(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.
Want to see more full solutions like this?
Chapter 12 Solutions
CODE/CALC ET 3-HOLE
- Simplify the following Boolean functions using four-variable maps. PLEASE EXPLAIN IN WRITING THE STEPS AND WHAT IS GOINING ON F(A, B, C, D) =∑ (4, 6, 7, 15) F(A, B, C,D) =∑ (0, 2, 4, 5, 6, 7, 8, 10, 13, 15)arrow_forwardSimplify the following Boolean functions, using three-variable K-maps: F(A,B,C)= ∑(1,2,3,5,6,7)arrow_forwardSimplify the following Boolean functions, using three-variable K-maps: F(x, y, z)=Σ(0, 1, 2, 3, 5)arrow_forward
- Simplify the following Boolean functions using four-variable K-maps: C: F(w,x,y,z)=sum (1,3,4,5,6,7,9,11,13,15)arrow_forwardSimplify the following Boolean functions, using three-variable k-maps: a. F (x,y,z) = Σ(2,3,4,5) b. F (x,y,z) = Σ(0,2,4,6) c. F (x,y,z) = Σ(1,2,3,6,7) d. F (x,y,z) = Σ(1,2,3,5,6,7) e. F (x,y,z) = Σ(3,4,5,6,7)arrow_forwardSimplify the following Boolean functions using three-variable maps. a. F(x, y, z) = ∑ (0, 1, 5, 7) b. F(x, y, z) = ∑ (1, 2, 3, 6, 7)arrow_forward
- Simplify the following Boolean function , using three- variable maps: d) F(x,y,z) = Σ (3,5,6,7)arrow_forwardSimplify the following Boolean functions using three-variable maps. a. F(x, y, z) = ∑ (0, 1, 5, 7) b. F(x, y, z) = ∑ (1, 2, 3, 6, 7) c. F(x, y, z) = ∑ (3, 5, 6, 7) d. F(A, B, C) = ∑(0, 2, 3, 4. 6)arrow_forwardGiven A = {1,2,3} and B={u,v}, determine. a. A X B b. B X Barrow_forward
- What is the implicit equation of the plane through 3D points (1,0,0), (0, 1, 0), and (0, 0, 1)? What is the parametric equation? What is the nor- mal vector to this plane? Given four 2D points a0, a1, b0, and b1, design a robust procedure to determine whether the line segments a0a1 and b0b1 intersect.arrow_forwardCreate the K-maps and then simplify for the following functions: 1). F(x,y,z) = x′y′z′ + x′yz + x′yz′arrow_forwardSimplify the following Boolean functions, using K-maps: F (w, x, y, z)=Σ(11, 12, 13, 14, 15)arrow_forward
- Database System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. SudarshanPublisher:McGraw-Hill EducationStarting Out with Python (4th Edition)Computer ScienceISBN:9780134444321Author:Tony GaddisPublisher:PEARSONDigital Fundamentals (11th Edition)Computer ScienceISBN:9780132737968Author:Thomas L. FloydPublisher:PEARSON
- C How to Program (8th Edition)Computer ScienceISBN:9780133976892Author:Paul J. Deitel, Harvey DeitelPublisher:PEARSONDatabase Systems: Design, Implementation, & Manag...Computer ScienceISBN:9781337627900Author:Carlos Coronel, Steven MorrisPublisher:Cengage LearningProgrammable Logic ControllersComputer ScienceISBN:9780073373843Author:Frank D. PetruzellaPublisher:McGraw-Hill Education