As detailed on the syllabus, your grade in this course will be determined by your proficiency on a variety of standards. (This is known as Standards Based Grading.)
Here is the list of standards that form the basis for this class, along with guiding questions that address each standard. You will be assessed on these standards throughout the semester. Certain standards are labeled core; these are the standards that everyone who passes Calculus II must master at their desired level of comprehension.
The standards are categorized and color-coded based on what types of skill you are learning:
- Mathematical Maturity. These standards are important ideas and concepts that help you to improve your learning of mathematical concepts.
- Your Mathematical Toolbox. These standards are basic skills upon which we build a deeper knowledge base.
- Key Concepts and Theory. These standards represent the fundamental ideas that underlie this area of mathematics.
- Evaluating Integrals. These standards test your ability to perform a variety of techniques when faced with an integral.
- Applications of Calculus. These standards ask you to take the knowledge from this class and apply it to a variety of real-world scenarios.
The Standards
Core standards are labeled by (**) and Reasonable standards are labeled by (*).
Standard 1. Differentiation Skills. Are you able to apply your differentiation skills from Calculus I? Are you able to evaluate standard derivatives involving polynomials? Involving trigonometric functions? Involving exponential and logarithmic functions? Can you apply the product rule? The quotient rule? The chain rule?
(*) Standard 2. Definitions. (core)
Do you understand what a definition is? Are you able to write an explicit definition statement? Are you able to explain what the definition means in your own words? Are you able to give examples and non-examples of the definition? This standard only involves the definitions for antiderivative and differential equation.
Standard 3. Sigma Notation. (previously Standard 4) Can you convert from sums to sigma noation? Can you convert from sigma notation to a sum? Can you evaluate sums involving Σ i, Σ i2, Σ i3?
(*) Standard 4. Riemann Sums. (previously Standard 3) Can you use rectangles to approximate area under a curve? Can you set up a Riemann sum, both theoretically and for a given function? What is the midpoint rule?
Standard 5. Calculator Skills. Can you use your calculator to take definite integrals? Can you use your calculator to calculate Riemann sums?
(**) Standard 6. The Concept of an Integral. (core) Do you understand what an antiderivative is? A definite integral? Do you understand when it exists? Do you understand what it means for a function to be integrable? Do you understand the difference between definite and indefinite integrals?
Standard 7. Properties of Integrals. Do you know and can you apply the algebraic properties that definite and indefinite integrals obey? Given a definite integral, can you determine simple upper and lower bounds for it?
(**) Standard 8. Basic Integrals. (core) Can you evaluate standard antiderivatives, definite integrals, and indefinite integrals involving polynomials? Involving trigonometric functions?
(**) Standard 9. Mathematical Experience. (core) Can you approach problems in multiple ways? Are you willing to make mistakes? Can you learn from your mistakes? Are you able to discuss mathematical concepts with your classmates? (Includes both talking and actively listening!) Can you contribute to a class discussion constructively and supportively?
(*) Standard 10. Particle Motion. Do you understand how position, velocity, and acceleration functions are related? Do you understand the difference between displacement and total distance traveled? Can you calculate displacement and distance traveled from a given velocity or acceleration? Do you understand how the relationship between position and velocity can be understood as the area under a curve?
(*) Standard 11. Theorems. Do you understand what a theorem is? Do you understand what a hypothesis is? Do you understand what a conclusion is? Are you able to determine the hypotheses and conclusions of a given theorem? Are you able to write an explicit theorem statement? Are you able to explain what a theorem means in your own words? Are you able to understand the consequences and non-consequences of a theorem?
Standard 12. Key Theorems. Can you state and apply the Fundamental Theorem of Calculus, parts I and II? Can you state and apply the Mean Value Theorem for Integrals? Do you understand the interpretations of these theorems?
Standard 13. Average Value. Do you understand what the average value of a function is? Can you set up an integral to calculate the average value of a function?
Standard 14. FTC Integrals. Can you evaluate expressions involving both derivatives and integrals? Can you apply FTC I when there are derivatives of integrals? And use the chain rule when necessary? Can you use FTC II when there are integrals of derivatives?
(*) Standard 15. Net Change. Do you understand how to apply FTC II to calculate net change of a quantity?
(**) Standard 16. Basic Substitution Rule. (core) Can you correctly apply the substitution rule for integrals involving polynomials and trigonometric functions?
(*) Standard 17. Inverse Functions. Do you understand what the inverse of a function is? Do you understand inverse trigonometric functions? Can you determine the domains and ranges of the inverse of a function? Can you draw the graph of the inverse of a function? Do you understand what the properties of the inverse of a function are?
Standard 18. Inverse Function Calculus. Do you know the general form of a derivative of an inverse function? Can you take the derivative of an inverse trigonometric function? Can you apply the substitution rule for expressions that involve inverse trigonometric functions?
(**) Standard 19. Logarithmic and Exponential Functions. (core) Do you understand what a logarithmic function is? An exponential function? ex, ln x, ax, loga x? Can you convert between expressions involving exponentials and logarithms of different bases? Do you understand the inverse relationship between exponential and logarithmic functions? Do you know the shapes of the curves, the domains and ranges of these functions?
(**) Standard 20. Exponential and Logarithm Calculus. (core) Can you differentiate and integrate expressions involving exponential functions and logarithms including ex, ln x, ax, loga x? Can you apply the substitution rule for expressions that involve exponential functions and logarithms?
(*) Standard 21. Exponential Growth and Decay. Do you understand Exponential Growth and Decay? Can you apply Newton's Law of Cooling? Can you calculate the half-life of a decaying dample? Can you determine the value of a bank account calculated via continuously compounded interest?
(*) Standard 22. Area between curves. In order to calculate the area between curves, can you set up and evaluate an integral with respect to x? Can you set up and evaluate an integral with respect to y? Can you convert between the two? This involves determining the correct bounds of integration.
Standard 23. Project Management.
Can you work together on your project as a group? Can you follow project instructions? Can you work toward a common goal with others? Can you work within a given timeframe and meet deadlines? Can you produce a well-written project summary?
Standard 24. Calculator and Computer Skills.
Can you use Mathematica to plot functions? Can you use Mathematica's Documentation Center? Can you use Mathematica to compute the volume of the goblet in your project?
Standard 25. Disk/Washer Method.
Can you set up and calculate the necessary integrals for calculating volume of the goblet?
(*) Standard 25. Solids of Revolution. Can you set up and calculate the necessary integrals for calculating a volume of revolution using the disk/washer method? Using the method of cylindrical shells?
(**) Standard 26. Integrals as Slicing Mechanism, Theory. (core)
Can you explain / derive the formulas that arise for area, volume, and arc length as applications of the idea that A= ∫ dA, V= ∫ dV, and L= ∫ dL?
For area, this involves area between curves with respect to x or y (See Theorems 7.1.2 (p 372) and the Equation at the top of p 374). For volume, this involves the definition of volume on p 378 and the disk/washer method on page 382 and the cylindrical shell method on page 388. For arc length, this follows the explanation of the formula on page 393.
Standard 27. Integrals as Slicing Mechanism, Application.
Can you set up integrals to calculate Area/Volume/Arc length by slicing? Can you identify the slices involved in A= ∫ dA? (Area) V= ∫ dV? (Volume) L= ∫ dL? (Arc length) Can you calculate volume of solids of revolution using cylindrical shells?
(*) Standard 28. Differential Equations. Are you able to write and apply the definitions of the following concepts? Differential equation. Initial-value problem. General solution. Particular solution. The order of a differential equation. Separable differential equation. Direction field.
(*) Standard 29. Solving Differential Equations. Can you solve separable differential equations? Can you solve an initial-value problem? Can you sketch a solution curve in a given direction field with a given initial condition?
