Coverage: Lessons 1-16 (Sections 13.1-15.7)
Exam Date: Tuesday, February 25, 2026
Time: 6:30 PM - 7:30 PM
Location: Announced on Brightspace
Review of Vectors [Lesson 1] (§13.1-13.4)
Vector Notation and Operations
A vector has both magnitude (length) and direction. Vector notation:
where \(\vec{i} = \langle 1,0,0 \rangle\), \(\vec{j} = \langle 0,1,0 \rangle\), \(\vec{k} = \langle 0,0,1 \rangle\) are the standard basis vectors.
Basic Operations
- Addition: \(\vec{u} + \vec{v} = \langle u_1+v_1, u_2+v_2, u_3+v_3 \rangle\)
- Scalar multiplication: \(c\vec{u} = \langle cu_1, cu_2, cu_3 \rangle\)
- Vector from P to Q: \(\overrightarrow{PQ} = \langle x_2-x_1, y_2-y_1, z_2-z_1 \rangle\)
Magnitude and Unit Vectors
Magnitude (Length) of vector \(\vec{v} = \langle v_1, v_2, v_3 \rangle\):
A unit vector is a vector of length 1. The unit vector in the direction of \(\vec{v}\) is:
Dot Product
Dot Product of \(\vec{u} = \langle u_1, u_2, u_3 \rangle\) and \(\vec{v} = \langle v_1, v_2, v_3 \rangle\):
where \(\theta\) is the angle between the vectors. Result is a scalar.
- \(\vec{u} \cdot \vec{v} = 0 \Longleftrightarrow\) vectors are perpendicular (orthogonal)
- Angle between vectors: \(\cos\theta = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}||\vec{v}|}\)
Cross Product
Cross Product of \(\vec{u} = \langle u_1, u_2, u_3 \rangle\) and \(\vec{v} = \langle v_1, v_2, v_3 \rangle\):
Result is a vector perpendicular to both \(\vec{u}\) and \(\vec{v}\).
- \(|\vec{u} \times \vec{v}| = |\vec{u}||\vec{v}|\sin\theta\) = area of parallelogram formed by \(\vec{u}\) and \(\vec{v}\)
- \(\vec{u} \times \vec{v} = \vec{0} \Longleftrightarrow\) vectors are parallel
- \(\vec{u} \times \vec{v} = -(\vec{v} \times \vec{u})\) (anti-commutative)
Practice Problems
Problem 1. Find the projection of \(\vec{u} = \langle 2, -1, 3 \rangle\) onto \(\vec{v} = \langle 1, 2, 2 \rangle\).
▼ Show answer\(\left\langle \dfrac{2}{3},\, \dfrac{4}{3},\, \dfrac{4}{3}\right\rangle\)
Problem 2. Find the area of the triangle with vertices \(P(1,0,0)\), \(Q(0,2,0)\), and \(R(0,0,3)\).
▼ Show answer\(\dfrac{7}{2}\)
Problem 3. If \(\vec{u} = \langle 3, -1, 2 \rangle\) and \(\vec{v} = \langle 1, 4, -1 \rangle\), find \(\vec{u} \times \vec{v}\).
▼ Show answer\(\langle -7,\, 5,\, 13\rangle\)
Problem 4. For what value of \(c\) are the vectors \(\langle 2, c, -1 \rangle\) and \(\langle 3, -6, c \rangle\) orthogonal?
▼ Show answer\(c = \dfrac{6}{7}\)
Problem 5. Find the angle between \(\vec{u} = \langle 1, 1, 0 \rangle\) and \(\vec{v} = \langle 0, 1, 1 \rangle\).
▼ Show answer\(\theta = \dfrac{\pi}{3}\)
Lines and Planes in Space [Lesson 2] (§13.5)
Equations of a Line
Line through point \(P_0(x_0, y_0, z_0)\) with direction vector \(\vec{v} = \langle a, b, c \rangle\):
Vector Equation
Parametric Equations
Symmetric Equations (if \(a, b, c \neq 0\))
Equation of a Plane
Plane through point \(P_0(x_0, y_0, z_0)\) with normal vector \(\vec{n} = \langle a, b, c \rangle\):
or equivalently: \(ax + by + cz = d\), where \(d = ax_0 + by_0 + cz_0\)
Finding Normal Vectors
- If two direction vectors \(\vec{v}_1\) and \(\vec{v}_2\) lie in the plane, then \(\vec{n} = \vec{v}_1 \times \vec{v}_2\)
- If given three points, use two vectors formed by the points
Note: Angle between two planes equals the angle between their normal vectors.
Practice Problems
Problem 1. The line through \((1, 3, -1)\) and \((3, 1, 2)\) intersects the plane \(x + y + z = 6\) at what point?
▼ Show answer\((3,\, 1,\, 2)\)
Problem 2. Find the equation of the plane containing \((2, 1, -1)\) and parallel to the vectors \(\langle 1, 0, 3 \rangle\) and \(\langle 2, 1, 0 \rangle\).
▼ Show answer\(-3x + 6y + z + 1 = 0\)
Problem 3. Find the line of intersection between the planes \(x + y = 2\) and \(y + z = 3\).
▼ Show answer\(x = 2+t,\quad y = -t,\quad z = 3+t\)
Problem 4. Find the angle between the planes \(x + y + z = 1\) and \(x - y = 0\).
▼ Show answer\(\theta = \dfrac{\pi}{2}\) (planes are perpendicular)
Problem 5. For what value of \(a\) do the lines \(\vec{r}_1(t)=\langle 1+t,\, 2t,\, 3+t\rangle\) and \(\vec{r}_2(s)=\langle 3-s,\, s+1,\, 2s+a\rangle\) intersect?
▼ Show answer\(a = 2\)
Quadric Surfaces [Lessons 3-4] (§13.6)
A quadric surface is the graph of a second-degree equation in three variables.
| Surface | Standard Equation | Key Features |
|---|---|---|
| Ellipsoid | \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\) | All traces are ellipses |
| Elliptic Paraboloid | \(z = \frac{x^2}{a^2} + \frac{y^2}{b^2}\) | Opens upward (or downward if negative) |
| Hyperbolic Paraboloid | \(z = \frac{x^2}{a^2} - \frac{y^2}{b^2}\) | Saddle shape |
| Hyperboloid (1 sheet) | \(\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\) | Connected, one piece |
| Hyperboloid (2 sheets) | \(\frac{z^2}{c^2} - \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) | Two separate pieces |
| Elliptic Cone | \(\frac{z^2}{c^2} = \frac{x^2}{a^2} + \frac{y^2}{b^2}\) | Double cone, vertex at origin |
Identifying Quadric Surfaces
- Look at which variables are squared
- Check the signs (all positive, mixed signs, or one isolated)
- Check what the equation equals (0, 1, or a variable)
- Sketch traces in the coordinate planes (xy, xz, yz)
Traces are curves of intersection with planes parallel to coordinate planes.
Practice Problems
Problem 1. Identify the surface: \(\displaystyle -x^2 + \frac{y^2}{4} - \frac{z^2}{9} = 1\).
▼ Show answerHyperboloid of two sheets (axis along \(y\))
Problem 2. Identify the surface defined by \(x^2 + 4x + y^2 - 2y + z^2 = -4\). (Hint: complete the square.)
▼ Show answerSphere centered at \((-2,\, 1,\, 0)\) with radius \(1\)
Problem 3. Identify the surface: \(z = 4x^2 + 9y^2\).
▼ Show answerElliptic paraboloid opening upward
Problem 4. Describe the trace of \(\displaystyle \frac{x^2}{4} + \frac{y^2}{9} - z^2 = 1\) in the plane \(z = 2\).
▼ Show answerEllipse: \(\dfrac{x^2}{20} + \dfrac{y^2}{45} = 1\)
Problem 5. Identify the surface: \(\displaystyle x^2 + \frac{y^2}{4} = \frac{z^2}{9}\).
▼ Show answerElliptic cone with axis along the \(z\)-axis
Vector-Valued Functions [Lesson 5] (§14.1)
Definition
A vector-valued function (or vector function) has the form:
The graph is a space curve traced out as t varies.
Limits and Continuity
The limit of a vector function is computed component-wise:
Practice Problems
Problem 1. Find the domain on which \(\vec{r}(t) = \left\langle \sqrt{t-1},\, \ln(4-t),\, e^t \right\rangle\) is continuous.
▼ Show answer\([1,\, 4)\)
Problem 2. Evaluate \(\displaystyle \lim_{t \to 0} \left\langle \frac{\sin(3t)}{t},\, e^{2t},\, \frac{t^2+1}{t+1} \right\rangle\).
▼ Show answer\(\langle 3,\, 1,\, 1\rangle\)
Problem 3. Find the domain on which \(\vec{r}(t) = \left\langle \frac{1}{t^2-4},\, \sqrt{t},\, \cos(t) \right\rangle\) is continuous.
▼ Show answer\([0,2)\cup(2,\infty)\)
Problem 4. Describe the curve traced by \(\vec{r}(t) = \langle 2\cos t,\, 2\sin t,\, 0 \rangle\) for \(0 \le t \le 2\pi\).
▼ Show answerCircle of radius 2 in the \(xy\)-plane centered at the origin
Problem 5. Evaluate \(\displaystyle \lim_{t \to 0}\left\langle \frac{e^t - 1}{t},\, \frac{1-\cos t}{t^2},\, t\sin\!\left(\frac{1}{t}\right) \right\rangle\).
▼ Show answer\(\left\langle 1,\, \dfrac{1}{2},\, 0\right\rangle\)
Calculus of Vector-Valued Functions & Motion in Space [Lessons 6-7] (§14.2-14.3)
Derivative
Integral
Differentiation Rules
For vector functions \(\vec{u}(t)\) and \(\vec{v}(t)\):
- \(\frac{d}{dt}[\vec{u}(t) + \vec{v}(t)] = \vec{u}'(t) + \vec{v}'(t)\)
- \(\frac{d}{dt}[c\vec{u}(t)] = c\vec{u}'(t)\)
- \(\frac{d}{dt}[f(t)\vec{u}(t)] = f'(t)\vec{u}(t) + f(t)\vec{u}'(t)\)
- \(\frac{d}{dt}[\vec{u}(t) \cdot \vec{v}(t)] = \vec{u}'(t) \cdot \vec{v}(t) + \vec{u}(t) \cdot \vec{v}'(t)\)
- \(\frac{d}{dt}[\vec{u}(t) \times \vec{v}(t)] = \vec{u}'(t) \times \vec{v}(t) + \vec{u}(t) \times \vec{v}'(t)\)
Motion in Space
If \(\vec{r}(t)\) is the position vector at time t:
- Velocity: \(\vec{v}(t) = \vec{r}'(t)\)
- Speed: \(|\vec{v}(t)| = |\vec{r}'(t)|\)
- Acceleration: \(\vec{a}(t) = \vec{v}'(t) = \vec{r}''(t)\)
Unit Tangent Vector
Practice Problems
Problem 1. A particle has velocity \(\vec{v}(t) = \langle 2t,\, 3,\, -1 \rangle\) and at \(t=0\) passes through \((1,-2,5)\). Find \(\vec{r}(t)\).
▼ Show answer\(\vec{r}(t) = \langle t^2+1,\, 3t-2,\, -t+5\rangle\)
Problem 2. The tangent line to \(\vec{r}(t) = \langle t^2,\, 2t+1,\, t^3 \rangle\) at \(t=1\) intersects the \(xz\)-plane. Find the point of intersection.
▼ Show answer\(\left(-2,\, 0,\, -\dfrac{7}{2}\right)\)
Problem 3. Find the unit tangent vector \(\vec{T}(0)\) for \(\vec{r}(t) = \langle \cos t,\, \sin t,\, t \rangle\).
▼ Show answer\(\vec{T}(0) = \left\langle 0,\, \dfrac{1}{\sqrt{2}},\, \dfrac{1}{\sqrt{2}}\right\rangle\)
Problem 4. Find the speed of a particle with position \(\vec{r}(t) = \langle e^t,\, e^{-t},\, \sqrt{2}\,t \rangle\) at \(t = 0\).
▼ Show answerSpeed \(= 2\)
Problem 5. If \(\vec{r}(t) = \langle t,\, t^2,\, t^3 \rangle\), compute \(\vec{r}'(t) \times \vec{r}''(t)\).
▼ Show answer\(\langle 6t^2,\, -6t,\, 2\rangle\)
Problem 6. A particle travels along \(\vec{r}(t) = \langle 3\cos t,\, 4\sin t,\, 0 \rangle\). Find the speed at \(t = \frac{\pi}{2}\).
▼ Show answerSpeed \(= 3\)
Arc Length and Curvature [Lesson 8] (§14.4-14.5)
Arc Length
Arc Length of curve \(\vec{r}(t)\) from \(t = a\) to \(t = b\):
Arc Length Function
Curvature
Curvature measures how quickly the curve changes direction:
For a plane curve \(y = f(x)\):
Practice Problems
Problem 1. Find the arc length of \(\vec{r}(t) = \langle 3\cos t,\, 3\sin t,\, 4t \rangle\) for \(0 \le t \le 2\pi\).
▼ Show answer\(L = 10\pi\)
Problem 2. Find the curvature \(\kappa(t)\) of \(\vec{r}(t) = \langle 2\cos t,\, 2\sin t,\, t \rangle\).
▼ Show answer\(\kappa = \dfrac{2}{5}\)
Problem 3. Find the curvature of the plane curve \(y = x^3\) at the point \((0,0)\).
▼ Show answer\(\kappa = 0\)
Problem 4. Find the arc length of \(\vec{r}(t) = \langle 3t,\, 4t,\, 0 \rangle\) for \(1 \le t \le 5\).
▼ Show answer\(L = 20\)
Problem 5. A particle moves along \(\vec{r}(t) = \langle 5\sin t,\, 3\sin t,\, 4\cos t \rangle\). Find the distance traveled from \(t=0\) to \(t=\pi\).
▼ Show answer\(L = \displaystyle\int_0^{\pi}\sqrt{16+18\cos^2 t}\,dt\)
Functions of Several Variables [Lesson 9] (§15.1)
Definition
A function of two variables assigns to each ordered pair (x, y) in a set D a unique real number \(z = f(x,y)\).
Visualization Methods
- Graph: Surface in 3D space
- Level Curves: Curves in xy-plane where \(f(x,y) = k\) (constant)
- Contour Map: Collection of level curves
Functions of Three Variables
For \(w = f(x,y,z)\):
- Level Surfaces: Surfaces where \(f(x,y,z) = k\)
Practice Problems
Problem 1. Find the domain of \(f(x, y) = \sqrt{9 - x^2 - y^2}\).
▼ Show answer\(\{(x,y):\, x^2+y^2 \le 9\}\)
Problem 2. Describe the nonempty level curves of \(f(x, y) = x^2 + 4y^2\).
▼ Show answerEllipses \(\dfrac{x^2}{k}+\dfrac{y^2}{k/4}=1\) centered at origin (\(k>0\))
Problem 3. Describe the level curves of \(f(x, y) = y - x^2\).
▼ Show answerParabolas \(y = x^2 + k\)
Problem 4. Find the domain of \(f(x, y) = \ln(x + y - 1)\).
▼ Show answer\(\{(x,y):\, x+y > 1\}\)
Problem 5. Describe the level surfaces of \(f(x, y, z) = x^2 + y^2 + z^2\).
▼ Show answerSpheres of radius \(\sqrt{k}\) centered at origin (\(k>0\))
Limits and Continuity [Lesson 10] (§15.2)
Limits
We write \(\displaystyle\lim_{(x,y) \to (a,b)} f(x,y) = L\), if f(x,y) approaches L as (x,y) approaches (a,b) from any direction.
Proving a Limit Does Not Exist
Find two different paths that give different limits:
- Along x-axis: \(y = 0\)
- Along y-axis: \(x = 0\)
- Along \(y = x\) or \(y = mx\)
- Along parabolic paths: \(y = x^2\)
Continuity
f is continuous at (a,b) if:
Practice Problems
Problem 1. Evaluate \(\displaystyle\lim_{(x,y) \to (0,0)} \frac{x^3 y}{x^4 + y^2}\), or show it does not exist.
▼ Show answerlimit does not exist
Problem 2. Evaluate \(\displaystyle\lim_{(x,y) \to (0,0)} \frac{3x^2 y}{x^2 + y^2}\), or show it does not exist.
▼ Show answerlimit does not exist
Problem 3. Find the value of \(k\) such that \(\displaystyle\lim_{(x,y) \to (0,0)} \frac{x^4 - 2k(x^2+y^2) - y^4}{x^2+y^2} = 6\).
▼ Show answer\(k = -3\)
Problem 4. Evaluate \(\displaystyle\lim_{(x,y) \to (0,0)} \frac{\sin(x^2 + y^2)}{x^2 + y^2}\).
▼ Show answer\(1\)
Problem 5. Evaluate \(\displaystyle\lim_{(x,y) \to (0,0)} \frac{x^3 - y^3}{x^2 + y^2}\), or show it does not exist.
▼ Show answer\(0\)
Partial Derivatives [Lesson 11] (§15.3)
First-Order Partial Derivatives
Partial derivative with respect to x (treat y as constant):
Partial derivative with respect to y (treat x as constant):
Higher-Order Partial Derivatives
- \(f_{xx} = \frac{\partial^2 f}{\partial x^2}\)
- \(f_{yy} = \frac{\partial^2 f}{\partial y^2}\)
- \(f_{xy} = \frac{\partial^2 f}{\partial y \partial x}\)
- \(f_{yx} = \frac{\partial^2 f}{\partial x \partial y}\)
Clairaut's Theorem
If \(f_{xy}\) and \(f_{yx}\) are both continuous:
Practice Problems
Problem 1. Let \(f(x,y,z)=\cos(xy)+e^{y^2z}+\ln(z^3)\). Compute \(f_{yxz}(0,1,1)\).
▼ Show answer\(f_{yxz}(0,1,1) = 0\)
Problem 2. If \(f(x, y) = x^3 y^2 + \sin(xy)\), find \(f_{xy}(0, 0)\).
▼ Show answer\(f_{xy}(0,0) = 1\)
Problem 3. If \(f(x,y) = e^{x^2 y}\), find \(f_x(1, 2)\).
▼ Show answer\(f_x(1,2) = 4e^2\)
Problem 4. Let \(f(x,y)=x^2\ln(y)\). Find \(f_{xy}\) and verify Clairaut's theorem by also computing \(f_{yx}\).
▼ Show answer\(f_{xy} = f_{yx} = \dfrac{2x}{y}\)
Problem 5. If \(f(x,y) = \arctan\!\left(\frac{y}{x}\right)\), find \(f_x + f_y\) at \((1, 1)\).
▼ Show answer\(f_x + f_y = 0\) at \((1,1)\)
The Chain Rule [Lesson 12] (§15.4)
Chain Rule - Case 1
If \(z = f(x,y)\) where \(x = g(t)\) and \(y = h(t)\):
Chain Rule - Case 2
If \(z = f(x,y)\) where \(x = g(s,t)\) and \(y = h(s,t)\):
Implicit Differentiation
If \(F(x,y,z) = 0\) defines z implicitly:
Practice Problems
Problem 1. If \(z = x^2 y\), where \(x = t^2\) and \(y = t^3\), find \(\frac{dz}{dt}\) at \(t = 1\).
▼ Show answer\(\left.\dfrac{dz}{dt}\right|_{t=1} = 7\)
Problem 2. Suppose \(xz^2 + y^2z = 2\) defines \(z\) implicitly as a function of \(x\) and \(y\). Find \(\frac{\partial z}{\partial x}\) at the point \((1, 1, 1)\).
▼ Show answer\(\dfrac{\partial z}{\partial x} = -\dfrac{1}{3}\)
Problem 3. If \(w=xy+yz\), \(x=s+t\), \(y=st\), \(z=s-t\), find \(\frac{\partial w}{\partial s}\) at \(s=1,\, t=0\).
▼ Show answer\(\left.\dfrac{\partial w}{\partial s}\right|_{s=1,t=0} = 0\)
Problem 4. If \(\sin(xy) = e^{2z}\), use implicit differentiation to find \(\frac{\partial z}{\partial x}\) at \(\left(1, \frac{\pi}{2}, 0\right)\).
▼ Show answer\(\dfrac{\partial z}{\partial x} = 0\)
Problem 5. If \(z = f(x,y)\) with \(x = r\cos\theta\) and \(y = r\sin\theta\), express \(\frac{\partial z}{\partial r}\) in terms of \(f_x\) and \(f_y\).
▼ Show answer\(\dfrac{\partial z}{\partial r} = f_x\cos\theta + f_y\sin\theta\)
Problem 6. If \(F(x,y,z) = x^2+y^2+z^2-3xyz=0\) defines \(z\) implicitly, find \(\frac{\partial z}{\partial x}\) at \((1,1,1)\), or explain why it does not exist.
▼ Show answer\(\left.\dfrac{\partial z}{\partial x}\right|_{(1,1,1)} = -1\)
Directional Derivatives & Gradient [Lesson 13] (§15.5)
Directional Derivative
The directional derivative \(D_{\vec{u}}f(a,b)\) measures rate of change in direction of unit vector \(\vec{u}\):
Gradient Vector
Properties of the Gradient
- \(\nabla f\) points in direction of maximum rate of increase
- \(|\nabla f|\) gives the maximum rate of increase
- \(-\nabla f\) points in direction of maximum decrease
- \(\nabla f\) is orthogonal to level curves
Practice Problems
Problem 1. Find the directional derivative of \(f(x,y)=x^2y-3y^2\) at the point \((2,1)\) in the direction of the vector \(\vec{v}=\langle 3,4\rangle\).
▼ Show answer\(D_{\hat{u}}f(2,1) = \dfrac{4}{5}\)
Problem 2. Find the maximum rate of change of \(f(x,y,z)=x\sqrt{yz}\) at the point \((4,1,4)\), and state the direction in which it occurs.
▼ Show answerMax rate of change: \(\sqrt{21}\), in direction \(\langle 2,\, 4,\, 1\rangle\)
Problem 3. At \((1,2)\), in what direction does \(f(x,y)=xe^{-y}\) decrease most rapidly? What is that rate of decrease?
▼ Show answerDirection: \(\langle -1,\, 1\rangle\); rate of decrease: \(\sqrt{2}\,e^{-2}\)
Problem 4. Find the directional derivative of \(f(x,y,z)=x^2+2y^2+3z^2\) at \((1,1,1)\) in the direction of \(\langle 1,-1,0\rangle\).
▼ Show answer\(D_{\hat{u}}f(1,1,1) = -\sqrt{2}\)
Problem 5. Suppose \(\nabla f(2,3)=\langle 4,-3\rangle\). Find \(D_{\vec{u}}f(2,3)\) where \(\vec{u}\) is the unit vector in the direction of \(\langle 3,4\rangle\).
▼ Show answer\(D_{\hat{u}}f(2,3) = 0\)
Tangent Planes & Linear Approximation [Lesson 14] (§15.6)
Tangent Plane
Equation of tangent plane to \(z = f(x,y)\) at \((a, b, f(a,b))\):
Linear Approximation
Differential
Normal Vector
For \(z = f(x,y)\): \(\vec{n} = \langle f_x, f_y, -1 \rangle\)
For \(F(x,y,z) = 0\): \(\vec{n} = \nabla F = \langle F_x, F_y, F_z \rangle\)
Practice Problems
Problem 1. Consider the surface \(2x^2-4x+y^3+z+2=0\). Find the point(s) on the surface where the tangent plane is parallel to the \(xy\)-plane.
▼ Show answer\((1,\, 0,\, 0)\)
Problem 2. Find the equation of the tangent plane to \(z = x^2 + 2xy\) at the point \((1, 1, 3)\).
▼ Show answer\(4x + 2y - z = 3\)
Problem 3. Use the linear approximation of \(f(x,y)=\sqrt{x^2+y^2}\) at \((3,4)\) to estimate \(f(3.1, 3.9)\).
▼ Show answer\(L(3.1, 3.9) \approx 4.98\)
Problem 4. If \(z = x^3 y^2\), find the differential \(dz\).
▼ Show answer\(dz = 3x^2y^2\,dx + 2x^3y\,dy\)
Problem 5. Find the normal vector to the surface \(x^2+y^2-z = 0\) at the point \((1, 2, 5)\).
▼ Show answer\(\vec{n} = \langle 2,\, 4,\, -1\rangle\)
Maximum and Minimum Problems [Lessons 15-16] (§15.7)
Critical Points
Point (a,b) is a critical point if:
Second Derivative Test
Let the discriminant be:
Classification
- If \(D > 0\) and \(f_{xx} > 0\): local minimum
- If \(D > 0\) and \(f_{xx} < 0\): local maximum
- If \(D < 0\): saddle point
- If \(D = 0\): inconclusive
Absolute Max/Min on Closed Region
Step 1: Find all critical points inside region D
Step 2: Find extreme values on the boundary of D
Step 3: Compare all values - largest is absolute max, smallest is absolute min
Practice Problems
Problem 1. Find the critical points of \(f(x,y) = x^2y - y + 3y^2\).
▼ Show answer\((0,\,\tfrac{1}{6}),\;(1,0),\;(-1,0)\)
Problem 2. Classify the critical point of \(f(x,y) = x^2+xy+y^2+3x-3y+4\) using the second derivative test.
▼ Show answerLocal minimum at \((-3,\, 3)\)
Problem 3. Let \(M\) and \(m\) be the absolute maximum and minimum values of \(f(x,y) = x^2 - 2x + y^2 + 4\) on the disk \(x^2 + y^2 \le 4\). Find \(M + m\).
▼ Show answer\(M + m = 15\)
Problem 4. Find and classify all critical points of \(f(x,y) = xy - x^3 - y^2\).
▼ Show answerSaddle point at \((0,0)\); local max at \((\tfrac{1}{6}, \tfrac{1}{12})\)
Problem 5. Show that \(f(x,y) = x^2 - y^2\) has a saddle point at \((0,0)\) using the second derivative test.
▼ Show answerSaddle point at \((0,0)\)
Problem 6. Find the absolute maximum and absolute minimum of \(f(x,y) = 4xy - x^2 - 2y^2 + 2\) on the region \(0 \le x \le 2\), \(0 \le y \le 1\).
▼ Show answerAbs. max \(= 4\) at \((2,1)\); abs. min \(= -2\) at \((2,0)\)