Chapter 29 — Exercises

40 problems on functions of several variables: domains and ranges, level curves and surfaces, multivariable limits, partial derivatives, higher and mixed partials with Clairaut's theorem, tangent planes and linearization, and applications. ⭐ to ⭐⭐⭐⭐.

These exercises follow the arc of the chapter: from domains and graphs (Sections 29.2–29.3), through level curves and surfaces (29.4–29.5), into multivariable limits and continuity (29.6–29.7), then the heart of the chapter — partial derivatives, higher partials, and Clairaut's theorem (29.8–29.9) — and finally tangent planes, linearization, and applications (29.10–29.13).

Work each problem by hand first, then check with sympy where the chapter showed you how. Star ratings mark difficulty:

• ⭐ Direct application of one definition or rule.
• ⭐⭐ Combines two ideas or requires a short setup.
• ⭐⭐⭐ Multi-step; requires judgment about method.
• ⭐⭐⭐⭐ Challenge: proof, subtle path-dependence, or open-ended modeling.

Part A — Domains and Ranges (Section 29.2)

A1. ⭐ Find and describe the domain of $f(x, y) = \sqrt{x^2 + y^2 - 4}$.

A2. ⭐ Find the domain of $f(x, y) = \ln(4 - x - y)$ and describe it as a region of the plane.

A3. ⭐ Find the domain of $f(x, y) = \dfrac{x + y}{x - y}$.

A4. ⭐ Find the range of $f(x, y) = 4 - x^2 - y^2$.

A5. ⭐⭐ Find both the domain and the range of $f(x, y) = \sqrt{16 - x^2 - y^2}$, and name the surface that is its graph.

Part B — Graphs, Level Curves, and Level Surfaces (Sections 29.3–29.5)

B1. ⭐ Name the surface $z = 3 - x - y$ and describe its shape.

B2. ⭐ For $f(x, y) = x^2 + y^2$, describe the level curve $f = 9$.

B3. ⭐⭐ For $f(x, y) = y - x^2$, sketch the level curves $f = -1, 0, 1$. What family of curves are they?

B4. ⭐⭐ For $f(x, y) = x^2 - y^2$, describe the level curves $f = 4$, $f = -4$, and $f = 0$. Which feature of the surface does the $f = 0$ curve reveal?

B5. ⭐⭐⭐ A contour map of $g(x, y)$ shows tightly bunched, nearly straight contours in the southeast corner and widely spaced, gently curved contours near the center, with a single small nested closed loop in the northwest. In plain words, describe the terrain in each region (steepness, and any peak or pit). Recall from Section 29.4 that the magnitude of the steepest slope — formalized as the gradient in Chapter 30 — is encoded by contour spacing.

Part C — Multivariable Limits (Section 29.6)

C1. ⭐⭐ Evaluate $\displaystyle\lim_{(x,y)\to(1,2)} (x^2 y + 3xy - 5)$, and justify why no path analysis is needed (Section 29.7).

C2. ⭐⭐ Show that $\displaystyle\lim_{(x,y)\to(0,0)} \frac{x^2 - y^2}{x^2 + y^2}$ does not exist by testing the $x$-axis and the $y$-axis.

C3. ⭐⭐⭐ Show that $\displaystyle\lim_{(x,y)\to(0,0)} \frac{xy}{x^2 + y^2}$ does not exist using the family of lines $y = mx$.

C4. ⭐⭐⭐⭐ Use polar coordinates to prove that $\displaystyle\lim_{(x,y)\to(0,0)} \frac{x^2 y}{x^2 + y^2} = 0$. State explicitly the bounding function $g(r)$ and explain why the squeeze theorem (Chapter 3) applies.

Part D — Partial Derivatives (Section 29.8)

D1. ⭐ Find $f_x$ and $f_y$ for $f(x, y) = 3x^2 - 4xy + y^3$.

D2. ⭐ Find $f_x$ and $f_y$ for $f(x, y) = x^2 y^3$.

D3. ⭐⭐ Find $f_x$ and $f_y$ for $f(x, y) = e^{xy}$.

D4. ⭐⭐ Find $f_x$ and $f_y$ for $f(x, y) = \dfrac{x}{y} + \dfrac{y}{x}$.

D5. ⭐⭐⭐ Find $f_x$ and $f_y$ for $f(x, y) = \sin(x^2 y) + \ln(x + y)$.

D6. ⭐⭐⭐⭐ For $f(x, y) = x^y$ (with $x > 0$), find $f_x$ and $f_y$. (Hint: for $f_y$, write $x^y = e^{y \ln x}$.)

Part E — Higher and Mixed Partials; Clairaut (Section 29.9)

E1. ⭐ For $f(x, y) = x^3 y^2$, find $f_{xx}$ and $f_{yy}$.

E2. ⭐⭐ For $f(x, y) = x^3 y^2$, find both mixed partials $f_{xy}$ and $f_{yx}$, verify they agree, and name the theorem that guarantees it.

E3. ⭐⭐ For $f(x, y) = \cos(xy)$, compute $f_{xy}$.

E4. ⭐⭐⭐ A function satisfies $f_x = 2xy + \cos x$ and $f_y = x^2 + e^y$. Use Clairaut's theorem as a consistency check: compute $f_{xy}$ from the first expression and $f_{yx}$ from the second. Could such an $f$ exist?

Part F — Tangent Planes, Linearization, Total Differential (Section 29.10)

F1. ⭐ Write the tangent plane to $z = x^2 + y^2$ at the point $(1, 2)$.

F2. ⭐⭐ Find the linearization $L(x, y)$ of $f(x, y) = \sqrt{x^2 + y^2}$ at $(3, 4)$ and use it to estimate $f(3.1, 3.9)$.

F3. ⭐⭐⭐ Use the total differential to estimate $(2.01)^2 (2.98)$ by taking $f(x, y) = x^2 y$ with base point $(2, 3)$. Compare with the exact value.

F4. ⭐⭐⭐⭐ The single-variable linearization of Chapter 11 is the $y$-frozen slice of the multivariable one. Show this: starting from the tangent plane $z = L(x, y)$ of $f$ at $(a, b)$, set $y = b$ and verify the result is exactly the Chapter 11 tangent line to the slice $x \mapsto f(x, b)$ at $x = a$.

Part G — Functions of Three Variables (Section 29.11)

G1. ⭐⭐ For $f(x, y, z) = x^2 y + y^2 z + z^2 x$, find $f_x$, $f_y$, $f_z$.

G2. ⭐⭐⭐ For $f(x, y, z) = x^2 + y^2 + z^2$, describe the level surface $f = 25$ (Section 29.5) and compute $f_x$, $f_y$, $f_z$ at $(3, 4, 0)$.

G3. ⭐⭐⭐ For the gravitational potential $\Phi(x, y, z) = -\dfrac{GM}{\sqrt{x^2 + y^2 + z^2}}$, compute $\partial \Phi / \partial x$ and describe the level surfaces.

Part H — Applied: Economics and Thermodynamics (Section 29.13)

H1. ⭐ For the Cobb–Douglas output $Q(L, K) = 10\,L^{0.4} K^{0.6}$, write the formulas for the marginal products $Q_L$ and $Q_K$.

H2. ⭐⭐ Using $Q(L, K) = 10\,L^{0.4} K^{0.6}$, evaluate the marginal product of labor $Q_L$ at $(L, K) = (100, 100)$. Interpret the number in one sentence.

H3. ⭐⭐⭐ For the ideal gas pressure $P(V, T) = \dfrac{nRT}{V}$, compute $\partial P / \partial V$ and $\partial P / \partial T$, state the sign of each, and explain in words what each sign means physically.

Part I — Applied: Biology and Data Science (Section 29.13)

I1. ⭐⭐ A logistic-style growth rate is modeled by $g(N, F) = rN\!\left(1 - \dfrac{N}{F}\right)$, where $N$ is population and $F$ is the food-determined capacity. Compute $g_N$ and $g_F$.

I2. ⭐⭐⭐ A linear model $\hat{y} = w_0 + w_1 x$ is fit to a single data point $(x, y) = (2, 5)$ by the squared loss $L(w_0, w_1) = (w_0 + 2 w_1 - 5)^2$. Compute $\partial L / \partial w_0$ and $\partial L / \partial w_1$, then evaluate both at $(w_0, w_1) = (1, 1)$.

I3. ⭐⭐⭐ Continuing I2, gradient descent moves each weight opposite its partial. At $(1, 1)$, which way (increase or decrease) does each of $w_0$ and $w_1$ move, and why does that lower the loss? (These are the exact quantities bundled into the gradient $\nabla L$ in Chapter 30.)

Part J — Synthesis and Challenge

J1. ⭐⭐⭐ A surface has $f_x(2, 1) = 3$ and $f_y(2, 1) = -1$ with $f(2, 1) = 5$. Estimate $f(2.2, 0.7)$ with the linearization and state the one assumption that makes the estimate trustworthy.

J2. ⭐⭐⭐⭐ Construct a function $f(x, y)$ whose limit at the origin is $0$ along every straight line $y = mx$ but fails to exist along the parabola $y = x^2$. (Hint: the Math Major Sidebar of Section 29.6 supplies the shape.) Show both computations.

J3. ⭐⭐⭐⭐ Let $u(x, y) = \ln\sqrt{x^2 + y^2}$. Show that $u$ satisfies Laplace's equation $u_{xx} + u_{yy} = 0$ away from the origin. (Such $u$ is called harmonic; it foreshadows the potential theory of Chapters 34–37.)

Answers to selected problems appear in appendices/answers-to-selected.md.