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Points and vectors

“Describe the set of all points $(x, y, z)$ in $\mathbb R^3$ that satisfy …”: Complete the square and name the shape, like "the sphere of radius $r$ centered at …". With $<$ , it is the inside of the sphere.Vectors →
“Determine whether or not the given points are collinear”: Collinear means on one line: $\overrightarrow{AC}$ is a multiple of $\overrightarrow{AB}$ . $\overrightarrow{AC} = c\,\overrightarrow{AB}$ Vectors →
“Find the component of $\mathbf b$ in the direction $\mathbf a$ ”: A number, not a vector: how far $\mathbf b$ reaches along $\mathbf a$ . $\dfrac{\mathbf a\cdot\mathbf b}{|\mathbf a|}$ Vectors →
“Compute the volumes of the parallelepipeds determined by the following vectors”: It is the slanted box with the three vectors as edges from one corner. Take the triple product and drop its sign. $|\mathbf a\cdot(\mathbf b\times\mathbf c)|$ Vectors →

Lines, planes and curves

“Find the vector parametric, scalar parametric and symmetric equations for the line”: One line, written three ways. Vector: $\langle x, y, z\rangle = P + t\,\mathbf d$ . Scalar: $x = x_0 + d_1t$ , and so on. Symmetric: solve each for $t$ and set them equal.Lines →
“the line of intersection of the given planes / the line through … and parallel to each of the two planes”: Either way, the direction is perpendicular to both normals, so cross them. For the intersection, also find one point on both planes. $\mathbf d = \mathbf n_1\times\mathbf n_2$ Lines →
“the plane containing $A$ and perpendicular to $L$ ”: The line's direction is the plane's normal. $\mathbf n = \mathbf d_L$ Planes →
“the plane $P$ that contains $L_1$ and is parallel to $L_2$ ”: Both directions lie along the plane, so their cross product is the normal. Use a point of $L_1$ . $\mathbf n = \mathbf d_1\times\mathbf d_2$ Planes →
“the unit tangent vector … pointing in the direction of increasing $t$ ”: Use $+\mathbf r'(t)$ (not its negative) and divide by its length. $\hat{\mathbf T} = \dfrac{\mathbf r'(t)}{|\mathbf r'(t)|}$ Curves →
“the particle flies off the path on a line $L$ in the direction tangent to the path”: It continues along the tangent line at that moment: $\mathbf r(t_0) + u\,\mathbf r'(t_0)$ .Curves →

Surfaces and level curves

“Draw a contour map of …, showing all types of level curves that occur”: Draw the curves $f(x, y) = c$ in the $xy$ -plane. Split into cases of $c$ , often $c > 0$ , $c = 0$ and $c < 0$ . Label at least one curve of each type. $f(x, y) = c$ Surfaces →
“Describe the level surfaces of $f(x, y, z)$ ”: Name the surface $f(x, y, z) = c$ for each case of $c$ , including empty cases and single points. $f(x, y, z) = c$ Surfaces →
“Sketch and describe the following surfaces”: Slice across and along the axis, name the surface, and give its axis, center or vertex.Surfaces →

Limits and partial derivatives

“Evaluate each of the following limits or show that it does not exist”: First try plugging in. If that fails, simplify or use polar coordinates to find the value. Or find two paths that give different limits.Limits & Continuity →
“Express the function in terms of the polar coordinates $r$ and $\theta$ , and simplify”: Substitute $x = r\cos\theta$ , $y = r\sin\theta$ and cancel powers of $r$ . $x = r\cos\theta,\ y = r\sin\theta$ Limits & Continuity →
“Compute, directly from the definitions”: Use the limit of the difference quotient, not the derivative rules. The rules usually fail at that special point. $f_x(a, b) = \lim_{h\to 0}\dfrac{f(a + h, b) - f(a, b)}{h}$ Partial Derivatives →
“Find all first partial derivatives of the following functions and evaluate them at the given point”: Find one partial per variable, holding the other variables constant. Plug in the point last.Partial Derivatives →
“A surface $z(x, y)$ is defined by … Compute $\dfrac{\partial z}{\partial x}$ , $\dfrac{\partial z}{\partial y}$ in terms of $x$ , $y$ , $z$ ”: Implicit differentiation: differentiate both sides, treating $z$ as a function of $x$ and $y$ . $\dfrac{\partial z}{\partial x} = -\dfrac{F_x}{F_z}$ Partial Derivatives →
“Which of the following functions are harmonic?”: Harmonic means it solves Laplace's equation: check $u_{xx} + u_{yy} = 0$ everywhere $u$ is defined. $u_{xx} + u_{yy} = 0$ Partial Derivatives →

Chain rule

“Write out the chain rule for each of the following functions”: Draw the . Write one product per path, and show where each partial is evaluated.Chain Rule →
“You may assume that $f$ is a smooth function so that the Chain Rule and Clairaut's Theorem … apply”: This is a second-derivative chain rule question. $f_1$ is the partial in $f$ 's first slot. $f_{12}$ is first slot, then second. You may combine $f_{12}$ and $f_{21}$ into one term. $f_{12} = f_{21}$ Chain Rule →
“For an arbitrary $F(z)$ show that $u(x, y)$ satisfies …”: $F$ has no formula: keep $F'$ (or $F_u$ , $F_v$ ) as a symbol and show the equation holds anyway.Chain Rule →

Tangent planes and normal lines

“Find the tangent plane and normal line to the surface … at …”: Find a normal vector $\mathbf n$ at the point. The plane is $\mathbf n \cdot \langle x - x_0, y - y_0, z - z_0\rangle = 0$ . The line is the point plus $t\,\mathbf n$ . $\langle x_0, y_0, z_0\rangle + t\,\mathbf n$ Tangent Planes →
“Use a 3-dimensional gradient vector to find the equation of the tangent plane”: Write the surface as $F(x, y, z) = c$ and use $\nabla F$ at the point as the normal. $\mathbf n = \nabla F(x_0, y_0, z_0)$ Tangent Planes →
“where the tangent plane is parallel to the plane …”: The surface's normal is a multiple of the plane's normal: solve $\nabla F = \lambda\,\mathbf n$ with the surface equation. $\nabla F = \lambda\,\mathbf n$ Tangent Planes →
“Find all horizontal planes that are tangent to the surface”: Horizontal means normal $\langle 0, 0, 1\rangle$ : on $z = f(x, y)$ , solve $f_x = 0$ and $f_y = 0$ . $f_x = f_y = 0$ Tangent Planes →
“tangent to the curve of intersection of the surfaces …”: The curve's direction is perpendicular to both normals: cross the two gradients. $\nabla F \times \nabla G$ Tangent Planes →

Linear approximation and error

“Find an approximate value … without using a calculator or computer”: Use the linear approximation at a nearby point where $f$ is easy to compute. $f(a, b) + f_x\,\Delta x + f_y\,\Delta y$ Linear Approximation →
“Use differentials to …”: Estimate a change with $df = f_x\,dx + f_y\,dy$ : each partial times its own small change. $df = f_x\,dx + f_y\,dy$ Linear Approximation →
“estimate the maximum possible error in calculating …”: Add the absolute value of each error's contribution, so nothing cancels. $|f_x|\,\Delta x + |f_y|\,\Delta y$ Linear Approximation →
“find the approximate percentage error in calculating …”: The maximum error divided by the value, times $100$ . $100\,\dfrac{|\Delta f|}{|f|}$ Linear Approximation →
“near this point the surface determines $x$ as a function … of $y$ and $z$ ”: $x$ is the dependent variable. Differentiate implicitly in $y$ and in $z$ to get $x_y$ and $x_z$ .Linear Approximation →

Directional derivatives and the gradient

“Find the rate of change of the given function at the given point in the given direction”: The directional derivative: dot the gradient with the unit vector of the direction. $D_{\mathbf u}f = \nabla f \cdot \mathbf u$ Gradient →
“In what direction should the hiker proceed from $P$ to ascend along the steepest path?”: Along $\nabla f$ . The slope of that path is $|\nabla f|$ . $\nabla f,\ |\nabla f|$ Gradient →
“Find the maximum rate of change of $f(x, y)$ at the point …”: The length of the gradient there. $|\nabla f|$ Gradient →
“In which direction does the temperature decrease most rapidly?”: Opposite the gradient. $-\nabla T$ Gradient →
“She wishes to stay at the same temperature … Find a direction in which the initial rate of change of the temperature is $0$ ”: Any direction perpendicular to the gradient. $\nabla T \cdot \mathbf u = 0$ Gradient →
“Find a unit vector $u$ that is tangent to the level curve of $f(x, y)$ ”: Tangent to a level curve means perpendicular to $\nabla f$ : swap the components and change one sign. $\pm\dfrac{\langle -f_y,\ f_x\rangle}{|\nabla f|}$ Gradient →

Critical points

“Find and classify all critical points of …”: Solve $f_x = 0$ and $f_y = 0$ together, then sort each point with the second derivative test. $D = f_{xx}f_{yy} - f_{xy}^2$ Critical Points →
“Use the Second Derivative Test to find all values of the constant $c$ for which … has a saddle point”: Write $D$ in terms of $c$ and solve $D < 0$ . The test says nothing about values with $D = 0$ . $D < 0$ Critical Points →
“points of indeterminate type”: Critical points with $D = 0$ , where the second derivative test gives no answer. $D = 0$ Critical Points →

Absolute extrema and Lagrange multipliers

“Find the absolute maximum and minimum values of the function … on the rectangular region …”: Check the critical points inside, each of the four edges and the four corners, then compare values.Absolute Extrema →
“… on the closed bounded domain defined by $x^2 + y^2 \le 1$ ”: The boundary circle belongs to the region, so check it too. A continuous function is sure to have a max and a min here.Absolute Extrema →
“Find the location and value of the absolute maximum and minimum”: Two answers for each: the point $(x, y)$ and the number $f(x, y)$ .Absolute Extrema →
“Use the method of Lagrange Multipliers to find the maximum and minimum values of … subject to the constraint …”: Solve $\nabla f = \lambda\nabla g$ together with the constraint, then compare $f$ at every solution. $\nabla f = \lambda\nabla g,\quad g = k$ Lagrange Multipliers →
“Give the system of equations that must be solved …”: Write $\nabla f = \lambda\nabla g$ out in components, plus the constraint. No solving in that part.Lagrange Multipliers →
“find the points on the sphere … closest to and farthest from the point …”: Minimize and maximize the squared distance to that point, with the sphere as the constraint. $D = (x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2$ Lagrange Multipliers →
“… on the curve of intersection of the plane … and the paraboloid …”: Two constraints, so two . $\nabla f = \lambda\nabla g + \mu\nabla h$ Lagrange Multipliers →
“find the dimensions of the least expensive box”: Minimize the cost with the fixed volume as the constraint. The answer is the side lengths, with units.Lagrange Multipliers →

Double integrals and polar coordinates

“evaluate the given double integral without using iteration. Instead, interpret the integral as … an area or a volume”: Do not find antiderivatives. Name the region or solid and use a geometry formula. Odd symmetry can make some pieces $0$ .Double Integrals →
“Sketch the region of integration”: Draw the region the limits describe. Label every boundary curve and where the curves meet.Double Integrals →
“write an equivalent double integral with the order of integration reversed”: Same region, sliced the other way: new constant outer limits, new curves inside.Double Integrals →
“Combine the sum of the two iterated double integrals into a single iterated double integral with the order of integration reversed”: Sketch both pieces. Together they form one region, so one integral in the other order covers it.Double Integrals →
“Express $I$ as an integral where we integrate first with respect to $x$ ”: $x$ is the inner variable: the order is $dx\,dy$ , with curves $x = \dots$ inside.Double Integrals →
“find polar coordinates $r$ and $\theta$ , with $r > 0$ and $0 \le \theta < 2\pi$ ”: One standard pair: $r = \sqrt{x^2 + y^2}$ and the angle in the point's own quadrant.Polar Coordinates →
“Use polar coordinates to evaluate … / Evaluate the integral by transforming to polar coordinates”: The Cartesian limits only describe the region. Redraw it in $r$ and $\theta$ , substitute, and replace $dA$ with $r\,dr\,d\theta$ . $dA = r\,dr\,d\theta$ Polar Coordinates →
“Find the volume lying inside the sphere … and above the paraboloid …”: Integrate the top surface minus the bottom one over the disk inside the circle where they meet.Polar Coordinates →

Mass, centroids and surface area

“A thin plate of uniform density”: A flat region with constant density $\rho$ : its center of mass is the centroid, and $\rho$ cancels.Double Integral Applications →
“Find $\bar y$ , the $y$ -coordinate of the center of mass”: Compute $\iint_D y\rho\,dA$ and divide by the mass $M$ . So you still need $M$ . $\bar y = \frac{1}{M}\iint_D y\,\rho\,dA$ Double Integral Applications →
“The average distance of a point in $D$ to …”: The average value of the distance function: divide its integral over $D$ by the area of $D$ . $\frac{1}{A(D)}\iint_D \text{distance}\,dA$ Double Integral Applications →
“Find the area of the part of the surface $z = f(x, y)$ that lies above …”: Surface area: integrate $\sqrt{1 + f_x^2 + f_y^2}$ over the region it lies above. $\iint_D \sqrt{1 + f_x^2 + f_y^2}\,dA$ Double Integral Applications →

Triple integrals and their coordinates

“in the order: first $z$ , then $y$ , then $x$ ”: $z$ is the innermost integral and $x$ the outermost. $dz\,dy\,dx$ Triple Integrals →
“in the first octant”: $x \ge 0$ , $y \ge 0$ and $z \ge 0$ .Triple Integrals →
“bounded by the coordinate planes and the plane …”: The planes $x = 0$ , $y = 0$ , $z = 0$ are faces. If the last plane crosses all three positive axes, the solid is a .Triple Integrals →
“Find the centroid of $E$ ”: The center of mass with constant density: divide each of $\iiint_E x\,dV$ , $\iiint_E y\,dV$ , $\iiint_E z\,dV$ by the volume.Cylindrical Coordinates →
“A hole of diameter … is drilled through the sphere along the $z$ -axis”: Remove the points closer to the $z$ -axis than the hole's radius. In cylindrical coordinates, $r$ starts at that radius.Cylindrical Coordinates →
“the ice cream cone”: Inside a sphere centered at the origin and above a cone with its tip there: $0 \le \rho \le a$ , $0 \le \varphi \le \varphi_0$ .Spherical Coordinates →
“the eighth of the sphere $x^2 + y^2 + z^2 \le 1$ with $x, y, z \ge 0$ ”: $0 \le \rho \le 1$ , $0 \le \varphi \le \tfrac{\pi}{2}$ , $0 \le \theta \le \tfrac{\pi}{2}$ .Spherical Coordinates →
“the hemispherical shell bounded by … and above the plane $z = 0$ ”: $a \le \rho \le b$ between the two spheres, with $0 \le \varphi \le \tfrac{\pi}{2}$ .Spherical Coordinates →
“by changing to spherical coordinates”: Read the solid from the Cartesian limits and describe it with $\rho$ , $\varphi$ , $\theta$ . Then replace $dV$ with $\rho^2\sin\varphi\,d\rho\,d\varphi\,d\theta$ .Spherical Coordinates →

What MATH 200 Exam Questions Are Really Asking

Points and vectors

Lines, planes and curves

Surfaces and level curves

Limits and partial derivatives

Chain rule

Tangent planes and normal lines

Linear approximation and error

Directional derivatives and the gradient

Critical points

Absolute extrema and Lagrange multipliers

Double integrals and polar coordinates

Mass, centroids and surface area

Triple integrals and their coordinates

Which topics are you reviewing?

Navigation

Points and vectors

Lines, planes and curves

Surfaces and level curves

Limits and partial derivatives

Chain rule

Tangent planes and normal lines

Linear approximation and error

Directional derivatives and the gradient

Critical points

Absolute extrema and Lagrange multipliers

Double integrals and polar coordinates

Mass, centroids and surface area

Triple integrals and their coordinates