Question

Let Q1 be a constant so that Q1 = L(20, 12), where z = L(x, y) is the equation of the tangent plane to the surface z = ln(19x + 8y) at the point (x0, y0) = (7, 11). Let Q = ln(3 + |Q1|). Then T = 5 sin2 (100Q)

satisfies:— (A) 0 ≤ T < 1. — (B) 1 ≤ T < 2. — (C) 2 ≤ T < 3. — (D) 3 ≤ T < 4. — (E) 4 ≤ T ≤ 5

Answer #1

Let Q1 be a constant so that Q1 = L(−3, 2), where z = L(x, y) is
the equation of the tangent plane to the surface z = ln(5x − 7y) at
the point (x0, y0) = (2, 1). Let Q = ln(3 + |Q1|). Then T = 5 sin2
(100Q)
satisfies:— (A) 0 ≤ T < 1. — (B) 1 ≤ T < 2. — (C) 2 ≤ T
< 3. — (D) 3 ≤ T < 4. —...

Let Q1 be a constant so that Q1 = L(5, 17), where z = L(x, y) is
the equation of the tangent plane to the surface z = x 6 + (y − x)
4 at the point (x0, y0) = (3, 4). Let Q = ln(3 + |Q1|). Then T = 5
sin2 (100Q)
satisfies:— (A) 0 ≤ T < 1. — (B) 1 ≤ T < 2. — (C) 2 ≤ T
< 3. — (D) 3 ≤...

Let Q1, Q2, Q3 be constants so that (Q1, Q2) is the critical
point of the function f(x, y) = xy + y − x, and Q3 = 1 if f has a
local minimum at (Q1, Q2), Q3 = 2 if f has a local maximum at (Q1,
Q2), Q3 = 3 if f has a saddle point at (Q1, Q2), and Q3 = 4
otherwise. Let Q = ln(3 + |Q1| + 2|Q2| + 3|Q3|). Then T =...

Let Q1, Q2, Q3 be constants so that (Q1, Q2) is the critical
point of the function f(x, y) = xy − 5x − 5y + 25, and Q3 = 1 if f
has a local minimum at (Q1, Q2), Q3 = 2 if f has a local maximum at
(Q1, Q2), Q3 = 3 if f has a saddle point at (Q1, Q2), and Q3 = 4
otherwise. Let Q = ln(3 + |Q1| + 2|Q2| + 3|Q3|). Then...

2. Let Q1 = y(2), Q2 = y(3), where y = y(x) solves y' + 2xy =
2x^3 , y(0) = 1. Let Q = ln(3 + |Q1| + 2|Q2|). Then T = 5 sin^2
(100Q) satisfies:— (A) 0 ≤ T < 1. — (B) 1 ≤ T < 2. — (C) 2 ≤
T < 3. — (D) 3 ≤ T < 4. — (E) 4 ≤ T ≤ 5.

Consider the surfaces x^2 + y^2 + z^2 = 1 and (z +√2)^2 = x^2 +
y^2, and let (x0, y0, z0) be a point
in their intersection. Show that the surfaces are tangent at this
point, that is, show that they
have a common tangent plane at (x0, y0, z0).

Topic: Calculus 3 / Differential Equation
Q1) Let (x0, y0,
z0) be a point on the curve C described by the following
equations
F1(x,y,z)=c1 , F2(x,y,z)=c2 .
Show that the vector [grad F1(x0,
y0, z0)] X [grad F2(x0, y0,
z0)] is tangent to C at (x0, y0,
z0)
Q2) (I've posted this question before but
nobody answered, so please do)
Find a vector tangent to the space circle
x2 + y2 + z2 = 1 , x + y +...

prove that the equation of the plane tangent to the sphere x^2 +
y^2 + z^2 = a^2 at the point (x0, y0, z0) on the sphere is x*x0 +
y*y0 + z*z0 = a^2

1. Let T(x, y, z) = (x + z, y − 2x, −z + 2y) and S(x, y, z) =
(2y − z, x − z, y + 3x). Use matrices to find the composition S ◦
T.
2. Find an equation of the tangent plane to the graph of x 2 − y
2 − 3z 2 = 5 at (6, 2, 3).
3. Find the critical points of f(x, y) = (x 2 + y 2 )e −y...

4. Consider the function z = f(x, y) = x^(2) + 4y^(2)
(a) Describe the contour corresponding to z = 1.
(b) Write down the equation of the curve obtained as the
intersection of the graph of z and the plane x = 1.
(c) Write down the equation of the curve obtained as the
intersection of the graph of z and the plane y = 1.
(d) Write down the point of intersection of the curves in (b)
and...

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