Calculus: Functions and Derivatives Quiz

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Questions and Answers

Which statement about the scalar function f(x, y, z) = xyz and vector field F(x, y, z) = (x, y, z) is correct at the point (1, 1, 1)?

∇ · (∇ f) = 3

∇ · (∇ × F) = 0 (correct)

∇ × ∇ f = 1

F is compressible

What property does the vector field F represent if it is said to be incompressible?

The divergence of F is equal to one.

The curl of F is equal to zero.

The divergence of F is equal to zero. (correct)

The field has no flow lines.

If C is the parameterized curve r(t) = (4 sin t, 4 cos t) for t ∈ [0, π], what is the orientation of the curve?

Counterclockwise direction on the xy-plane. (correct)

Clockwise direction on the xy-plane.

Vertical direction along the z-axis.

No specific orientation defined.

Which mathematical concept guarantees the existence of a potential function for the vector field F?

F is curl-free. Signup and view all the answers

What does the gradient ∇f of a scalar function represent in a physical context?

The rate of change of the function in all directions. Signup and view all the answers

Considering the function f(x, y) = xy, what is ∂f/∂y when evaluated at the point (1, 1)?

1 Signup and view all the answers

Which equation represents the curl operation applied to a vector field F = (P, Q, R)?

∇ × F = (∂R/∂y - ∂Q/∂z, ∂P/∂z - ∂R/∂x, ∂Q/∂x - ∂P/∂y) Signup and view all the answers

What is the significance of the directional derivative of a scalar function?

It measures the rate of change in a specific direction. Signup and view all the answers

What is the correct expression for f'(t) when f(t) = h(x(t), y(t)) using the chain rule?

f'(t) = ∂h/∂x * x'(t) + ∂h/∂y * y'(t) Signup and view all the answers

Which expression evaluates f'(1) given that x(t) = t and y(t) = t^2 when using provided partial derivatives?

f'(1) = 6 Signup and view all the answers

To ensure the vector field F(x, y, z) = (y, x + g(z), 7yz^6) satisfies ∇ × F = (0, 0, 0), which function g(z) should be chosen?

g(z) = 3z^7 Signup and view all the answers

What is the directional derivative of the function f at the point (2, 2, 2) in the direction of the vector (2, 0, 0)?

4 Signup and view all the answers

How does the divergence of a vector field relate to its physical interpretation?

It measures how much the vector field spreads out from a point. Signup and view all the answers

Which of the following statements about tangent planes is true?

A tangent plane at a given point is determined by the gradient at that point. Signup and view all the answers

What is the correct interpretation of a vector field being curl-free?

It indicates that the vector field is conservative. Signup and view all the answers

Which of the following best represents the relation between path independence and conservative vector fields?

In a conservative field, path independence is guaranteed for single-valued functions. Signup and view all the answers

What is the result of calculating the curl of the vector field defined by F = (x, -y, 0)?

(0, 0, 1) Signup and view all the answers

If the function f(x, y, z) has a level surface defined by f(x, y, z) = k, what can be inferred about the gradient of f at that surface?

It is orthogonal to the surface. Signup and view all the answers

Given a conservative vector field F, which of the following statements is true?

The line integral over any path is zero. Signup and view all the answers

What is the expression for the directional derivative of a function f at a point P in the direction of a unit vector u?

∇f · u Signup and view all the answers

In the context of double integrals, what does the term 'dA' represent?

A differential area element. Signup and view all the answers

Which of the following best describes the tangent plane at a point on a surface defined by a function z = f(x, y)?

It is defined by the normal vector at that point. Signup and view all the answers

What is the primary interpretation of the divergence of a vector field F?

The rate of expansion of the field at a point. Signup and view all the answers

What is the outcome of applying the chain rule to the function z = g(x, y) where both x and y are functions of t?

dz/dt = ∂g/∂x dx/dt + ∂g/∂y dy/dt Signup and view all the answers

Study Notes

Function Definition and Differentiation

Let ( f: \mathbb{R} \to \mathbb{R} ) be defined as ( f(t) = h(x(t), y(t)) ), where ( h: \mathbb{R}^2 \to \mathbb{R} ) is differentiable.
The derivative ( f'(t) ) can be calculated using the chain rule.

Differentiation Using Chain Rule

For ( x(t) ) and ( y(t) ):
- ( f'(t) = \frac{\partial h}{\partial x}(x(t), y(t)) \cdot x'(t) + \frac{\partial h}{\partial y}(x(t), y(t)) \cdot y'(t) )

Specific Function Evaluation

Given ( x(t) = t ) and ( y(t) = t^2 ):
- Calculate derivatives: ( x'(t) = 1 ) and ( y'(t) = 2t )
The evaluation point ( t = 1 ):
- ( f'(1) = \frac{\partial h}{\partial x}(1,2) \cdot 1 + \frac{\partial h}{\partial y}(1,1) \cdot 2 )

Provided Partial Derivatives

( \frac{\partial h}{\partial x}(1, 2) = 3 )
( \frac{\partial h}{\partial x}(1, 1) = 2 )
( \frac{\partial h}{\partial y}(1, 2) = -4 )
( \frac{\partial h}{\partial y}(1, 1) = 1 )

Value Calculation for ( f'(1) )

Substituting values:
- ( f'(1) = 3 \cdot 1 + 2 \cdot 2 )
- ( f'(1) = 3 + 4 = 7 )

Vector Field and Curl Condition

Given vector field ( F(x, y, z) = (y, x + g(z), 7yz^6) )
Condition for curl ( \nabla \times F = (0, 0, 0) )
Find function ( g(z) ) that meets this criterion.

Directional Derivative

Compute directional derivative of ( f(x, y, z) = xy^2 z + g(z) ) at point ( (2, 2, 2) ) along vector ( (2, 0, 0) ).
Requires calculating gradients and projecting along the direction of the vector.

Gradient and Divergence Properties

For scalar function ( f(x, y, z) = xyz ) and vector field ( F(x, y, z) = (x, y, z) ):
- Verify properties at point ( (1, 1, 1) ):
  - ( \nabla \times \nabla f = 0 ) (not equal to 1).
  - ( \nabla \cdot (\nabla \times F) = 0 ).
  - ( F ) is deemed incompressible if ( \nabla \cdot F = 0 ).

Parametric Curve and Orientation

Curve ( r(t) = (4 \sin t, 4 \cos t) ) represents circular motion for ( t \in [0, \pi] ).
Check options for the curve's orientation.

Line Integral Evaluation

Evaluate the line integral ( \int_C (x , dy - y , dx) ) over the curve ( C ).
Use parametrization of the curve to compute the integral value.

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Description

Test your understanding of differentiable functions and the chain rule in this quiz. Determine which statement regarding the derivative of the function f(t) is true based on the given conditions. Enhance your calculus knowledge with practical application!