show that ∂ r ∂ x = cos θ and ∂ θ ∂ x = − sin θ r (b) View r and θ as functions of x and y and use implicit differentiation to show that ∂ r ∂ y = sin θ and ∂ θ ∂ y = cos θ r (c) Use the results in parts (a) and (b) to show that ∂ z ∂ x = ∂ z ∂ r cos θ − 1 r ∂ z ∂ θ = sin θ ∂ z ∂ y = ∂ z ∂ r sin θ + 1 r ∂ z ∂ θ = cos θ (d) Use the result in part (c) to show that ∂ z ∂ x 2 + ∂ z ∂ y 2 = ∂ z ∂ r 2 + 1 r 2 ∂ z ∂ θ 2 (e) Use the result in part (c) to show that if z = f x , y satisfies Laplace’s equation ∂ 2 z ∂ x 2 + ∂ 2 z ∂ y 2 = 0 then z = g r , θ satisfies the equation ∂ 2 z ∂ r 2 + 1 r 2 ∂ 2 z ∂ θ 2 + 1 r ∂ z ∂ r = 0 and conversely. The latter equation is called the polar form of Laplace’s equation.
Suppose that the equation
z
=
f
x
,
y
is expressed in the polar form
z
=
g
r
,
θ
by making the substitution
x
=
r
cos
θ
and
y
=
r
sin
θ
.
(a) View r and
θ
as functions of x and y and use implicit differentiation to show that
∂
r
∂
x
=
cos
θ
and
∂
θ
∂
x
=
−
sin
θ
r
(b) View r and
θ
as functions of x and y and use implicit differentiation to show that
∂
r
∂
y
=
sin
θ
and
∂
θ
∂
y
=
cos
θ
r
(c) Use the results in parts (a) and (b) to show that
∂
z
∂
x
=
∂
z
∂
r
cos
θ
−
1
r
∂
z
∂
θ
=
sin
θ
∂
z
∂
y
=
∂
z
∂
r
sin
θ
+
1
r
∂
z
∂
θ
=
cos
θ
(d) Use the result in part (c) to show that
∂
z
∂
x
2
+
∂
z
∂
y
2
=
∂
z
∂
r
2
+
1
r
2
∂
z
∂
θ
2
(e) Use the result in part (c) to show that if
z
=
f
x
,
y
satisfies Laplace’s equation
∂
2
z
∂
x
2
+
∂
2
z
∂
y
2
=
0
then
z
=
g
r
,
θ
satisfies the equation
∂
2
z
∂
r
2
+
1
r
2
∂
2
z
∂
θ
2
+
1
r
∂
z
∂
r
=
0
and conversely. The latter equation is called the polar form of Laplace’s equation.
The equation below defines y implicitly as a function of x:
2x2 + xy = 3y2
Use the equation to answer the questions below.
A) Find dy/dx using implicit differentiation. SHOW WORK.
B) What is the slope of the tangent line at the point (1, 1)? SHOW WORK.
C) What is the equation of the tangent line to the graph at the point (1, 1)? Put answer in the form y = mx + b and SHOW WORK.
The motion of a cam is defined by the relation @=0.30t(12-t2) where @ is expressed in radians and t in seconds. Determine the angular displacement, angular velocity and angular acceleration of the cam when t=4seconds
Show that the curves r = a sin θ and r = a cos θ intersectat right angles.
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