4.1.3 Practice set: partial derivatives

Functions of two variables

Functions of two variables: surfaces and level curves (lesson 4.1.1)

0 of 6 correct

1.
For $f(x, y) = 2x^2 - xy + y$ , find $f(-1, 3)$
2.
The domain of $\sqrt{4 - x^2 - y^2}$ is a disc. How far along the positive $x$ -axis does it reach?
3.
Write the slice $x = 2$ of $x^2 - 3xy + y^2$ as a function of $y$
4.
Which level curve of $x^2 + 4y^2$ passes through $(2, 1)$ ? Give its height $c$
5.
Along the $x$ -axis, how far apart are the level curves $f = 0$ and $f = 6$ of the plane $3x + 2y$ ?
6.
Where does the level curve $x^2 + 2y^2 = 9$ cross the positive $x$ -axis?

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Partial derivatives

Partial derivatives (lesson 4.1.2)

0 of 9 correct

1.
For $f(x, y) = x^3 + 4xy - y^2$ , find $f_x$
2.
For $g(x, y) = \ln(x^2 + y^2)$ , find $g_x$ at $(1, 1)$
3.
A rectangle has area $A = wh$ cm². Find $\frac{\partial A}{\partial w}$ when $h = 5$ , in cm² per cm
4.
For $f(x, y) = x^3y^2$ , find $f_{xx}$ at $(1, 2)$
5.
For $f(x, y) = x^2y^3$ , find $f_{xy}$ at $(1, 1)$
6.
Estimate $f_y$ for $f(x, y) = xy^2$ at $(2, 1)$ by the central difference with $h = 0.1$
7.
For $f(x, y, z) = xy^2z^3$ , find $f_z$ at $(2, 1, 1)$
8.
Find $\frac{\partial}{\partial x}(x^2y + y^3)$
9.
Does $f_x$ exist at $(0, 3)$ for $f(x, y) = |x| + y^2$ ? Type 1 for yes, 0 for no

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Coding exercise

Exercise

Partial derivatives by central differences

Write two functions. In both, f takes one argument per variable, so f(*point) gives its value. point is a list such as [1, 2] or [2, 3, 4].

partials(f, point, h=1e-5) returns a list with one central difference for each variable, as in lesson 4.1.2. Move that coordinate by $\pm h$ and keep the others fixed.
second_partial(f, point, i, j, h=1e-4) returns the second partial in variable i, then variable j (counting from 0). For i == j use the second central difference; otherwise use the four corner points.

Raise ValueError if h is not positive. Do not change the list point: build a new list for each moved point.

The tests use the bowl $x^2 + 2y^2$ , the box volume $xyz$ , $e^{xy}$ and $x^2y + 3y^2$ from lessons 4.1.1 and 4.1.2. One test needs 8 correct digits, which a one-sided difference cannot reach.

See the tests

import math


def test_bowl():
    """x² + 2y² at (1, 1) has partials 2 and 4"""
    expect_close(partials(lambda x, y: x**2 + 2 * y**2, [1, 1]), [2, 4], what="partials of x² + 2y² at (1, 1)", rel=1e-8)


def test_three_variables():
    """The box volume xyz at (2, 3, 4) has partials 12, 8 and 6"""
    expect_close(partials(lambda x, y, z: x * y * z, [2, 3, 4]), [12, 8, 6], what="partials of xyz at (2, 3, 4)", rel=1e-8)


def test_central_accuracy():
    """e^(xy) at (1, 2) is accurate to 8 digits, which a one-sided difference is not"""
    e2 = math.exp(2)
    expect_close(partials(lambda x, y: math.exp(x * y), [1, 2]), [2 * e2, e2], what="partials of e^(xy) at (1, 2)", rel=1e-8)


def test_point_unchanged():
    """The point passed in is left as it was"""
    point = [1.0, 2.0]
    partials(lambda x, y: x * y, point)
    second_partial(lambda x, y: x * y, point, 0, 1)
    expect_equal(point, [1.0, 2.0], what="the point after the calls")


def test_second_partials():
    """x²y + 3y² at (2, 1): fxx = 2, fyy = 6 and fxy = 4"""
    f = lambda x, y: x**2 * y + 3 * y**2
    expect_close(second_partial(f, [2, 1], 0, 0), 2, what="fxx", rel=1e-5)
    expect_close(second_partial(f, [2, 1], 1, 1), 6, what="fyy", rel=1e-5)
    expect_close(second_partial(f, [2, 1], 0, 1), 4, what="fxy", rel=1e-5)


def test_mixed_symmetric():
    """For e^(xy) at (1, 2) the mixed partial is 3e², in either order"""
    f = lambda x, y: math.exp(x * y)
    expect_close(second_partial(f, [1, 2], 0, 1), 3 * math.exp(2), what="fxy", rel=1e-5)
    expect_close(second_partial(f, [1, 2], 1, 0), 3 * math.exp(2), what="fyx", rel=1e-5)


def test_three_variable_mixed():
    """For xyz at (2, 3, 4) the mixed partial in x and z is y = 3"""
    expect_close(second_partial(lambda x, y, z: x * y * z, [2, 3, 4], 0, 2), 3, what="the xz partial of xyz", rel=1e-6)


def test_bad_step():
    """A step of 0 raises ValueError"""
    try:
        partials(lambda x, y: x + y, [1, 1], h=0)
    except ValueError:
        return
    raise AssertionError("expected ValueError for h = 0")

Functions of two variables

Partial derivatives

Coding exercise

Free further reading