← Calculus III (Multivariable)

Unit 3: Functions of Several Variables and Partial Derivatives

A surface sliced by planes turns partial derivatives into ordinary derivatives you can watch, building to gradients, saddle points, and Lagrange multipliers.

31 topics · 118 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

4.1 Functions of Two Variables and Their Graphs

  1. How it worksHow does plotting z = f(x, y) above every point in the xy-plane build up a surface?
  2. Worked exampleFind the domain of f(x, y) = square root of (9 - x2 - y2) and describe it.
  3. Common mistakeWhy isn't the graph of a two-variable function just two curves stacked on top of each other?
Your turn1 of 2
For f(x, y) = √(82 - x2 - y2), find f(1, 2).

4.1 Level Curves and Contour Maps

  1. How it worksHow does slicing a surface at height z = k and looking straight down produce a level curve?
  2. Worked exampleSketch the level curves of f(x, y) = x2 + y2 for k = 1, 4, and 9.
  3. Common mistakeWhy doesn't a single level curve tell you how high up the surface actually sits there?
  4. Worked exampleDraw the contour map of f(x, y) = y - x2 and label three of its level curves.
Your turn1 of 2
For f(x, y) = x2 + y2, write the radius of the level curve at height k, as a function of k.

4.1 Level Surfaces of Functions of Three Variables

  1. How it worksHow does fixing f(x, y, z) equal to a constant carve out a surface instead of a curve?
  2. Worked exampleFind and describe the level surface of f(x, y, z) = x2 + y2 + z2 when f = 16.
  3. Ties togetherHow does a level surface of a three-variable function play the same role a level curve plays for a two-variable one?
  4. Common mistakeWhy doesn't the level surface for f = 0 have to pass through the origin?
Your turn1 of 2
For f(x, y, z) = x2 + y2 + z2, write the radius of the level surface, as a function of f's value k.

4.1 Traces: Slicing a Surface with a Plane

  1. How it worksHow does fixing y at a constant value slice a surface z = f(x, y) into a curve in x and z alone?
  2. Worked exampleFind the trace of z = 4 - x2 - y2 in the plane y = 1.
  3. Worked examplePicture what the cross section of z = x2 - y2 looks like once you fix x = 2.
  4. Going furtherWhat happens to the family of traces of a saddle-shaped surface as the cutting plane slides across it?
Your turn1 of 2
Find z on the trace of z = 14 - x2 - y2 in the plane y = 1, as a function of x.

4.2 Limits of Functions of Two Variables

  1. How it worksHow do you test a two-variable limit by switching to polar coordinates centered at that point?
  2. Worked exampleFind the limit of (x2 y) / (x2 + y2) as (x, y) approaches (0, 0).
  3. Going furtherWhat happens to a two-variable limit calculation once the point being approached isn't the origin?
  4. Common mistakeWhy doesn't getting the same answer along infinitely many straight lines through a point guarantee a two-variable limit exists?
Your turn1 of 2
Find the limit of f(x, y) = x2 + 4y2 as (x, y) approaches (1, 4).

4.2 Why the Path Test Shows a Limit Fails to Exist

  1. Common mistakeWhy doesn't checking a two-variable limit along just the x-axis and the y-axis prove it exists?
  2. How it worksHow does plugging in y = mx and getting an answer that depends on m prove a limit doesn't exist?
  3. Worked exampleShow that the limit of xy / (x2 + y2) as (x, y) approaches (0, 0) does not exist using y = mx.
  4. Worked exampleShow that the limit of (x2 - y2) / (x2 + y2) as (x, y) approaches (0, 0) does not exist using two different paths.
Your turn1 of 2
Along the path y = mx, f(x, y) = xy/(x2+y2) approaches a constant value as (x,y) approaches (0,0) along that line. Write that value as a function of the slope m.

4.2 Using the Squeeze Theorem for Two-Variable Limits

  1. How it worksHow does bounding |f(x, y)| between 0 and an expression that shrinks to 0 force the limit itself to be 0?
  2. Worked exampleShow that the limit of (x2 y) / (x2 + y2) as (x, y) approaches (0, 0) equals 0 using the squeeze theorem.
  3. Ties togetherHow does the squeeze theorem give a positive proof that complements the path test's negative one?
Your turn1 of 2
Suppose 0 ≤ |f(x,y)| ≤ √(x2+y2) for all (x,y). Find the upper bound this inequality gives for |f(3, 2)|.

4.2 Continuity of Functions of Several Variables

  1. How it worksHow does continuity at a point require the limit there to exist and equal the function's actual value?
  2. Worked exampleFind whether f(x, y) = (x2 y)/(x2 + y2) for (x,y) not (0,0), with f(0,0) = 0, is continuous at the origin.
  3. Going furtherWhat happens to continuity at a boundary point where a two-variable function switches formulas?
  4. Worked exampleFind whether f(x, y) = sin(xy)/(xy) for xy not 0, with f(x,y) = 1 when xy = 0, is continuous at the origin.
Your turn1 of 2
Consider f(x, y) = sin(4xy)/(xy) for xy not 0. Find the value f would need to take at points where xy = 0 for f to be continuous there.

4.3 Partial Derivatives as Slopes of Slice Curves

  1. How it worksHow does holding y fixed and differentiating with respect to x measure the slope of one slice curve?
  2. Worked exampleFind the slope of the trace of z = x2 + 3y2 in the plane y = 1 at x = 2, using a partial derivative.
  3. Common mistakeWhy doesn't a single partial derivative at a point tell you the surface's steepness in every direction?
  4. Ties togetherHow does this partial derivative equal the ordinary derivative of the trace curve found by slicing the surface with a plane?
Your turn1 of 2
Find the slope of the trace of z = x2 + 4y2 in the plane y = 1, as a function of x, using a partial derivative.

4.3 Computing Partial Derivatives

  1. Worked exampleFind fx and fy for f(x, y) = x3 y2 + sin(xy).
  2. Worked exampleFind the partial derivatives of f(x, y) = exy with respect to x and to y.
  3. Going furtherWhat changes about computing a partial derivative once the variable you're differentiating with respect to sits inside a nested function?
  4. Worked exampleFind the partial derivatives of f(x, y) = ln(x2 + y2).
Your turn1 of 2
Find partial f partial x for f(x, y) = 4x3 y2 + sin(xy).

4.3 Partial Derivatives of Functions of Three or More Variables

  1. Worked exampleFind fx, fy, and fz for f(x, y, z) = x2 yz + y3 z2.
  2. Common mistakeWhy doesn't computing partial derivatives get any harder in principle once a fourth variable is added?
  3. Ties togetherHow does the set of partial derivatives of a three-variable function generalize the gradient idea from two variables?
  4. Worked exampleFind fx, fy, and fz for f(x, y, z) = exyz.
Your turn1 of 2
Find partial f partial x for f(x, y, z) = x2 yz + y3 z2.

4.3 Higher-Order Partial Derivatives

  1. How it worksHow do fxx, fxy, fyx, and fyy each come from a different order of two single-variable differentiations?
  2. Worked exampleFind fxx, fyy, and fxy for f(x, y) = x3 y - 2xy2.
  3. Going furtherWhat happens to the number of distinct higher partial derivatives once you move from second order to third order?
  4. Common mistakeWhy doesn't a function need infinitely many derivatives defined for Clairaut's theorem to apply at just second order?
Your turn1 of 2
Find f_xx for f(x, y) = 4x3y - 2xy2.

4.3 Clairaut's Theorem and Mixed Partials

  1. Common mistakeWhy isn't it always true that fxy equals fyx no matter what function you start with?
  2. How it worksHow does Clairaut's theorem guarantee fxy equals fyx as long as both are continuous?
  3. Worked exampleShow that fxy equals fyx for f(x, y) = x2 sin(y) + y3.
Your turn1 of 2
Find f_xy for f(x, y) = 4x2 sin(y) + y3.

4.4 Tangent Planes to a Surface

  1. How it worksHow do the two partial derivatives at a point supply the normal vector for that point's tangent plane?
  2. Worked exampleFind the tangent plane to z = x2 + y2 at the point (1, 2, 5).
  3. Common mistakeWhy doesn't a tangent plane at one point tell you anything about the surface far away from that point?
  4. Worked exampleBuild the equation of the plane tangent to the surface z = x2 y at (1, 2, 2).
Your turn1 of 2
The surface z = x2 + 4y2 has a tangent plane at (1, 3, z0). Write that tangent plane's z as a function of x and y.

4.4 Linear Approximation in Two Variables

  1. How it worksHow does the tangent plane's own equation double as the linear approximation formula?
  2. Worked exampleFind the linear approximation to f(x, y) = x2 + y2 near (2, 3) and use it to estimate f(2.1, 2.9).
  3. Going furtherWhat happens to a linear approximation's accuracy as the point moves farther from the point of tangency?
  4. Worked exampleFind the linear approximation of f(x, y) = square root of (x2 + y2) near (3, 4).
Your turn1 of 2
Find the linear approximation L(x, y) to f(x, y) = x2 + y2 near (4, 4).

4.4 Differentials and Error Estimation

  1. How it worksHow does the total differential dz combine dx and dy, each weighted by a partial derivative?
  2. Worked exampleFind the estimated change in f(x, y) = x2 y as x goes from 2 to 2.05 and y goes from 3 to 2.96, using differentials.
  3. Worked exampleFind the estimated maximum error in the computed volume of a cylinder of radius 4 cm and height 10 cm if each measurement is off by up to 0.1 cm.
Your turn1 of 2
Using differentials, write the estimated change in f(x, y) = x2y at the point (4, 4), as a function of the small changes dx and dy.

4.5 The Chain Rule for One Independent Variable

  1. How it worksHow does dz/dt combine both partial derivatives with the rates dx/dt and dy/dt?
  2. Worked exampleFind dz/dt for z = x2 + y2 where x = cos t and y = sin t.
  3. Common mistakeWhy isn't dz/dt just fx times dx/dt, ignoring the y-dependence entirely?
  4. Worked exampleFind dz/dt for z = xey where x = t2 and y = t3, at t = 1.
Your turn1 of 2
Find dz/dt, as a function of t, for z = x2 + 4y2 where x = cos(t) and y = sin(t).

4.5 The Chain Rule with Tree Diagrams for Several Variables

  1. How it worksHow does a tree diagram turn every path from the final output back to an independent variable into one term to add up?
  2. Worked exampleFind dw/dt for w = x2 + y2 + z2 where x = t, y = t2, and z = et, using a tree diagram.
  3. Worked exampleFind dz/ds for z = f(x, y) where x = s + t and y = s - t, in terms of fx and fy.
  4. Going furtherWhat happens to the tree diagram once one intermediate variable doesn't actually depend on a particular independent variable?
Your turn1 of 2
Find dw/dt, as a function of t, for w = x2 + y2 + z2 where x = t, y = 2t2, and z = e1.15t.

4.5 Implicit Differentiation with Partial Derivatives

  1. How it worksHow does the chain rule turn the implicit function theorem into the formula dy/dx = -Fx/Fy?
  2. Worked exampleFind dy/dx if x3 + y3 = 6xy, using implicit partial differentiation.
  3. Going furtherWhat happens to the implicit differentiation formula once the equation involves three variables instead of two?
  4. Worked exampleFind dz/dx for the surface x2 + y2 + z2 = 9 near the point (1, 2, 2).
Your turn1 of 2
Find dy/dx if x3 + y3 = 7xy, using implicit partial differentiation.

4.6 Directional Derivatives

  1. How it worksHow does the directional derivative formula combine the partial derivatives with a unit direction vector?
  2. Worked exampleFind the directional derivative of f(x, y) = x2 y at (1, 2) in the direction (3, 4).
  3. Common mistakeWhy doesn't a directional derivative use just any vector, without normalizing it to length 1 first?
  4. Going furtherWhat happens to the directional derivative when the chosen direction points exactly opposite the gradient?
Your turn1 of 2
Find the directional derivative of f(x, y) = 4x2y in the direction (4, 6), as a function of x and y.

4.6 The Gradient Vector

  1. How it worksHow does stacking a function's partial derivatives into a single vector give you its gradient?
  2. Worked exampleFind the gradient of f(x, y) = x2 y + y3 at the point (1, 2).
  3. Worked exampleFind the directional derivative of f(x, y) = ex sin(y) at (0, π/2) using the gradient shortcut.
  4. Common mistakeWhy isn't the gradient just the two partial derivatives written side by side with no special meaning attached?
Your turn1 of 2
Find the y-component of the gradient of f(x, y) = 4x2y + y3, as a function of x and y.

4.6 Why the Gradient Points Uphill, Not at the Max

  1. Common mistakeWhy doesn't the gradient at a point aim straight at the function's overall maximum?
  2. How it worksHow does maximizing the dot product of the gradient with a unit vector prove the gradient gives the steepest ascent?
  3. Going furtherWhat happens to the direction of steepest ascent at a point that sits exactly on top of a local maximum?
Your turn1 of 2
A student stands at a point on a hillside, faces the direction the gradient points, and starts walking. Why doesn't that direction lead straight to the mountain's overall summit?

4.6 The Gradient Perpendicular to Level Curves

  1. How it worksHow does differentiating along a level curve prove the gradient must be orthogonal to it at every point?
  2. Worked exampleFind a vector normal to the level curve of f(x, y) = x2 + 4y2 at the point (2, 1).
  3. Ties togetherHow does the gradient's perpendicularity to level curves connect to a normal vector's role in a tangent plane?
Your turn1 of 2
Find the x-component of a vector normal to the level curve of f(x, y) = 4x2 + 4y2, as a function of x.

4.6 Tangent Planes to Level Surfaces Using the Gradient

  1. How it worksHow do you build the tangent plane to a level surface F(x, y, z) = c using the gradient at a point?
  2. Worked exampleFind the tangent plane to the surface x2 + y2 + z2 = 9 at the point (2, 2, 1).
  3. Going furtherWhat happens to this tangent-plane method once the surface is given as z = f(x, y) instead of F(x, y, z) = c?
  4. Worked exampleBuild the tangent plane to x2 - y2 + z2 = 4 at the point (2, 2, 2), using the gradient.
Your turn1 of 2
A point (x, y, z) lies on a sphere centered at the origin. Write the x-coefficient A of the tangent plane at this point (Ax + By + Cz = D, using the gradient), as a function of x.

4.7 Local Maxima, Minima, and Critical Points

  1. How it worksHow does setting both fx and fy equal to zero locate every candidate for a local max or min?
  2. Worked exampleFind the critical points of f(x, y) = x2 + y2 - 4x + 6y + 13.
  3. Common mistakeWhy doesn't every critical point of a two-variable function have to be a local max or min?
  4. Worked exampleFind the critical points of f(x, y) = x3 + y3 - 3xy.
Your turn1 of 2
Find the x-coordinate of the critical point of f(x, y) = x2 + y2 - 4x + 3y + 9.

4.7 The Second Derivative Test and the Discriminant

  1. How it worksHow does the discriminant D = fxx fyy - (fxy)2 combine curvature information from three second partials?
  2. Worked exampleFind the type of critical point f(x, y) = x3 - 3xy + y3 has at (1, 1) using the second derivative test.
  3. Common mistakeWhy doesn't a critical point with D = 0 have to be a saddle point?
Your turn1 of 2
Find the discriminant D = fxx·fyy - (fxy)2 for f(x, y) = 3x3 - 3xy + y3 at the point (1, 3).

4.7 Saddle Points

  1. How it worksHow does a saddle point rotate to look like a minimum from one direction and a maximum from a perpendicular one?
  2. Worked exampleShow that f(x, y) = y2 - x2 has a saddle point at the origin by checking its traces along the x- and y-axes.
  3. Worked exampleFind whether the critical point of f(x, y) = xy at (0, 0) is a saddle, using the second derivative test.
  4. Common mistakeWhy isn't a saddle point simply a small, weak version of a local max or min?
  5. Ties togetherHow does rotating a saddle-shaped surface connect to viewing the same critical point along two different traces?
Your turn1 of 2
Find the discriminant D = fxx·fyy - (fxy)2 for f(x, y) = 6xy at the origin.

4.7 Absolute Extrema on Closed and Bounded Regions

  1. How it worksHow does the extreme value theorem guarantee a continuous function on a closed bounded region has both an absolute max and min?
  2. Worked exampleFind the absolute maximum and minimum of f(x, y) = x2 + y2 - 2x on the closed disk x2 + y2 ≤ 4.
  3. Going furtherWhat happens to the search for absolute extrema once the region is open instead of closed?
  4. Worked exampleFind the absolute maximum and minimum of f(x, y) = xy on the closed triangle with vertices (0,0), (2,0), and (0,2).
Your turn1 of 2
Find the absolute maximum of f(x, y) = x2 + y2 - 4x on the closed disk x2+y2 ≤ 62 (with a > 0).

4.8 Lagrange Multipliers with One Constraint

  1. How it worksHow do the level curves of f and the constraint curve slide against each other until they touch tangentially?
  2. Worked exampleFind the maximum of f(x, y) = xy subject to x + y = 10, using Lagrange multipliers.
  3. Worked exampleFind the minimum of f(x, y) = x2 + y2 subject to 2x + y = 5, using Lagrange multipliers.
  4. Common mistakeWhy doesn't a tangential touch between a level curve and the constraint mean the two curves are equal there?
  5. Going furtherWhat happens to the Lagrange multiplier method when the constraint curve is unbounded, like a hyperbola?
Your turn1 of 2
Find the maximum of f(x, y) = xy subject to x + y = 21, using Lagrange multipliers.

4.8 Lagrange Multipliers with Two Constraints

  1. How it worksHow does the gradient of f become a combination of both constraints' gradients at the optimal point?
  2. Worked exampleFind the extrema of f(x, y, z) = x + y + z subject to x2 + y2 = 2 and x + z = 1, using Lagrange multipliers.
  3. Ties togetherHow does adding a second constraint to a Lagrange multiplier problem echo intersecting two planes to get a line?
Your turn1 of 2
Find the maximum value of f(x, y, z) = x + y + z subject to x2 + y2 = 42 and x + z = 2, using Lagrange multipliers.

4.7 Optimization Applications: Minimizing Cost or Maximizing Volume

  1. How it worksHow does translating a word problem into a two-variable function turn it into a critical-point search?
  2. Worked exampleFind the dimensions of an open-top rectangular box with volume 32 cubic feet that minimize the amount of material used.
  3. Worked exampleFind the dimensions of the rectangular box of maximum volume that fits inside the sphere x2 + y2 + z2 = 12.
  4. Ties togetherHow does the box-in-a-sphere problem turn into a Lagrange multiplier problem once the sphere is framed as a constraint?
Your turn1 of 2
An open-top rectangular box with a square base is to have volume 140 cubic feet, minimizing the material used. Find the optimal base side length x.