Question 1

Find an equation of the plane tangent to the surface xy³z² = 2 at the point (2, 1, -1). A) x = (2 + t) i + (1 + 6t) j + (-1 - 4t) k, t R B) x + 6y + 4z = 4 C) x + 6y - 4z = 12 D) x + 6y - 4z = 0 E) x + 5y - 4z = 11

Accepted Answer

The answer of Find an equation of the plane tangent...

Question 2

The directional derivative of a function F(x,y) at a point P in the direction of the unit vector-  i +  j is equal to -4, while the directional derivative at P in the direction of the unit vector  i+  j is equal to0. The value F(P) is equal to:
A) 20
B) ± 2 
C)  i -  j
D)  i -  j
E) 4 i - 2 j

Accepted Answer

The answer of The directional derivative of a function F(x,y)...

Question 3

Find equations of the tangent line at the point (-2, 1, 5) to the hyperbola that is the intersection of the surface z = 2x² - 3y² and the plane z = 5. A) 3x + 4y = 2, z = 5 B) 4x + 3y = 5, z = -5 C) 4x + 3y = -5, z = 5 D) 4x - 3y = -5, z = -5 E) 3x + 4y = -2, z = 5

Accepted Answer

The surface equation is z = 2x^2 - 3y^2. At the point (-2, 1, 5), the plane z = 5 intersects the surface, giving the equation 5 = 2(-2)^2 - 3(1)^2, which holds true. The gradient of the surface at this point is ∇f = (∂f/∂x, ∂f/∂y) = (8x, -6y). At (-2, 1), this is (-16, -6). The tangent line in the xy-plane is perpendicular to this gradient, giving the equation 4x + 3y = -5. The z-coordinate remains constant at z = 5.

Question 4

Find the two unit vectors tangent at the point (1, 1, 1) to the curve of intersection of the surfaces xy² + x²y + z³ = 3 and x³ - y³ - xyz = -1. A) ± ( i + j - 2 k ) B) ± ( i - j - 2 k ) C) ± ( i - 2 j - 2 k ) D) ± ( i + j - k ) E) ± ( i + j - 2 k )

Accepted Answer

The answer of Find the two unit vectors tangent at...

Question 5

A particle is moving so that at time t its position is given by  . Find the derivative of f(x, y, z) = xyz at the location of the particle at time t =  in the direction in which the particle is moving at that time.
A) - 
B) 
C) - $\pi$
D) 
E) 3 $\pi$

Accepted Answer

The answer of A particle is moving so that at...

Question 6

The electric potential V at time t at any point (x, y) in the plane is given by V = sin . An observer is moving in the plane so that at time t her position is given by x = t³, y = 2 -t⁴. How fast does she experience V changing at time t = 1? A) - + B) + C) - - D) - - E) +

Accepted Answer

The answer of The electric potential V at time t...

Question 7

Find the second directional derivative of f(x, y) = e^2x (1 + y²) at (0, 2) in the direction of the vector i - j. A) 1 B) 2 C) 3 D) 4 E) 0

Accepted Answer

The answer of Find the second directional derivative of f(x,...

Question 8

Find the second directional derivative of f(x, y) = cos(xy) + sin(xy) at ( $\pi$, 1/4) in the direction of the vector  i + j.
A) -  - 
B) -  - 
C) -  + 
D) -  - 
E) -  -

Accepted Answer

The answer of Find the second directional derivative of f(x,...

Question 9

Near what points can the equation 2x² + xy + y² = 7 not be solved for y as a function of x? A) (2, 1) and (-2, -1) B) (2, -1) and (-2, 1) C) (2, 1), (-2, -1), (2, -1), and (-2, 1) D) (2, 1) and (2, -1) E) (2, -1), (-2, 1), and (0, 0)

Accepted Answer

The answer of Near what points can the equation 2x²...

Question 10

The equation x + y² + sin(xy) = 1 defines y as a function of x near the point (0, 1). Find the value of at that point. A) -2 B) -1 C) 1 D) 2 E) 0

Accepted Answer

The answer of The equation x + y² + sin(xy)...

Question 11

A) 
B) - 
C) 
D) - 
E) -

Accepted Answer

The answer of 
A) 
B) - 
C) 
D) - 
E)...

Question 12

Do the equations x² + y²s + yt² =13 and y² + x²s + xt² = 9 define s and t as functions of x and y near the point (x, y, s, t) = (1, 2, 1, -2)? If so, find at that point. A) 2 B) 1 C) 0 D) -2 E) The equations do not define s and t as functions of x and y near the given point.

Accepted Answer

The answer of Do the equations x² + y²s +...

Question 13

If the equations x = u² - v² and y = 2uv define u and v as functions of x and y, find and . A) = , = - B) = , = - C) = , = D) = - , = E) = - , = -

Accepted Answer

The answer of If the equations x = u² -...

Question 14

The equations x = u² - v² and y = 2uv define u and v as functions of x and y in a neighbourhood of the point where u = 2 and v = 1. Evaluate at that point. A) - B) C) D) - E)

Accepted Answer

The answer of The equations x = u² - v²...

Question 15

Suppose the system of equations can be solved for u and v as functions of x, y, and z near the point P₀ where (x, y, z, u, v) = (1, 1, 1, 1, 1).Compute at (x, y, z) = (1, 1, 1). A) 2 B) C) - 2 D) - E) -

Accepted Answer

The answer of Suppose the system of equations  can...

Question 16

Given that the relation y² + y = 14 - sin(xz²) + implicitly defines x as a differentiable function of y and z, find at the point (0, 3, 4). A) - B) 2 C) -8 D) - E) 8

Accepted Answer

The answer of Given that the relation y² + y...

Question 17

Assume that the relation  - 65 +  = 0 defines z as a differentiable functionof x and y near the point (x , y) = (4 , 0).
(a) If z = f(x , y) , find  and  at (x,y) = (4 , 0).
(b) If x =  +  , y =  , find  at (u , v) =(0 , 2)\.
Hints :Part (a): First , find the value of z at (x , y) =(4 , 0).
Part (b): Use the chain rule!

Accepted Answer

The answer of Assume that the relation  - 65...

Question 18

The Implicit Function Theorem implies that the system of equations u = x² - xy, v = 2xy - y² has a solution for x and y as functions of u and v valid in a neighbourhood of the point where x = y = 1.

Accepted Answer

The Implicit Function Theorem states that if a system of equations can be locally expressed with continuously differentiable functions and the Jacobian matrix of the system (with respect to the variables we want to solve for) is non-singular at a point, then there exists a neighborhood around that point where solutions for the variables as functions of the others can be found. In this case, without explicitly calculating the Jacobian, the presence of x and y in both equations in a non-trivial manner (indicating dependence on x and y) and the assumption of differentiability (implicit in the context of the theorem's application) suggest that, at least near the point (x, y) = (1, 1), the system meets the criteria for the Implicit Function Theorem to guarantee local solutions for x and y as functions of u and v.

Question 19

Find the Taylor series of the function e^xy about the point (2, 3), by using Taylor series for functions of one variable. A) B) C) D) E)

Accepted Answer

The answer of Find the Taylor series of the function...

Question 20

Find the Taylor series of the function  about the point (-1, 0), by using Taylor series for functions of one variable.
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the Taylor series of the function...

Quiz 13: Partial Differentiation

Find an equation of the plane tangent to the surface xy³z² = 2 at the point (2, 1, -1) .

The directional derivative of a function F(x,y) at a point P in the direction of the unit vector- i + j is equal to -4, while the directional derivative at P in the direction of the unit vector i+ j is equal to0. The value ofF(P) is equal to:

Find equations of the tangent line at the point (-2, 1, 5) to the hyperbola that is the intersection of the surface z = 2x² - 3y² and the plane z = 5.

Find the two unit vectors tangent at the point (1, 1, 1) to the curve of intersection of the surfaces xy² + x²y + z³ = 3 and x³ - y³ - xyz = -1.

The electric potential V at time t at any point (x, y) in the plane is given by V = sin . An observer is moving in the plane so that at time t her position is given by x = t³, y = 2 -t⁴. How fast does she experience V changing at time t = 1?

Find the second directional derivative of f(x, y) = e^2x (1 + y²) at (0, 2) in the direction of the vector i - j.

Find the second directional derivative of f(x, y) = cos(xy) + sin(xy) at ( $\pi$ , 1/4) in the direction of the vector $Find the second directional derivative of f(x, y) = cos(xy) + sin(xy) at ( \pi , 1/4) in the direction of the vector i + j. A) - - B) - - C) - + D) - - E) - -$ i + j.

Near what points can the equation 2x² + xy + y² = 7 not be solved for y as a function of x?

The equation x + y² + sin(xy) = 1 defines y as a function of x near the point (0, 1) . Find the value of at that point.

Do the equations x² + y²s + yt² =13 and y² + x²s + xt² = 9 define s and t as functions of x and y near the point (x, y, s, t) = (1, 2, 1, -2) ? If so, find at that point.

If the equations x = u² - v² and y = 2uv define u and v as functions of x and y, find and .

The equations x = u² - v² and y = 2uv define u and v as functions of x and y in a neighbourhood of the point where u = 2 and v = 1. Evaluate at that point.

Suppose the system of equations can be solved for u and v as functions of x, y, and z near the point P₀ where (x, y, z, u, v) = (1, 1, 1, 1, 1) .Compute at (x, y, z) = (1, 1, 1) .

Given that the relation y² + y = 14 - sin(xz²) + implicitly defines x as a differentiable function of y and z, find at the point (0, 3, 4) .

The Implicit Function Theorem implies that the system of equations u = x² - xy, v = 2xy - y² has a solution for x and y as functions of u and v valid in a neighbourhood of the point where x = y = 1.

Find the Taylor series of the function e^xy about the point (2, 3) , by using Taylor series for functions of one variable.

Find the Taylor series of the function about the point (-1, 0) , by using Taylor series for functions of one variable.

Quiz 13: Partial Differentiation

Find an equation of the plane tangent to the surface xy3z2 = 2 at the point (2, 1, -1) .

The directional derivative of a function F(x,y) at a point P in the direction of the unit vector- i + j is equal to -4, while the directional derivative at P in the direction of the unit vector i+ j is equal to0. The value ofF(P) is equal to:

Find equations of the tangent line at the point (-2, 1, 5) to the hyperbola that is the intersection of the surface z = 2x2 - 3y2 and the plane z = 5.

Find the two unit vectors tangent at the point (1, 1, 1) to the curve of intersection of the surfaces xy2 + x2y + z3 = 3 and x3 - y3 - xyz = -1.

A particle is moving so that at time t its position is given by . Find the derivative of f(x, y, z) = xyz at the location of the particle at time t = in the direction in which the particle is moving at that time.

The electric potential V at time t at any point (x, y) in the plane is given by V = sin . An observer is moving in the plane so that at time t her position is given by x = t3, y = 2 -t4. How fast does she experience V changing at time t = 1?

Find the second directional derivative of f(x, y) = e2x (1 + y2) at (0, 2) in the direction of the vector i - j.

Find the second directional derivative of f(x, y) = cos(xy) + sin(xy) at ( π\piπ , 1/4) in the direction of the vector i + j.

Near what points can the equation 2x2 + xy + y2 = 7 not be solved for y as a function of x?

The equation x + y2 + sin(xy) = 1 defines y as a function of x near the point (0, 1) . Find the value of at that point.

Do the equations x2 + y2s + yt2 =13 and y2 + x2s + xt2 = 9 define s and t as functions of x and y near the point (x, y, s, t) = (1, 2, 1, -2) ? If so, find at that point.

If the equations x = u2 - v2 and y = 2uv define u and v as functions of x and y, find and .

The equations x = u2 - v2 and y = 2uv define u and v as functions of x and y in a neighbourhood of the point where u = 2 and v = 1. Evaluate at that point.

Suppose the system of equations can be solved for u and v as functions of x, y, and z near the point P0 where (x, y, z, u, v) = (1, 1, 1, 1, 1) .Compute at (x, y, z) = (1, 1, 1) .

Given that the relation y2 + y = 14 - sin(xz2) + implicitly defines x as a differentiable function of y and z, find at the point (0, 3, 4) .

The Implicit Function Theorem implies that the system of equations u = x2 - xy, v = 2xy - y2 has a solution for x and y as functions of u and v valid in a neighbourhood of the point where x = y = 1.

Find the Taylor series of the function exy about the point (2, 3) , by using Taylor series for functions of one variable.

Find the Taylor series of the function about the point (-1, 0) , by using Taylor series for functions of one variable.

Find an equation of the plane tangent to the surface xy³z² = 2 at the point (2, 1, -1) .

Find equations of the tangent line at the point (-2, 1, 5) to the hyperbola that is the intersection of the surface z = 2x² - 3y² and the plane z = 5.

Find the two unit vectors tangent at the point (1, 1, 1) to the curve of intersection of the surfaces xy² + x²y + z³ = 3 and x³ - y³ - xyz = -1.

The electric potential V at time t at any point (x, y) in the plane is given by V = sin . An observer is moving in the plane so that at time t her position is given by x = t³, y = 2 -t⁴. How fast does she experience V changing at time t = 1?

Find the second directional derivative of f(x, y) = e^2x (1 + y²) at (0, 2) in the direction of the vector i - j.

Find the second directional derivative of f(x, y) = cos(xy) + sin(xy) at ( $\pi$ , 1/4) in the direction of the vector $Find the second directional derivative of f(x, y) = cos(xy) + sin(xy) at ( \pi , 1/4) in the direction of the vector i + j. A) - - B) - - C) - + D) - - E) - -$ i + j.

Near what points can the equation 2x² + xy + y² = 7 not be solved for y as a function of x?

The equation x + y² + sin(xy) = 1 defines y as a function of x near the point (0, 1) . Find the value of at that point.

Do the equations x² + y²s + yt² =13 and y² + x²s + xt² = 9 define s and t as functions of x and y near the point (x, y, s, t) = (1, 2, 1, -2) ? If so, find at that point.

If the equations x = u² - v² and y = 2uv define u and v as functions of x and y, find and .

The equations x = u² - v² and y = 2uv define u and v as functions of x and y in a neighbourhood of the point where u = 2 and v = 1. Evaluate at that point.

Suppose the system of equations can be solved for u and v as functions of x, y, and z near the point P₀ where (x, y, z, u, v) = (1, 1, 1, 1, 1) .Compute at (x, y, z) = (1, 1, 1) .

Given that the relation y² + y = 14 - sin(xz²) + implicitly defines x as a differentiable function of y and z, find at the point (0, 3, 4) .

The Implicit Function Theorem implies that the system of equations u = x² - xy, v = 2xy - y² has a solution for x and y as functions of u and v valid in a neighbourhood of the point where x = y = 1.

Find the Taylor series of the function e^xy about the point (2, 3) , by using Taylor series for functions of one variable.