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)

Video Notes Practice

Vector Notation and Operations

A vector has both magnitude (length) and direction. Vector notation:

\[\vec{v} = \langle v_1, v_2, v_3 \rangle \quad \text{or} \quad \vec{v} = v_1\vec{i} + v_2\vec{j} + v_3\vec{k}\]

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

Magnitude and Unit Vectors

Magnitude (Length) of vector \(\vec{v} = \langle v_1, v_2, v_3 \rangle\):

\[|\vec{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}\]

A unit vector is a vector of length 1. The unit vector in the direction of \(\vec{v}\) is:

\[\hat{v} = \frac{\vec{v}}{|\vec{v}|}\]

Dot Product

Dot Product of \(\vec{u} = \langle u_1, u_2, u_3 \rangle\) and \(\vec{v} = \langle v_1, v_2, v_3 \rangle\):

\[\vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + u_3v_3 = |\vec{u}||\vec{v}|\cos\theta\]

where \(\theta\) is the angle between the vectors. Result is a scalar.

Cross Product

Cross Product of \(\vec{u} = \langle u_1, u_2, u_3 \rangle\) and \(\vec{v} = \langle v_1, v_2, v_3 \rangle\):

\[\vec{u} \times \vec{v} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix} = \langle u_2v_3-u_3v_2, u_3v_1-u_1v_3, u_1v_2-u_2v_1 \rangle\]

Result is a vector perpendicular to both \(\vec{u}\) and \(\vec{v}\).

Practice Problems

Problem 1. Find the projection of \(\vec{u} = \langle 2, -1, 3 \rangle\) onto \(\vec{v} = \langle 1, 2, 2 \rangle\).

Problem 2. Find the area of the triangle with vertices \(P(1,0,0)\), \(Q(0,2,0)\), and \(R(0,0,3)\).

Problem 3. If \(\vec{u} = \langle 3, -1, 2 \rangle\) and \(\vec{v} = \langle 1, 4, -1 \rangle\), find \(\vec{u} \times \vec{v}\).

Problem 4. For what value of \(c\) are the vectors \(\langle 2, c, -1 \rangle\) and \(\langle 3, -6, c \rangle\) orthogonal?

Problem 5. Find the angle between \(\vec{u} = \langle 1, 1, 0 \rangle\) and \(\vec{v} = \langle 0, 1, 1 \rangle\).

Lines and Planes in Space [Lesson 2] (§13.5)

Video Notes Practice

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

\[\vec{r}(t) = \vec{r}_0 + t\vec{v} = \langle x_0, y_0, z_0 \rangle + t\langle a, b, c \rangle\]

Parametric Equations

\[x = x_0 + at, \quad y = y_0 + bt, \quad z = z_0 + ct\]

Symmetric Equations (if \(a, b, c \neq 0\))

\[\frac{x-x_0}{a} = \frac{y-y_0}{b} = \frac{z-z_0}{c}\]

Equation of a Plane

Plane through point \(P_0(x_0, y_0, z_0)\) with normal vector \(\vec{n} = \langle a, b, c \rangle\):

\[a(x-x_0) + b(y-y_0) + c(z-z_0) = 0\]

or equivalently: \(ax + by + cz = d\), where \(d = ax_0 + by_0 + cz_0\)

Finding Normal Vectors

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?

Problem 2. Find the equation of the plane containing \((2, 1, -1)\) and parallel to the vectors \(\langle 1, 0, 3 \rangle\) and \(\langle 2, 1, 0 \rangle\).

Problem 3. Find the line of intersection between the planes \(x + y = 2\) and \(y + z = 3\).

Problem 4. Find the angle between the planes \(x + y + z = 1\) and \(x - y = 0\).

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?

Quadric Surfaces [Lessons 3-4] (§13.6)

Video (Lesson 3) Notes (Lesson 3) Video (Lesson 4) Notes (Lesson 4) Practice

A quadric surface is the graph of a second-degree equation in three variables.

Standard Quadric Surfaces
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

Traces are curves of intersection with planes parallel to coordinate planes.

Practice Problems

Problem 1. Identify the surface: \(\displaystyle -x^2 + \frac{y^2}{4} - \frac{z^2}{9} = 1\).

Problem 2. Identify the surface defined by \(x^2 + 4x + y^2 - 2y + z^2 = -4\). (Hint: complete the square.)

Problem 3. Identify the surface: \(z = 4x^2 + 9y^2\).

Problem 4. Describe the trace of \(\displaystyle \frac{x^2}{4} + \frac{y^2}{9} - z^2 = 1\) in the plane \(z = 2\).

Problem 5. Identify the surface: \(\displaystyle x^2 + \frac{y^2}{4} = \frac{z^2}{9}\).

Vector-Valued Functions [Lesson 5] (§14.1)

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Definition

A vector-valued function (or vector function) has the form:

\[\vec{r}(t) = \langle f(t), g(t), h(t) \rangle = f(t)\vec{i} + g(t)\vec{j} + h(t)\vec{k}\]

The graph is a space curve traced out as t varies.

Limits and Continuity

The limit of a vector function is computed component-wise:

\[\lim_{t \to a} \vec{r}(t) = \left\langle \lim_{t \to a} f(t), \lim_{t \to a} g(t), \lim_{t \to a} h(t) \right\rangle\]

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.

Problem 2. Evaluate \(\displaystyle \lim_{t \to 0} \left\langle \frac{\sin(3t)}{t},\, e^{2t},\, \frac{t^2+1}{t+1} \right\rangle\).

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.

Problem 4. Describe the curve traced by \(\vec{r}(t) = \langle 2\cos t,\, 2\sin t,\, 0 \rangle\) for \(0 \le t \le 2\pi\).

Problem 5. Evaluate \(\displaystyle \lim_{t \to 0}\left\langle \frac{e^t - 1}{t},\, \frac{1-\cos t}{t^2},\, t\sin\!\left(\frac{1}{t}\right) \right\rangle\).

Calculus of Vector-Valued Functions & Motion in Space [Lessons 6-7] (§14.2-14.3)

Video (Lesson 6) Notes (Lesson 6) Video (Lesson 7) Notes (Lesson 7) Practice

Derivative

\[\vec{r}'(t) = \lim_{h \to 0} \frac{\vec{r}(t+h) - \vec{r}(t)}{h} = \langle f'(t), g'(t), h'(t) \rangle\]

Integral

\[\int \vec{r}(t)\,dt = \left\langle \int f(t)\,dt, \int g(t)\,dt, \int h(t)\,dt \right\rangle + \vec{C}\]

Differentiation Rules

For vector functions \(\vec{u}(t)\) and \(\vec{v}(t)\):

Motion in Space

If \(\vec{r}(t)\) is the position vector at time t:

Unit Tangent Vector

\[\vec{T}(t) = \frac{\vec{r}'(t)}{|\vec{r}'(t)|} = \frac{\vec{v}(t)}{|\vec{v}(t)|}\]

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 \(\vec{r}(t)\).

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.

Problem 3. Find the unit tangent vector \(\vec{T}(0)\) for \(\vec{r}(t) = \langle \cos t,\, \sin t,\, t \rangle\).

Problem 4. Find the speed of a particle with position \(\vec{r}(t) = \langle e^t,\, e^{-t},\, \sqrt{2}\,t \rangle\) at \(t = 0\).

Problem 5. If \(\vec{r}(t) = \langle t,\, t^2,\, t^3 \rangle\), compute \(\vec{r}'(t) \times \vec{r}''(t)\).

Problem 6. A particle travels along \(\vec{r}(t) = \langle 3\cos t,\, 4\sin t,\, 0 \rangle\). Find the speed at \(t = \frac{\pi}{2}\).

Arc Length and Curvature [Lesson 8] (§14.4-14.5)

Video Notes Practice

Arc Length

Arc Length of curve \(\vec{r}(t)\) from \(t = a\) to \(t = b\):

\[L = \int_a^b |\vec{r}'(t)|\,dt = \int_a^b \sqrt{[f'(t)]^2 + [g'(t)]^2 + [h'(t)]^2}\,dt\]

Arc Length Function

\[s(t) = \int_{t_0}^t |\vec{r}'(u)|\,du\]

Curvature

Curvature measures how quickly the curve changes direction:

\[\kappa = \left|\frac{d\vec{T}}{ds}\right| = \frac{|\vec{T}'(t)|}{|\vec{r}'(t)|} = \frac{|\vec{r}'(t) \times \vec{r}''(t)|}{|\vec{r}'(t)|^3}\]

For a plane curve \(y = f(x)\):

\[\kappa(x) = \frac{|f''(x)|}{[1+(f'(x))^2]^{3/2}}\]

Practice Problems

Problem 1. Find the arc length of \(\vec{r}(t) = \langle 3\cos t,\, 3\sin t,\, 4t \rangle\) for \(0 \le t \le 2\pi\).

Problem 2. Find the curvature \(\kappa(t)\) of \(\vec{r}(t) = \langle 2\cos t,\, 2\sin t,\, t \rangle\).

Problem 3. Find the curvature of the plane curve \(y = x^3\) at the point \((0,0)\).

Problem 4. Find the arc length of \(\vec{r}(t) = \langle 3t,\, 4t,\, 0 \rangle\) for \(1 \le t \le 5\).

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 \(t=\pi\).

Functions of Several Variables [Lesson 9] (§15.1)

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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

Functions of Three Variables

For \(w = f(x,y,z)\):

Practice Problems

Problem 1. Find the domain of \(f(x, y) = \sqrt{9 - x^2 - y^2}\).

Problem 2. Describe the nonempty level curves of \(f(x, y) = x^2 + 4y^2\).

Problem 3. Describe the level curves of \(f(x, y) = y - x^2\).

Problem 4. Find the domain of \(f(x, y) = \ln(x + y - 1)\).

Problem 5. Describe the level surfaces of \(f(x, y, z) = x^2 + y^2 + z^2\).

Limits and Continuity [Lesson 10] (§15.2)

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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:

Continuity

f is continuous at (a,b) if:

\[\lim_{(x,y) \to (a,b)} f(x,y) = f(a,b)\]

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.

Problem 2. Evaluate \(\displaystyle\lim_{(x,y) \to (0,0)} \frac{3x^2 y}{x^2 + y^2}\), or show it does not exist.

Problem 3. Find the value of \(k\) such that \(\displaystyle\lim_{(x,y) \to (0,0)} \frac{x^4 - 2k(x^2+y^2) - y^4}{x^2+y^2} = 6\).

Problem 4. Evaluate \(\displaystyle\lim_{(x,y) \to (0,0)} \frac{\sin(x^2 + y^2)}{x^2 + y^2}\).

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.

Partial Derivatives [Lesson 11] (§15.3)

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First-Order Partial Derivatives

Partial derivative with respect to x (treat y as constant):

\[\frac{\partial f}{\partial x} = f_x = \lim_{h \to 0} \frac{f(x+h,y) - f(x,y)}{h}\]

Partial derivative with respect to y (treat x as constant):

\[\frac{\partial f}{\partial y} = f_y = \lim_{h \to 0} \frac{f(x,y+h) - f(x,y)}{h}\]

Higher-Order Partial Derivatives

Clairaut's Theorem

If \(f_{xy}\) and \(f_{yx}\) are both continuous:

\[f_{xy} = f_{yx}\]

Practice Problems

Problem 1. Let \(f(x,y,z)=\cos(xy)+e^{y^2z}+\ln(z^3)\). Compute \(f_{yxz}(0,1,1)\).

Problem 2. If \(f(x, y) = x^3 y^2 + \sin(xy)\), find \(f_{xy}(0, 0)\).

Problem 3. If \(f(x,y) = e^{x^2 y}\), find \(f_x(1, 2)\).

Problem 4. Let \(f(x,y)=x^2\ln(y)\). Find \(f_{xy}\) and verify Clairaut's theorem by also computing \(f_{yx}\).

Problem 5. If \(f(x,y) = \arctan\!\left(\frac{y}{x}\right)\), find \(f_x + f_y\) at \((1, 1)\).

The Chain Rule [Lesson 12] (§15.4)

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Chain Rule - Case 1

If \(z = f(x,y)\) where \(x = g(t)\) and \(y = h(t)\):

\[\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}\]

Chain Rule - Case 2

If \(z = f(x,y)\) where \(x = g(s,t)\) and \(y = h(s,t)\):

\[\frac{\partial z}{\partial s} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial s} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial s}\]

Implicit Differentiation

If \(F(x,y,z) = 0\) defines z implicitly:

\[\frac{\partial z}{\partial x} = -\frac{F_x}{F_z}, \quad \frac{\partial z}{\partial y} = -\frac{F_y}{F_z}\]

Practice Problems

Problem 1. If \(z = x^2 y\), where \(x = t^2\) and \(y = t^3\), find \(\frac{dz}{dt}\) at \(t = 1\).

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 \((1, 1, 1)\).

Problem 3. If \(w=xy+yz\), \(x=s+t\), \(y=st\), \(z=s-t\), find \(\frac{\partial w}{\partial s}\) at \(s=1,\, t=0\).

Problem 4. If \(\sin(xy) = e^{2z}\), use implicit differentiation to find \(\frac{\partial z}{\partial x}\) at \(\left(1, \frac{\pi}{2}, 0\right)\).

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 \(f_y\).

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.

Directional Derivatives & Gradient [Lesson 13] (§15.5)

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Directional Derivative

The directional derivative \(D_{\vec{u}}f(a,b)\) measures rate of change in direction of unit vector \(\vec{u}\):

\[D_{\vec{u}}f(a,b) = \nabla f(a,b) \cdot \vec{u} = f_x(a,b)u_1 + f_y(a,b)u_2\]

Gradient Vector

\[\nabla f(x,y) = \langle f_x(x,y), f_y(x,y) \rangle\]

Properties of the Gradient

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 \(\vec{v}=\langle 3,4\rangle\).

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.

Problem 3. At \((1,2)\), in what direction does \(f(x,y)=xe^{-y}\) decrease most rapidly? What is that rate of decrease?

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 \(\langle 1,-1,0\rangle\).

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 \(\langle 3,4\rangle\).

Tangent Planes & Linear Approximation [Lesson 14] (§15.6)

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Tangent Plane

Equation of tangent plane to \(z = f(x,y)\) at \((a, b, f(a,b))\):

\[z = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)\]

Linear Approximation

\[L(x,y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)\]

Differential

\[dz = f_x\,dx + f_y\,dy\]

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.

Problem 2. Find the equation of the tangent plane to \(z = x^2 + 2xy\) at the point \((1, 1, 3)\).

Problem 3. Use the linear approximation of \(f(x,y)=\sqrt{x^2+y^2}\) at \((3,4)\) to estimate \(f(3.1, 3.9)\).

Problem 4. If \(z = x^3 y^2\), find the differential \(dz\).

Problem 5. Find the normal vector to the surface \(x^2+y^2-z = 0\) at the point \((1, 2, 5)\).

Maximum and Minimum Problems [Lessons 15-16] (§15.7)

Video (Lesson 15) Notes (Lesson 15) Video (Lesson 16) Notes (Lesson 16) Practice

Critical Points

Point (a,b) is a critical point if:

\[f_x(a,b) = 0 \text{ and } f_y(a,b) = 0 \quad (\nabla f(a,b) = \vec{0})\]

Second Derivative Test

Let the discriminant be:

\[D = f_{xx}(a,b)f_{yy}(a,b) - [f_{xy}(a,b)]^2\]

Classification

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 \(f(x,y) = x^2y - y + 3y^2\).

Problem 2. Classify the critical point of \(f(x,y) = x^2+xy+y^2+3x-3y+4\) using the second derivative test.

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 \(M + m\).

Problem 4. Find and classify all critical points of \(f(x,y) = xy - x^3 - y^2\).

Problem 5. Show that \(f(x,y) = x^2 - y^2\) has a saddle point at \((0,0)\) using the second derivative test.

Problem 6. Find the absolute maximum and absolute minimum of \(f(x,y) = 4xy - x^2 - 2y^2 + 2\) on the region \(0 \le x \le 2\), \(0 \le y \le 1\).