Calculus III Memorization & Recognition Guide
It covers the material we've discussed for Chapter 10 (§§10.1–10.8), Chapter 11 (§§11.1–11.6), trigonometry, algebra, radicals, derivative rules, equation recognition, and exam-style question wording.
I've also included the less obvious things you mentioned forgetting, such as traces, cylinders, the orientation of quadrics, implicit differentiation, unit vectors, and the difference between velocity and speed.
# CALCULUS III — COMPLETE MEMORIZATION & RECOGNITION GUIDE
NYU MATH-UA 123 | Midterm 1 Preparation
Purpose: Memorize essential formulas, recognize mathematical shapes and problem types, understand their meanings, and develop fast recall for multiple-choice and free-response questions.
Important: This is a comprehensive study checklist for the sections we've discussed, not a guarantee that every item will appear on your particular exam. Some older practice exams include topics such as curvature that may not be in your current exam's coverage.
# PART I — TRIGONOMETRY AND ALGEBRA FOUNDATIONS
## 1. Unit Circle: Exact Values
Memorize the following values without a calculator.
| Angle (degrees) | Radians | Sine | Cosine | Tangent |
| --------------- | ------- | ----- | ------ | --------- |
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | Undefined |
| 120° | 2π/3 | √3/2 | −1/2 | −√3 |
| 135° | 3π/4 | √2/2 | −√2/2 | −1 |
| 150° | 5π/6 | 1/2 | −√3/2 | −√3/3 |
| 180° | π | 0 | −1 | 0 |
| 210° | 7π/6 | −1/2 | −√3/2 | √3/3 |
| 225° | 5π/4 | −√2/2 | −√2/2 | 1 |
| 240° | 4π/3 | −√3/2 | −1/2 | √3 |
| 270° | 3π/2 | −1 | 0 | Undefined |
| 300° | 5π/3 | −√3/2 | 1/2 | −√3 |
| 315° | 7π/4 | −√2/2 | √2/2 | −1 |
| 330° | 11π/6 | −1/2 | √3/2 | −√3/3 |
| 360° | 2π | 0 | 1 | 0 |
Unit circle rule:
\\[ (\cos\theta,\sin\theta) \\]
Cosine is the horizontal coordinate; sine is the vertical coordinate.
## 2. Quadrant Signs
| Quadrant | Sine | Cosine | Tangent |
| -------- | ---- | ------ | ------- |
| I | + | + | + |
| II | + | − | − |
| III | − | − | + |
| IV | − | + | − |
Memorize reference angles so you can evaluate expressions such as:
- \\(\sin(5\pi/6)=1/2\\)
- \\(\cos(7\pi/4)=\sqrt2/2\\)
- \\(\sin(3\pi/2)=-1\\)
- \\(\cos(2\pi/3)=-1/2\\)
## 3. Trigonometric Identities
Pythagorean identity:
\\[ \sin^2t+\cos^2t=1 \\]
Tangent:
\\[ \tan t=\frac{\sin t}{\cos t} \\]
Other useful identities:
\\[ 1+\tan^2t=\sec^2t \\]
\\[ \sin(2t)=2\sin t\cos t \\]
\\[ \cos(2t)=\cos^2t-\sin^2t \\]
Even/odd identities:
\\[ \cos(-t)=\cos t \\]
\\[ \sin(-t)=-\sin t \\]
These identities are useful for simplifying arc length, vector magnitudes, and trig expressions.
## 4. Trigonometric Derivatives
| Function | Derivative |
| ------------ | ------------------- |
| \\(\sin x\\) | \\(\cos x\\) |
| \\(\cos x\\) | \\(-\sin x\\) |
| \\(\tan x\\) | \\(\sec^2x\\) |
| \\(\sec x\\) | \\(\sec x\tan x\\) |
| \\(\csc x\\) | \\(-\csc x\cot x\\) |
| \\(\cot x\\) | \\(-\csc^2x\\) |
With the chain rule:
\\[ \frac{d}{dx}\sin(g(x))=\cos(g(x))g'(x) \\]
\\[ \frac{d}{dx}\cos(g(x))=-\sin(g(x))g'(x) \\]
Examples:
\\[ \frac{d}{dt}\cos(8t)=-8\sin(8t) \\]
\\[ \frac{d}{dx}\sin(x^2)=2x\cos(x^2) \\]
\\[ \frac{d}{dx}\cos(xy)=-y\sin(xy) \\]
The final example is a partial derivative with respect to \\(x\\), treating \\(y\\) as constant.
## 5. Other Derivatives to Memorize
| Function | Derivative |
| ----------------- | ------------------- |
| \\(x^n\\) | \\(nx^{n-1}\\) |
| \\(e^x\\) | \\(e^x\\) |
| \\(e^{g(x)}\\) | \\(e^{g(x)}g'(x)\\) |
| \\(\ln x\\) | \\(1/x\\) |
| \\(\ln\|g(x)\|\\) | \\(g'(x)/g(x)\\) |
| \\(\sqrt{x}\\) | \\(1/(2\sqrt{x})\\) |
| \\(1/x\\) | \\(-1/x^2\\) |
| \\(1/\sqrt{x}\\) | \\(-1/(2x^{3/2})\\) |
| \\(a^x\\) | \\(a^x\ln a\\) |
Product rule:
\\[ (uv)'=u'v+uv' \\]
Quotient rule:
\\[ \left(\frac uv\right)'=\frac{vu'-uv'}{v^2} \\]
Chain rule:
\\[ [f(g(x))]'=f'(g(x))g'(x) \\]
## 6. Radicals, Fractions, and Exponents
Convert radicals to powers before differentiating.
\\[ \sqrt{x}=x^{1/2} \\]
\\[ \frac1{\sqrt{x}}=x^{-1/2} \\]
\\[ \sqrt[3]{x}=x^{1/3} \\]
\\[ \frac1{x^3}=x^{-3} \\]
Examples:
\\[ \frac{d}{dx}x^{1/2}=\frac12x^{-1/2} \\]
\\[ \frac{d}{dx}x^{-1/2}=-\frac12x^{-3/2} \\]
\\[ \frac{d}{dx}\sqrt{9-x^2}=-\frac{x}{\sqrt{9-x^2}} \\]
Remember:
\\[ -x^2=-(x^2) \\]
It is not the same as \\((-x)^2\\).
## 7. Basic Integrals Worth Recognizing
\\[ \int x^n\\,dx=\frac{x^{n+1}}{n+1}+C,\qquad n\ne-1 \\]
\\[ \int\sin x\\,dx=-\cos x+C \\]
\\[ \int\cos x\\,dx=\sin x+C \\]
\\[ \int e^x\\,dx=e^x+C \\]
\\[ \int\frac1x\\,dx=\ln|x|+C \\]
You may need these when recovering position from velocity or calculating arc length.
# PART II — RECOGNIZING QUADRIC SURFACES
## 8. Sphere
Equation:
\\[ x^2+y^2+z^2=R^2 \\]
Characteristics:
- Closed, round surface.
- Three squared variables with equal coefficients.
- Radius \\(R\\).
- Center at the origin in this form.
Example:
\\[ x^2+y^2+z^2=25 \\]
Sphere of radius 5.
Shifted sphere:
\\[ (x-a)^2+(y-b)^2+(z-c)^2=R^2 \\]
Center \\((a,b,c)\\).
## 9. Ellipsoid
Equation:
\\[ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 \\]
Characteristics:
- Closed surface.
- Three positive squared terms.
- Different axis lengths.
- Looks like a stretched or compressed sphere.
Example:
\\[ \frac{x^2}{9}+\frac{y^2}{16}+\frac{z^2}{25}=1 \\]
Semiaxes: 3, 4, 5.
Longest along \\(z\\).
## 10. Elliptic Paraboloid
Equation:
\\[ z=x^2+y^2 \\]
Characteristics:
- Bowl shape.
- One variable is linear.
- Two squared variables have the same sign.
- Opens along the linear variable's axis.
Variations:
\\[ z=-x^2-y^2 \\]
Opens downward.
\\[ x=y^2+z^2 \\]
Opens toward positive \\(x\\).
\\[ y=-x^2-z^2 \\]
Opens toward negative \\(y\\).
\\[ z=4x^2+y^2 \\]
Narrower in the \\(x\\)-direction.
## 11. Hyperbolic Paraboloid
Equation:
\\[ z=x^2-y^2 \\]
Characteristics:
- Saddle shape.
- One variable linear.
- Two squared variables with opposite signs.
- Curves upward in one direction and downward in another.
Variation:
\\[ z=y^2-x^2 \\]
Saddle with upward and downward directions exchanged.
## 12. Hyperboloid of One Sheet
Equation:
\\[ x^2+y^2-z^2=1 \\]
Characteristics:
- One connected surface.
- Hourglass or cooling-tower appearance.
- Two positive squared terms.
- One negative squared term.
- The negative term identifies the main axis.
Variations:
\\[ \frac{x^2}{4}+\frac{y^2}{9}-\frac{z^2}{16}=1 \\]
Axis along \\(z\\).
\\[ x^2-y^2+z^2=1 \\]
Axis along \\(y\\).
## 13. Hyperboloid of Two Sheets
Equation:
\\[ z^2-x^2-y^2=1 \\]
Characteristics:
- Two disconnected pieces.
- One positive squared term.
- Two negative squared terms.
- The positive term identifies the axis.
Variation:
\\[ x^2-y^2-z^2=1 \\]
Two sheets along the \\(x\\)-axis.
## 14. Elliptic Cone
Equation:
\\[ z^2=x^2+y^2 \\]
Equivalent:
\\[ x^2+y^2-z^2=0 \\]
Characteristics:
- Double cone.
- Two pieces meet at a vertex.
- Squared terms have mixed signs.
- Right-hand side is zero.
- In this form, the axis is \\(z\\).
Variations:
\\[ z^2=4x^2+y^2 \\]
Elliptic cone.
\\[ x^2=y^2+z^2 \\]
Cone oriented along \\(x\\).
## 15. Circular Cylinder
Equation:
\\[ x^2+y^2=9 \\]
Characteristics:
- Circular cross-section of radius 3.
- Extends along the missing variable \\(z\\).
Variation:
\\[ y^2+z^2=16 \\]
Cylinder of radius 4 extending along \\(x\\).
## 16. Elliptic Cylinder
Equation:
\\[ \frac{x^2}{16}+\frac{y^2}{4}=1 \\]
Characteristics:
- Elliptical cross-section.
- Missing variable \\(z\\).
- Extends infinitely along \\(z\\).
- Semiaxes 4 and 2.
## 17. Parabolic Cylinder
Equation:
\\[ z=x^2 \\]
Characteristics:
- Parabolic trough.
- Missing variable \\(y\\).
- Extends along \\(y\\).
Variation:
\\[ y=z^2 \\]
Extends along \\(x\\).
## 18. Hyperbolic Cylinder
Equation:
\\[ x^2-y^2=1 \\]
Characteristics:
- Hyperbola as its cross-section.
- Missing variable \\(z\\).
- Extends along \\(z\\).
- Two disconnected curved surfaces.
## 19. Plane
Equation:
\\[ ax+by+cz=d \\]
Characteristics:
- Flat surface.
- First powers of variables.
- Normal vector \\(\langle a,b,c\rangle\\).
Example:
\\[ 2x+3y-z=6 \\]
## 20. Other Graphs to Recognize
\\[ z=\sqrt{x^2+y^2} \\]
Upper cone.
\\[ z=\sqrt{9-x^2-y^2} \\]
Upper hemisphere.
\\[ z=\cos(x^2+y^2) \\]
Radially oscillating surface with circular level curves.
\\[ z=e^{-(x^2+y^2)} \\]
Smooth hill with maximum at the origin.
\\[ z=x^2+y^2+5 \\]
Bowl shifted upward 5.
\\[ z=(x-2)^2+(y+3)^2 \\]
Bowl with vertex at \\((2,-3,0)\\).
\\[ x^2+y^2+z^2=0 \\]
Only the origin.
\\[ x^2+y^2=0 \\]
The \\(z\\)-axis.
## 21. Quadric Recognition Rules
When looking at an equation:
1. Count how many variables are squared.
2. Check whether the squared terms have the same or opposite signs.
3. Identify any variable appearing to the first power.
4. Identify missing variables.
5. Look at whether the equation equals 0, 1, or another constant.
6. Check the coefficients for stretching and orientation.
Quick rules:
- Three positive squared terms = sphere or ellipsoid, if equal to a positive constant.
- Two positive and one negative, equal to 1 = one-sheet hyperboloid.
- One positive and two negative, equal to 1 = two-sheet hyperboloid.
- Mixed squared signs, equal to 0 = cone, in standard form.
- Two squared terms of the same sign and one linear variable = elliptic paraboloid.
- Two squared terms of opposite signs and one linear variable = hyperbolic paraboloid.
- Missing variable = cylinder or another surface extending along that axis.
# PART III — CHAPTER 10 FORMULAS AND RECOGNITION
## 22. Three-Dimensional Coordinates
A point in space:
\\[ P=(x,y,z) \\]
Distance between two points:
\\[ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2} \\]
Sphere:
\\[ (x-a)^2+(y-b)^2+(z-c)^2=R^2 \\]
## 23. Vectors
Vector:
\\[ \mathbf v=\langle a,b,c\rangle \\]
Magnitude:
\\[ \\|\mathbf v\\|=\sqrt{a^2+b^2+c^2} \\]
Unit vector:
\\[ \mathbf u=\frac{\mathbf v}{\\|\mathbf v\\|} \\]
Vector from \\(P\\) to \\(Q\\):
\\[ \overrightarrow{PQ}=Q-P \\]
## 24. Dot Product
\\[ \mathbf a\cdot\mathbf b=a_1b_1+a_2b_2+a_3b_3 \\]
Also:
\\[ \mathbf a\cdot\mathbf b=\\|\mathbf a\\|\\|\mathbf b\\|\cos\theta \\]
Angle:
\\[ \cos\theta=\frac{\mathbf a\cdot\mathbf b}{\\|\mathbf a\\|\\|\mathbf b\\|} \\]
Perpendicular vectors:
\\[ \mathbf a\cdot\mathbf b=0 \\]
Projection:
\\[ \operatorname{proj}\_{\mathbf b}\mathbf a =\frac{\mathbf a\cdot\mathbf b}{\\|\mathbf b\\|^2}\mathbf b \\]
## 25. Cross Product
\\[ \mathbf a\times\mathbf b= \left\langle a_2b_3-a_3b_2,\\; a_3b_1-a_1b_3,\\; a_1b_2-a_2b_1 \right\rangle \\]
Magnitude:
\\[ \\|\mathbf a\times\mathbf b\\| =\\|\mathbf a\\|\\|\mathbf b\\|\sin\theta \\]
Uses:
- Find a vector perpendicular to two vectors.
- Find a normal vector to a plane.
- Find the area of a parallelogram.
- Find the area of a triangle.
Parallelogram area:
\\[ A=\\|\mathbf a\times\mathbf b\\| \\]
Triangle area:
\\[ A=\frac12\\|\mathbf a\times\mathbf b\\| \\]
Parallel vectors have cross product zero.
## 26. Lines in Space
Vector equation:
\\[ \mathbf r(t)=\mathbf r_0+t\mathbf v \\]
Parametric equations:
\\[ x=x_0+at,\quad y=y_0+bt,\quad z=z_0+ct \\]
Symmetric equations:
\\[ \frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c} \\]
When a direction component is zero, that coordinate stays constant.
## 27. Planes
Point-normal equation:
\\[ a(x-x_0)+b(y-y_0)+c(z-z_0)=0 \\]
General equation:
\\[ ax+by+cz=d \\]
Normal vector:
\\[ \mathbf n=\langle a,b,c\rangle \\]
To find a plane through three points, use two vectors in the plane and take their cross product.
Parallel planes have parallel normal vectors.
Perpendicular planes have perpendicular normal vectors.
## 28. Distances and Intersections
Point-to-plane distance:
\\[ d=\frac{|ax_0+by_0+cz_0-d|}{\sqrt{a^2+b^2+c^2}} \\]
To intersect a line and a plane:
1. Substitute the line's parametric equations into the plane equation.
2. Solve for \\(t\\).
3. Substitute \\(t\\) back into the line.
Recognize parallel, intersecting, identical, and skew lines.
## 29. Vector Functions
\\[ \mathbf r(t)=\langle x(t),y(t),z(t)\rangle \\]
Derivative:
\\[ \mathbf r'(t)=\langle x'(t),y'(t),z'(t)\rangle \\]
Velocity:
\\[ \mathbf v(t)=\mathbf r'(t) \\]
Acceleration:
\\[ \mathbf a(t)=\mathbf r''(t) \\]
Speed:
\\[ \text{speed}=\\|\mathbf r'(t)\\| \\]
Important: velocity is a vector; speed is a scalar.
## 30. Parametric Circles and Helices
Circle:
\\[ \mathbf r(t)=\langle a\cos t,a\sin t\rangle \\]
Helix:
\\[ \mathbf r(t)=\langle a\cos t,a\sin t,bt\rangle \\]
Characteristics of a helix:
- \\(x,y\\) trace a circle.
- \\(z\\) changes with \\(t\\).
- The curve spirals upward or downward.
For \\(\mathbf r(t)=\langle\cos t,\sin t,t\rangle\\):
\\[ \mathbf r'(t)=\langle-\sin t,\cos t,1\rangle \\]
## 31. Unit Tangent Vector
\\[ \mathbf T(t)=\frac{\mathbf r'(t)}{\\|\mathbf r'(t)\\|} \\]
This gives the direction of motion with magnitude 1.
## 32. Arc Length
Two dimensions:
\\[ L=\int_a^b\sqrt{(x'(t))^2+(y'(t))^2}\\,dt \\]
Three dimensions:
\\[ L=\int_a^b\sqrt{(x'(t))^2+(y'(t))^2+(z'(t))^2}\\,dt \\]
Equivalent vector form:
\\[ L=\int_a^b\\|\mathbf r'(t)\\|\\,dt \\]
Recognition: when a question asks for the length of a parametric curve, differentiate its components first.
## 33. Additional Vector Calculus Rules
\\[ \frac{d}{dt}[\mathbf u(t)\cdot\mathbf v(t)] =\mathbf u'(t)\cdot\mathbf v(t)+\mathbf u(t)\cdot\mathbf v'(t) \\]
\\[ \frac{d}{dt}[\mathbf u(t)\times\mathbf v(t)] =\mathbf u'(t)\times\mathbf v(t)+\mathbf u(t)\times\mathbf v'(t) \\]
Curvature, if included in the assigned coverage:
\\[ \kappa(t)=\frac{\\|\mathbf r'(t)\times\mathbf r''(t)\\|}{\\|\mathbf r'(t)\\|^3} \\]
For a plane curve:
\\[ \kappa(t)=\frac{|x'(t)y''(t)-y'(t)x''(t)|}{((x')^2+(y')^2)^{3/2}} \\]
# PART IV — CHAPTER 11 FORMULAS AND RECOGNITION
## 34. Functions of Several Variables
\\[ z=f(x,y) \\]
Two inputs determine one output.
Domain: all allowed input pairs \\((x,y)\\).
## 35. Domain Restrictions
Square root:
\\[ \sqrt{A(x,y)}\implies A(x,y)\geq0 \\]
Square root in denominator:
\\[ \frac1{\sqrt{A(x,y)}}\implies A(x,y)>0 \\]
Logarithm:
\\[ \ln(A(x,y))\implies A(x,y)>0 \\]
Ordinary denominator:
\\[ \frac1{A(x,y)}\implies A(x,y)\ne0 \\]
Example:
\\[ f(x,y)=\sqrt{9-x^2-y^2} \\]
Domain:
\\[ x^2+y^2\leq9 \\]
Closed disk.
## 36. Level Curves
Definition:
\\[ f(x,y)=c \\]
A level curve contains all points where the function has the same height.
Example:
\\[ f(x,y)=x^2+y^2 \\]
For \\(c=9\\):
\\[ x^2+y^2=9 \\]
Circle of radius 3.
For:
\\[ f(x,y)=x^2+4y^2 \\]
At \\(c=16\\):
\\[ \frac{x^2}{16}+\frac{y^2}{4}=1 \\]
Ellipse with semiaxes 4 and 2.
Important distinction:
- Level curves fix \\(z=c\\).
- Traces can fix \\(x=c\\), \\(y=c\\), or \\(z=c\\).
- Contour maps show several level curves.
## 37. Limits in Several Variables
\\[ \lim\_{(x,y)\to(a,b)}f(x,y) \\]
A limit exists only if the function approaches the same value along every path.
Common approaches to test:
\\[ y=0 \\]
\\[ x=0 \\]
\\[ y=mx \\]
\\[ y=x^2 \\]
If two paths give different limiting values, the limit does not exist.
Important: getting the same value along two paths does not prove the limit exists.
Polar substitution near the origin:
\\[ x=r\cos\theta,\quad y=r\sin\theta \\]
\\[ x^2+y^2=r^2 \\]
If you can bound the absolute value by a function of \\(r\\) tending to zero uniformly in angle, that can establish the limit.
## 38. Continuity
A function is continuous at \\((a,b)\\) if:
1. \\(f(a,b)\\) is defined.
2. The limit as \\((x,y)\to(a,b)\\) exists.
3. The limit equals \\(f(a,b)\\).
## 39. Partial Derivatives
\\[ f_x=\frac{\partial f}{\partial x} \\]
Differentiate with respect to \\(x\\), treating \\(y\\) as constant.
\\[ f_y=\frac{\partial f}{\partial y} \\]
Differentiate with respect to \\(y\\), treating \\(x\\) as constant.
Example:
\\[ f(x,y)=x^2y+3y^2 \\]
\\[ f_x=2xy \\]
\\[ f_y=x^2+6y \\]
## 40. Higher-Order Partial Derivatives
\\[ f\_{xx},\quad f\_{yy},\quad f\_{xy},\quad f\_{yx} \\]
For \\(f\_{xy}\\), first differentiate with respect to \\(x\\), then \\(y\\).
Under suitable continuity conditions:
\\[ f\_{xy}=f\_{yx} \\]
This is Clairaut's theorem.
## 41. Tangent Planes
For \\(z=f(x,y)\\), at \\((a,b)\\):
\\[ z=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b) \\]
Interpretation: the plane that best approximates the surface near the point.
## 42. Linear Approximation
\\[ L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b) \\]
Use:
\\[ f(x,y)\approx L(x,y) \\]
near \\((a,b)\\).
## 43. Total Differential
\\[ dz=f_x\\,dx+f_y\\,dy \\]
Approximate change:
\\[ \Delta z\approx f_x\Delta x+f_y\Delta y \\]
## 44. Multivariable Chain Rule: One Parameter
If:
\\[ z=f(x,y),\quad x=x(t),\quad y=y(t) \\]
then:
\\[ \frac{dz}{dt}=f_x\frac{dx}{dt}+f_y\frac{dy}{dt} \\]
For three intermediate variables:
\\[ w=f(x,y,z) \\]
\\[ \frac{dw}{dt}=f_xx'+f_yy'+f_zz' \\]
## 45. Multivariable Chain Rule: Two Parameters
If \\(x=x(s,t)\\) and \\(y=y(s,t)\\):
\\[ \frac{\partial z}{\partial s} =f_xx_s+f_yy_s \\]
\\[ \frac{\partial z}{\partial t} =f_xx_t+f_yy_t \\]
Tree diagrams help identify every path.
Rule: multiply along each path and add all paths leading to the requested variable.
## 46. Implicit Differentiation
If:
\\[ F(x,y)=0 \\]
then, when \\(F_y\ne0\\):
\\[ \frac{dy}{dx}=-\frac{F_x}{F_y} \\]
If:
\\[ F(x,y,z)=0 \\]
and \\(F_z\ne0\\):
\\[ z_x=-\frac{F_x}{F_z} \\]
\\[ z_y=-\frac{F_y}{F_z} \\]
Example:
\\[ x^2+y^2+z^2=25 \\]
\\[ z_x=-\frac{x}{z},\qquad z_y=-\frac{y}{z} \\]
## 47. Gradient
\\[ \nabla f=\langle f_x,f_y\rangle \\]
For three variables:
\\[ \nabla f=\langle f_x,f_y,f_z\rangle \\]
Interpretation:
- Points in the direction of fastest increase.
- Its magnitude gives the maximum rate of increase.
- Is perpendicular to a level curve or level surface at a regular point.
## 48. Directional Derivative
\\[ D\_{\mathbf u}f=\nabla f\cdot\mathbf u \\]
The direction vector \\(\mathbf u\\) must have length 1.
Given a non-unit vector \\(\mathbf v\\):
\\[ \mathbf u=\frac{\mathbf v}{\\|\mathbf v\\|} \\]
If a direction angle \\(\theta\\) is given:
\\[ \mathbf u=\langle\cos\theta,\sin\theta\rangle \\]
Therefore:
\\[ D\_{\mathbf u}f=f_x\cos\theta+f_y\sin\theta \\]
Interpretation:
- Positive = increasing in that direction.
- Negative = decreasing.
- Zero = no first-order change in that direction.
## 49. Maximum and Minimum Directional Derivatives
Maximum rate:
\\[ \\|\nabla f\\| \\]
Direction of fastest increase:
\\[ \frac{\nabla f}{\\|\nabla f\\|} \\]
Minimum rate:
\\[ -\\|\nabla f\\| \\]
Direction of fastest decrease:
\\[ -\frac{\nabla f}{\\|\nabla f\\|} \\]
These direction formulas require a nonzero gradient.
## 50. Tangent Line to a Level Curve
For:
\\[ f(x,y)=c \\]
the gradient is normal to the level curve.
At \\((a,b)\\):
\\[ f_x(a,b)(x-a)+f_y(a,b)(y-b)=0 \\]
This is the tangent line when the gradient is nonzero.
## 51. Tangent Plane to a Level Surface
For:
\\[ F(x,y,z)=c \\]
at \\((a,b,c_0)\\):
\\[ F_x(a,b,c_0)(x-a)+F_y(a,b,c_0)(y-b)+F_z(a,b,c_0)(z-c_0)=0 \\]
The gradient is the normal vector.
# PART V — EXAM QUESTION RECOGNITION
## 52. Recognize the Instruction
| Exam wording | Method |
| ---------------------------------- | ---------------------------------------------------------- |
| Find the domain | Identify denominator, radical, and log restrictions |
| Sketch level curves | Set \\(f(x,y)=c\\) |
| Identify the surface | Count squares, inspect signs, find linear/missing variable |
| Determine whether a limit exists | Test paths; prove existence if paths agree |
| Find first partial derivatives | Hold other variables constant |
| Find mixed partials | Differentiate in the indicated order |
| Find the tangent plane | Evaluate \\(f,f_x,f_y\\) at the point |
| Use linearization | Build \\(L(x,y)\\) and substitute nearby values |
| Find \\(dz/dt\\) | Chain rule |
| Find \\(z_s,z_t\\) | Two-parameter chain rule |
| Find \\(z_x,z_y\\) implicitly | Differentiate implicit equation or use \\(-F_x/F_z\\) |
| Find the gradient | Collect first partial derivatives |
| Find directional derivative | Gradient dot unit vector |
| Find maximum rate of increase | Magnitude of gradient |
| Find direction of steepest ascent | Unit gradient |
| Find tangent line to a level curve | Use gradient as normal |
| Find tangent vector | Differentiate \\(\mathbf r(t)\\) |
| Find speed | Magnitude of \\(\mathbf r'(t)\\) |
| Find unit tangent | Normalize \\(\mathbf r'(t)\\) |
| Find arc length | Integrate speed |
| Find angle between vectors | Dot product |
| Find area of triangle in space | Half cross-product magnitude |
| Find a plane through three points | Cross product for normal |
| Find line-plane intersection | Substitute parametric line into plane |
| Find distance to plane | Point-to-plane distance formula |
## 53. Exam-Style Trig Examples
Question 1: Evaluate a vector function.
\\[ \mathbf r(t)=\langle\cos t,\sin t,t\rangle \\]
Find \\(\mathbf r(\pi/6)\\).
Answer:
\\[ \left\langle\frac{\sqrt3}{2},\frac12,\frac{\pi}{6}\right\rangle \\]
Question 2: Find a tangent vector.
\\[ \mathbf r(t)=\langle3\cos t,3\sin t,t\rangle \\]
Answer:
\\[ \mathbf r'(t)=\langle-3\sin t,3\cos t,1\rangle \\]
Question 3: Find arc length.
\\[ \mathbf r(t)=\langle2\cos t,2\sin t,3t\rangle \\]
Speed:
\\[ \\|\mathbf r'(t)\\|=\sqrt{4\sin^2t+4\cos^2t+9}=\sqrt{13} \\]
Over \\(a\le t\le b\\):
\\[ L=\sqrt{13}(b-a) \\]
Question 4: Find a partial derivative.
\\[ f(x,y)=\sin(xy^2) \\]
Answer:
\\[ f_x=y^2\cos(xy^2) \\]
Question 5: Find a directional derivative.
\\[ f(x,y)=x^2+\sin(xy) \\]
At \\((1,0)\\), in direction angle \\(\pi/6\\):
\\[ \nabla f(1,0)=\langle2,1\rangle \\]
\\[ D\_{\mathbf u}f=2\frac{\sqrt3}{2}+\frac12 =\sqrt3+\frac12 \\]
Question 6: Recognize a parametric curve.
\\[ \mathbf r(t)=\langle\cos t,\sin t,t\rangle \\]
Answer: a helix of radius 1 around the \\(z\\)-axis.
# PART VI — FLASHCARD QUESTION BANK
Use each question as the front of a flashcard and the answer as the back.
## Shape Recognition
| Front | Back |
| -------------------------- | -------------------------------------------------- |
| \\(x^2+y^2+z^2=25\\) | Sphere, radius 5 |
| \\(x^2/4+y^2/9+z^2/16=1\\) | Ellipsoid, longest along \\(z\\) |
| \\(z=x^2+y^2\\) | Upward elliptic paraboloid |
| \\(z=-x^2-y^2\\) | Downward elliptic paraboloid |
| \\(z=x^2-y^2\\) | Hyperbolic paraboloid |
| \\(x^2+y^2-z^2=1\\) | Hyperboloid of one sheet |
| \\(z^2-x^2-y^2=1\\) | Hyperboloid of two sheets |
| \\(x^2+y^2-z^2=0\\) | Double cone |
| \\(x^2+y^2=16\\) | Circular cylinder, radius 4, along \\(z\\) |
| \\(z=x^2\\) | Parabolic cylinder, along \\(y\\) |
| \\(x^2-y^2=1\\) | Hyperbolic cylinder, along \\(z\\) |
| \\(x=y^2+z^2\\) | Elliptic paraboloid opening along positive \\(x\\) |
| \\(z=\sqrt{x^2+y^2}\\) | Upper cone |
| \\(z=\sqrt{9-x^2-y^2}\\) | Upper hemisphere |
| \\(z=e^{-(x^2+y^2)}\\) | Smooth radial hill |
| \\(z=\cos(x^2+y^2)\\) | Radially oscillating surface |
## Trigonometry
| Front | Back |
| ----------------------------- | ------------------------ |
| \\(\sin(\pi/6)\\) | \\(1/2\\) |
| \\(\cos(\pi/6)\\) | \\(\sqrt3/2\\) |
| \\(\sin(\pi/4)\\) | \\(\sqrt2/2\\) |
| \\(\cos(\pi/3)\\) | \\(1/2\\) |
| \\(\sin(5\pi/6)\\) | \\(1/2\\) |
| \\(\cos(5\pi/6)\\) | \\(-\sqrt3/2\\) |
| \\(\sin(3\pi/2)\\) | \\(-1\\) |
| \\(\cos(7\pi/4)\\) | \\(\sqrt2/2\\) |
| \\(\sin^2t+\cos^2t\\) | 1 |
| \\(\tan t\\) | \\(\sin t/\cos t\\) |
| \\(\frac d{dt}\cos(8t)\\) | \\(-8\sin(8t)\\) |
| \\(\frac d{dt}\sin(t^2)\\) | \\(2t\cos(t^2)\\) |
| \\(\frac d{dt}[t^2\sin t]\\) | \\(2t\sin t+t^2\cos t\\) |
| \\(\frac d{dx}\sqrt{x}\\) | \\(1/(2\sqrt{x})\\) |
| \\(\frac d{dx}(1/\sqrt{x})\\) | \\(-1/(2x^{3/2})\\) |
| \\(\frac d{dx}\ln x\\) | \\(1/x\\) |
| \\(\frac d{dx}e^{3x}\\) | \\(3e^{3x}\\) |
## Calculus III Formulas
| Front | Back |
| ----------------------------------------------------------- | ------------------------------------------------ |
| How do you find a unit vector? | Divide vector by its magnitude |
| When are vectors perpendicular? | Dot product equals zero |
| When are vectors parallel? | Cross product equals zero |
| How do you find triangle area? | Half the magnitude of the cross product |
| How do you find speed? | \\(\\\|\mathbf r'(t)\\\|\\) |
| How do you find unit tangent? | \\(\mathbf r'/\\\|\mathbf r'\\\|\\) |
| How do you find arc length? | \\(\int\\\|\mathbf r'(t)\\\|dt\\) |
| What is a level curve? | \\(f(x,y)=c\\) |
| What is a partial derivative? | Differentiate one variable, hold others constant |
| What is a tangent plane? | Local linear approximation to a surface |
| What is the total differential? | \\(dz=f_xdx+f_ydy\\) |
| What is the multivariable chain rule? | Multiply derivatives along each path and add |
| What is implicit \\(dy/dx\\)? | \\(-F_x/F_y\\) |
| What is implicit \\(z_x\\)? | \\(-F_x/F_z\\) |
| What is the gradient? | Vector of first partial derivatives |
| What is a directional derivative? | \\(\nabla f\cdot\mathbf u\\) |
| What is maximum rate of increase? | \\(\\\|\nabla f\\\|\\) |
| Which direction increases fastest? | Direction of gradient |
| What is a tangent line to a level curve perpendicular to? | Gradient |
| How do you prove a limit does not exist? | Find two paths with different limits |
| What does a missing variable in a surface equation suggest? | A cylinder extending along that axis |
# PART VII — WHAT YOU MUST PRACTICE ON PAPER
Flashcards alone are insufficient for the following skills. You need to solve full problems independently.
1. Cross products and finding planes through three points.
2. Lines and planes: intersections, distances, and geometric relationships.
3. Identifying surfaces from unfamiliar equations and sketches.
4. Finding domains of vector functions with multiple restrictions.
5. Vector derivatives, tangent lines, speed, and arc length.
6. Multivariable limits, including showing that limits do not exist.
7. Partial derivatives involving products, quotients, exponentials, logarithms, trig, and radicals.
8. Higher-order and mixed partial derivatives.
9. Tangent planes, linearization, and total differentials.
10. Multivariable chain-rule problems with one or two independent parameters.
11. Implicit differentiation, including equations with logarithms and trigonometry.
12. Gradient and directional derivative problems using angles and non-unit vectors.
13. Finding all directions with a specified directional derivative.
14. Finding locations where a function increases fastest in a specified direction.
15. Tangent lines to level curves and tangent planes to level surfaces.
16. Related-rates applications involving changing volume, temperature, or other quantities.
17. Proof-style questions and justifications.
18. Timed mixed multiple-choice and free-response practice without notes or calculators.
## Final Memory Strategy
Use three stages:
Recognition: Look at an equation, graph, or exam instruction and identify the topic within a few seconds.
Retrieval: Hide the answer and reproduce the relevant formula, rule, or geometric explanation from memory.
Independent application: Solve a new problem without hints, then repeat a related problem later in a different context.
For the best long-term retention, review missed cards later the same day, the following day, several days later, and again after a week. Prioritize the cards you repeatedly miss.
Your aim is not to memorize hundreds of isolated facts. It is to recognize the underlying structure of a problem so that unfamiliar exam questions begin to feel familiar.
You can copy this entire guide into Word, select all, choose Arial, size 12, and upload it to StudyFetch. For the visual surface cards, use the surface names in Part II to pair each equation with a corresponding 3D graph; the text above contains the recognition characteristics, but the diagrams themselves are not embedded in this copyable version.