(a)
Whether the equation
4 x − 3 y = 12
describes a line in
ℝ 3
.
(a)
Answer to Problem 1RE
The statement is false.
Explanation of Solution
Given:
The equation is
Calculation:
The graph of the given equation
From Figure 1, it is observed that the given equation represents a plane in
So, the given statement the equation
Therefore, the statement is false.
(b)
Whether the equation
z 2 = 2 x 2 − 6 y 2
satisfies z as a single function of x and y.
(b)
Answer to Problem 1RE
The statement is false.
Explanation of Solution
Given:
The equation is
Calculation:
The given equation is
If
Obtain the function in terms of x and y.
The functions are
Thus, z as a two function in terms of x and y.
Therefore, the statement is false.
(c)
Whether the function f satisfies the derivative
f x x y = f y y x
.
(c)
Answer to Problem 1RE
The statement is false.
Explanation of Solution
Let the function f has a continuous partial derivatives of all orders.
Then prove that
For example, assume
Obtain the value of
Take partial derivative of the function f with respect to x and obtain
Thus,
Take partial derivative of the equation (1) with respect to x and obtain
Hence,
Again, take partial derivative for the equation (2) with respect to y and obtain
Therefore,
Obtain the value of
Take partial derivative of the function f with respect to y and obtain
Thus,
Take partial derivative of the equation (1) with respect to y and obtain
Hence,
Again, take partial derivative for the equation (2) with respect to x and obtain
Therefore,
From above, it is concluded that
Thus,
Therefore, the statement is false.
(d)
Whether the gradient
∇ f ( a , b )
lies in the plane tangent to the surface at
( a , b , f ( a , b ) )
.
(d)
Answer to Problem 1RE
The statement is false.
Explanation of Solution
Given:
The surface is
Theorem used: The Gradient and Level Curves
“Given a function f differentiable at
Calculation:
The given surface is
Assume the critical point be
Then, the given function is differentiable at
Thus, the gradient of
By above theorem, it can be concluded that the line tangent to the level curve of f at
But it does not satisfy the given statement. Because, it is given that the the gradient
Since,
Therefore, the statement is false.
(e)
Whether the plane is always orthogonal to both the distinct intersecting planes.
(e)
Answer to Problem 1RE
The statement is true.
Explanation of Solution
Assume the equations of a plane.
The normal
Therefore, the statement is true.
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Chapter 12 Solutions
Student Solutions Manual, Single Variable for Calculus: Early Transcendentals
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