Chapter 29 — Quiz

Q: What is the domain of ?

We need the radicand positive (it sits under a square root and in a denominator, so it cannot be zero): , i.e. . The domain is the open disk of radius centered at the origin. (Section 29.2)

Q: For , the level curves are circles of radius . What does the *spacing* of these circles tell you about the surface?

The radii get closer together as you move outward, so equally spaced heights correspond to ever-smaller horizontal steps — the contours crowd together and the paraboloid gets steeper the farther out you go. Contour spacing (not count) encodes steepness. (Section 29.4)

Q: True or false: if along the -axis and along the -axis as , then .

False. Agreement along the two axes is necessary but nowhere near sufficient — the limit must agree along every path, of which there are infinitely many. The classic counterexample gives on both axes but along . (Section 29.6)

Q: Show that does not exist.

Approach along : . This depends on the slope — it is along the -axis () but along (). Different paths give different values, so the limit does not exist. (Section 29.6)

Q: Compute and for .

Hold fixed for : . Hold fixed for : . (Section 29.8)

Q: For , compute .

Hold constant; the chain rule supplies the derivative of the exponent in , which is : . (By symmetry .) (Section 29.8)

Q: State Clairaut's theorem, and verify it for .

Clairaut's theorem: if and are both continuous on an open region containing a point, they are equal there — the order of mixed differentiation does not matter. Check: ; and . They agree. (Section 29.9)

Q: Write the equation of the tangent plane to at the point above .

Here , , and . The tangent plane is $z = 2x + 4y - 5$. (Section 29.10)

Q: A level surface of at value is what geometric object? And what does look like along the level surface ?

For it is the sphere of radius centered at the origin. For the only solution is the single point — the sphere of radius zero. (Section 29.5)

Q: Use the linearization of at the base point to estimate . What is the exact value, and why does the error equal exactly that gap?

, , . With , : $1.1^2 + 0.9^2 = 1.21 + 0.81 = 2.020.02(\Delta x)^2 + (\Delta y)^2 = 0.01 + 0.01 = 0.02$, since the linearization keeps only the first-order terms. (Section 29.10)

DataField.Dev

Chapter 29 — Quiz

10 questions covering domains, level curves and surfaces, multivariable limits, partial derivatives, Clairaut's theorem, and tangent planes. Try each before opening the answer. Section references point back to the chapter.

1. What is the domain of $f(x, y) = \dfrac{1}{\sqrt{4 - x^2 - y^2}}$?

Answer

We need the radicand positive (it sits under a square root *and* in a denominator, so it cannot be zero): $4 - x^2 - y^2 > 0$, i.e. $x^2 + y^2 < 4$. The domain is the **open disk** of radius $2$ centered at the origin. *(Section 29.2)*

2. For $f(x, y) = x^2 + y^2$, the level curves $f = 1, 2, 3, 4$ are circles of radius $1, \sqrt2, \sqrt3, 2$. What does the spacing of these circles tell you about the surface?

Answer

The radii $1,\ 1.41,\ 1.73,\ 2$ get **closer together** as you move outward, so equally spaced heights correspond to ever-smaller horizontal steps — the contours crowd together and the paraboloid gets **steeper** the farther out you go. Contour *spacing* (not count) encodes steepness. *(Section 29.4)*

3. True or false: if $f(x, y) \to L$ along the $x$-axis and along the $y$-axis as $(x, y) \to (0, 0)$, then $\displaystyle\lim_{(x,y)\to(0,0)} f(x, y) = L$.

Answer

**False.** Agreement along the two axes is necessary but nowhere near sufficient — the limit must agree along *every* path, of which there are infinitely many. The classic counterexample $f = \dfrac{2xy}{x^2+y^2}$ gives $0$ on both axes but $1$ along $y = x$. *(Section 29.6)*

4. Show that $\displaystyle\lim_{(x,y)\to(0,0)} \frac{xy}{x^2 + y^2}$ does not exist.

Answer

Approach along $y = mx$: $\dfrac{x \cdot mx}{x^2 + m^2 x^2} = \dfrac{m x^2}{(1 + m^2)x^2} = \dfrac{m}{1 + m^2}$. This depends on the slope $m$ — it is $0$ along the $x$-axis ($m = 0$) but $\tfrac12$ along $y = x$ ($m = 1$). Different paths give different values, so the limit **does not exist**. *(Section 29.6)*

5. Compute $f_x$ and $f_y$ for $f(x, y) = x^3 y - 2x y^2 + 5y$.

Answer

Hold $y$ fixed for $f_x$: $f_x = 3x^2 y - 2y^2$. Hold $x$ fixed for $f_y$: $f_y = x^3 - 4xy + 5$. *(Section 29.8)*

6. For $g(x, y) = e^{xy}$, compute $g_x$.

Answer

Hold $y$ constant; the chain rule supplies the derivative of the exponent $xy$ in $x$, which is $y$: $g_x = y\,e^{xy}$. (By symmetry $g_y = x\,e^{xy}$.) *(Section 29.8)*

7. State Clairaut's theorem, and verify it for $f(x, y) = x^2 y^3$.

Answer

**Clairaut's theorem:** if $f_{xy}$ and $f_{yx}$ are both continuous on an open region containing a point, they are equal there — the order of mixed differentiation does not matter. Check: $f_x = 2x y^3 \Rightarrow f_{xy} = 6x y^2$; and $f_y = 3 x^2 y^2 \Rightarrow f_{yx} = 6x y^2$. They agree. *(Section 29.9)*

8. Write the equation of the tangent plane to $z = x^2 + y^2$ at the point above $(1, 2)$.

Answer

Here $f(1,2) = 1 + 4 = 5$, $f_x = 2x \Rightarrow f_x(1,2) = 2$, and $f_y = 2y \Rightarrow f_y(1,2) = 4$. The tangent plane is $$z = 5 + 2(x - 1) + 4(y - 2),$$ which simplifies to $z = 2x + 4y - 5$. *(Section 29.10)*

9. A level surface of $f(x, y, z) = x^2 + y^2 + z^2$ at value $c = 9$ is what geometric object? And what does $f$ look like along the level surface $c = 0$?

Answer

For $c = 9$ it is the **sphere** of radius $\sqrt 9 = 3$ centered at the origin. For $c = 0$ the only solution is the single point $(0,0,0)$ — the sphere of radius zero. *(Section 29.5)*

10. Use the linearization of $f(x, y) = x^2 + y^2$ at the base point $(1, 1)$ to estimate $f(1.1, 0.9)$. What is the exact value, and why does the error equal exactly that gap?

Answer

$f(1,1) = 2$, $f_x(1,1) = 2$, $f_y(1,1) = 2$. With $\Delta x = 0.1$, $\Delta y = -0.1$: $$L = 2 + 2(0.1) + 2(-0.1) = 2.$$ Exact: $1.1^2 + 0.9^2 = 1.21 + 0.81 = 2.02$. The error $0.02$ is exactly the discarded second-order piece $(\Delta x)^2 + (\Delta y)^2 = 0.01 + 0.01 = 0.02$, since the linearization keeps only the first-order terms. *(Section 29.10)*

Scoring Guide

Score	Interpretation
9–10	Excellent. You command surfaces, multivariable limits, partials, Clairaut, and tangent planes. Move confidently to Chapter 30 (gradient and directional derivatives).
7–8	Solid. Revisit any missed item — most likely the two-path limit logic (Q3–Q4) or the linearization error accounting (Q10).
5–6	Partial grasp. Re-read Sections 29.6 (limits) and 29.8–29.10 (partials, tangent plane) and redo the corresponding exercises.
0–4	Rework the chapter from Section 29.3 onward, doing every "Check Your Understanding" box before retrying the quiz.