Q: Which of the following is the correct definition of the partial derivative $$f_x(a, b)$$?

$$f_x(a, b) = \lim_{h \to 0} \frac{f(a + h, b) - f(a, b)}{h}$$

Question 1

Given $$f(x, y) = x^2$ + y^2$, what is the value of the partial derivative $$\frac{\partial f}{\partial x}$$ at the point (1$$, 2)$$?

Accepted Answer

$$2x$$ evaluated at $$(1,2) is$$ $$2$$

Question 2

Which of the following best describes the geometric meaning of the partial derivative $$\frac{\partial f}{\partial x} at a $$point $$(a, b)$$ for a surface $$z = f(x, y)$$?

Accepted Answer

It is the slope of the tangent line to the curve obtained by intersecting the surface with the plane $$y = b$$

Question 3

Let $$f(x, y) = 4 - x^2$ - 2y^2$. Find f_x(1$$, 1)$$ and interpret its meaning.

Accepted Answer

$$f_x(1, 1) = -2$$, which is the slope of the tangent to the curve $$y = 1 at$$ $$x = 1$$

Question 4

Given $$f(x, y) = 3x^2$ + 5xy^2$ + x^3$ + 4x^2$ + 10y$$, calculate $$\frac{\partial f}{\partial x}.$$

Accepted Answer

$$6x + 5y^2$ + 3x^2$ + 8x$$

Question 5

If $$f(x, y) = x^2$y^2$$ + $$2xy^2 + x$, find f$$_{xy}$.

Accepted Answer

$$4xy + 2$$

Question 6

Which of the following is the correct definition of the partial derivative $$f_x(a, b)$$?

Accepted Answer

$$f_x(a, b) = \lim_{h 	o 0} \frac{f(a + h, b) - f(a, b)}{h}$$

Question 7

Suppose $$f(x, y) = x^3$y^2$$ - $$2xy^2. $$Find $$f_{yy}$.

Accepted Answer

$$2x^3 - 4x$

Question 8

Clairaut's Theorem states that if $$f_{xy}$$ and $$f_{yx}$$ are both continuous on a disk $$D$$ containing $$(a, b)$$, then:

Accepted Answer

$$f_{xy}(a, b) = f_{yx}(a, b)$$

Question 9

Which of the following is a partial differential equation?

Accepted Answer

$$\frac{\$$partial^2$$ u}{\partial x^2$} + \frac{\$$partial^2$$ u}{\partial y^2$} = 0$$

Question 10

If $$u(x, y) = e^x \sin y$$, which of the following is true regarding Laplace's equation $$\frac{\$$partial^2$$ u}{\partial x^2$} + \frac{\$$partial^2$$ u}{\partial y^2$} = 0$$?

Accepted Answer

$$u(x, y) is a $$solution to Laplace's equation

Question 11

Given $$u(x, t) = \sin(x - at)$$, show that $$u is a $$solution to the wave equation $$\frac{\$$partial^2$$ u}{\partial t^2$} = a^2$ \frac{\$$partial^2$$ u}{\partial x^2$}.$$

Accepted Answer

Both sides equal $$-a^2$$ \sin(x - at)$$, so u$ is a solution

Question 12

If $$f(x, y) = x^2$y^2$$ + $$2xy^2 + x$, what is f$$_{yx}$?

Accepted Answer

$$4xy + 2$$

Question 13

Which notation is equivalent to $$\frac{\partial f}{\partial x}$$?

Accepted Answer

$$f_x$$

Question 14

For $$f(x, y) = x^2$ + y^2$, what is f$$_{yy}$$?

Accepted Answer

$$2$$

Question 15

If $$f(x, y) = x^2$y^2$$ - $$2xy^2$$, find $$f_{xy}(1, 2)$.

Accepted Answer

$$4$$

Question 16

Which of the following is NOT a rule for finding partial derivatives?

Accepted Answer

To find $$f_x$$, treat $$x as a $$constant and differentiate $$f(x, y)$$ with respect to $$y$$

Question 17

If $$f(x, y) = x^2$y^2$$ - $$2xy^2$$, what is $$f_{xx}$?

Accepted Answer

$$2y^2$$

Question 18

Which of the following statements about higher partial derivatives is correct?

Accepted Answer

If $$f_{xy}$$ and $$f_{yx}$$ are continuous, then $$f_{xy} = f_{yx}$$

Section 4.3 - Partial Derivatives and Applications

Study Guide - Practice Questions

Study Guide - Flashcards