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NOTICE: I will be offering “Captivating Calculus” in period 5 on Choice Fridays in 2403. This class is mandatory for those who scored below 80% on a unit test, but highly encouraged for anyone who is struggling or unsure about any of the material. Otherwise, please use this as an additional opportunity to get help with calculus, or to come ask deeper questions that we don’t have time for in class. If you don’t have any questions, feel free to come by anyway for your lunch hour to hang out in case any questions arise from others. Please let me know in advance if you intend to come; I will typically come for the first few minutes to see if anyone shows up.
Schedule
Tentative schedule for upcoming classes:
- Week of Oct. 14
- Monday, Oct. 14: No class
- Wednesday, Oct. 16: Unit 6 test
- Friday, Oct. 18: 7.1-7.3
- Week of Oct. 21
- Monday, Oct. 21: 7.2-7.4
- Wednesday, Oct. 23: 7.5-7.7
- Friday, Oct. 25: Mini-math (7.1-7.5)
- Week of Oct. 28
- Monday, Oct. 28: 7.8-7.9
- Wednesday, Oct. 30: Mini-math (7.6-7.9)
- Friday, Nov. 1: Unit 7 Review
- Week of Nov. 4
- Monday, Nov. 4: Unit 7 test
- Wednesday, Nov. 6: Pre-break activities
- Friday, Nov. 8: PTC
- Week of Nov. 11
- Monday, Nov. 11: November break
- Wednesday, Nov. 13: November break
- Friday, Nov. 15: November break
Homework
If you are consistently spending more than 1 hour per day on homework, please see me. (Nearly) Every section we cover will have an associated AP Classroom online homework for you to complete by a certain deadline, typically before the next class (or in the case of a test, the class after that). You will also receive paper assignments from Flipped Math to work on. Any additional homework outside of these two will be posted here.
Homework for Monday, September 9:
- Introduction Questionnaire
Exams
Unit 1-5 test is scheduled for Wednesday, September 11 in-class. You should be proficient in all material contained in Units 1-5. More precisely, you should be able to:
- Compute limits of a function graphically.
- Compute limits of a composite function graphically.
- Interpret tabular information for finding limits.
- Use limit properties and arithmetic to find limits.
- Compute a limit of a function algebraically via 4 techniques:
- Technique 0: rational-like function evaluation
- Technique 1: factor and reduce
- Technique 2: rationalize
- Technique 3: Fundamental Trigonometric Limit
- Technique 4: (for x approaching +/- infinity) divide by dominant terms in the denominator
- Apply this technique for expressions involving radicals
- Compute a one-sided limit, including checking for signs in absolute values and values close to 0.
- Find a limit via one-sided limits.
- Determine where a function is continuous, or find values for constants which give continuity.
- Identify the type of a discontinuity.
- Find vertical and horizontal asymptotes of a function.
- Use Squeeze Theorem.
- Use Intermediate Value Theorem.
- Describe the relationship between continuity and differentiability.
- Calculate the average rate of change from a table, graph, or function.
- Compute a derivative from first principles.
- Identify a limit as a derivative and use derivative rules to find the limit.
- Approximate the value of a derivative from either a table or graph.
- Sketch the derivative of a function given a graph of the function and vice versa, the graph of a function given the derivative.
- Apply the various derivative rules: sum/difference rule, constant multiple rule, power rule, product rule, quotient rule (this is of course the vast majority of the test, either directly or indirectly)
- Note: This is true both for given functions (involving powers, trigonometric functions, exponential functions, and logarithmic functions) as well as given a table of values for f(x), g(x), f’(x), g’(x).
- Use derivatives to find the equation of tangent/normal lines with a given slope and through a point or parallel/perpendicular to given lines.
- Apply chain rule
- Compute dy/dx implicitly, and find the slope of the tangent to an implicit curve at a point.
- Compute higher-order derivatives.
- Be able to compute higher-order implicit derivatives.
- Differentiate inverse functions, including the inverse trigonometric functions.
- Be able to compute derivatives via logarithmic differentiation.
- Determine the velocity and acceleration of a particle moving in a straight-line given its position/displacement function as well as compute the value of the velocity or acceleration at particular times.
- Determine when a particle is at rest as well as when it is moving in the positive or negative direction.
- Find the total distance travelled by a particle within a specified amount of time.
- Find rates of change in contexts other than motion including understanding the correct units
- Solve related rates problems (you should know standard formulas for area/volume and perimeter/surface area, and be able to use simple geometry such as similarity of triangles and Pythagorean Theorem)
- Approximate a function with local linearization and analyze a linearization
- Use l’Hôpital’s rule appropriately
- Apply the Extreme Value Theorem
- Find and classify critical points
- Find local and global extrema
- Determine where a function is increasing or decreasing
- Use the First Derivative Test
- Determine the concavity of a function on an interval
- Use the Second Derivative Test
- Determine information about f, f’, or f’’ given information about another one of these functions
- Solve optimization problems
- Analyze an implicit function using derivatives and second derivatives
- Apply the Mean Value Theorem
- Determine where a function satisfies the Mean Value Theorem
Unit 6 test is scheduled for Wednesday, October 16 in-class. There will be a calculator portion and a non-calculator portion. You should be proficient in all material contained in Unit 6. More precisely, you should be able to:
- Approximate the area under a curve using right and left Riemann sums, midpoint and trapezoidal rules, both with equal and unequal intervals, from a function, graph, or table of values
- Identify if an area estimate is an overestimate or underestimate
- Interpret or produce an integral given a contextual application
- Utilize the Fundamental Theorem of Calculus I to differentiate an accumulation function
- includes accumulation functions where the bounds are functions
- Analyze accumulation functions for: where they are increasing or decreasing, locate and classify critical points, determine concavity and points of inflection
- Use integral properties
- Solve indefinite integrals or utilize the Fundamental Theorem of Calculus II to solve definite integrals, including:
- algebraic techniques to simplify the integrand (splitting, expanding, long division, completing the square)
- Substitution Rule
- Integration by Parts (IBP)
- Partial Fraction Decomposition (PFD)
- Handle improper integrals
Exam weighting: 18.5
NOTE: any material from Units 1 to 5 is also fair game and may be useful/necessary.
Final exam information: 8:00AM, Monday, May 12, 5th floor
Part I tests are MCQ and Part II tests are FRQ
Online resources
- The Essence of Calculus YouTube series (brought to you by 3 Blue 1 Brown)
- AP Calculus BC course (Khan Academy)
- Online calculator (WolframAlpha)
- Brachistochrone curve
- Interactive visualization of the Chain Rule
- Optimization problems
- Concavity
- Trigonometry
Practice problems
- Unit 1
- Unit 3-5
- Derivative practice (not comprehensive, answers included)
- Unit 6
- Mini-maths
Challenges
Calculus-based:
- Unit 1: A weird limit - Find an example of a function whose limit at 0 from the right DNE.
- Unit 1: Continuous nowhere - Does there exist a function which is continuous nowhere?
- Unit 1: Continuous on the irrationals - Does there exist a function which is continuous only on the irrationals?
- Unit 1: Continuous on the rationals - Does there exist a function which is continuous only on the rationals?
- Unit 2: Continuous but not differentiable - Does there exist a function which is continuous everywhere but not differentiable infinitely often?
- Unit 2: Differentiability for piecewise - Prove you can use derivatives from the left and right to determine differentiability for piecewise functions.
- Unit 3: Continuous but nowhere differentiable - Does there exist a function which is continuous everywhere but differentiable nowhere?
Assorted: