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.
With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.
4) Find the directional derivative of at the given point in thedirection indicated by the angle theta.
Find all the points where r = 2 + sin theta has vertical and horizontal tangent lines.
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.
Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.