Solution Found!
Explain why or why not Determine whether the
Chapter 10, Problem 41E(choose chapter or problem)
Explain why or why not Determine whether the following state ments are true and give an explanation or counterexample.
a. The planes tangent to the cylinder \(x^{2}+z^{2}=1 \text { in } \mathbf{R}^{3}\) all have the form ax + bz + c = 0.
b. Suppose w = xy/z for x > 0, y > 0, and 2 > 0. A decrease in z with x and y fixed results in an increase in w.
c. The gradient \(\nabla F(a, b, c)\) lies in the plane tangent to the surface F(x, y, z) = 0 at (a, b,c).
Questions & Answers
QUESTION:
Explain why or why not Determine whether the following state ments are true and give an explanation or counterexample.
a. The planes tangent to the cylinder \(x^{2}+z^{2}=1 \text { in } \mathbf{R}^{3}\) all have the form ax + bz + c = 0.
b. Suppose w = xy/z for x > 0, y > 0, and 2 > 0. A decrease in z with x and y fixed results in an increase in w.
c. The gradient \(\nabla F(a, b, c)\) lies in the plane tangent to the surface F(x, y, z) = 0 at (a, b,c).
ANSWER:Solution 41EStep 1a.The plane tangent to the cylinder in all has the form Now, consider that: Implicit form is: Gradient is: Consider a point at which the tangent plane lies. So, So, the equation of tangent plane is: Consider the constants as: So, the equation comes to the form .So, the statement is true.