## Tuesday, August 16, 2016

### #MTBoSBlaugust Day 11: AP Calculus AB Learning Targets

Here's my first draft of my AP Calculus AB Learning Targets for this year. My goal is to add these to my Student Unit Organizers so students have a road map as we go through the unit. In addition, these will help me design my assessments and my study guides- the next 2 tasks on my list. (How do I already feel behind when there are still 3 weeks left until I see kids?!?!?!)

These are a compilation of my own 2015-2016 course calendar, information from my APSI last month, the AP Curriculum Framework, and reading through other teacher's course calendars for student-friendly wording (This one in particular was helpful). They align to Calculus of a Single Variable, 8th Ed. by Larson, Hostetler, and Edwards.

I've made a few big changes from other teacher's that I've read, the book, and some things I didn't like last year. The biggest ones are:

• Weaving transcendentals and trig throughout the units, rather than separating them like the textbooks and many other teachers do. I am still trying to wrap my brain around late transcendentals- I feel like the more practice kids can get with these intricate functions the better. If someone wants to convince me otherwise, please please please do. I am interested in hearing the other side.
• I switched the order I taught integration. Last year I started with the "area problem" as a motivation and then switched back to general antiderivatives, but starting with antiderivatives seems more of a natural flow and allows us to revisit u-substitution multiple times, rather than just once. It's my major change for this year, but I'm excited about it. This also would logically put differential equations in the middle of the integration unit, which I think will be a nice break for the kids.
• https://www.lookhuman.com/design/73407-girls-just-want-to-have-differential-equations/6733-heathered_blue_nl-md
• (Side note: I just ordered the shirt at right...while I was writing these learning targets.... I need to start buying back to school clothes, but this was so much more fun.)
• I moved volumes of known cross sections to after disks and washers, but I'm not sure if I'm going to keep it that way. Disk and washer were simpler cases for my kiddos last year since they are always the formula for a circular cross section; I saw so much more struggle in using the formulas for other shapes.  I'm considering testing it out this year to see how it changes the understanding of the kids. Nothing set in stone here yet.
• L'Hopital's isn't an "after exam" topic anymore- it's right in there, during my derivatives unit! Excited to get to spiral in some limits there to review. Also excited that it's only the simplest case, not all the other wacky indeterminant forms we'd normally get into after the exam.
Let me know if you see anything I missed or have any sequencing tips that made a huge difference in your students' understanding!
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AP Calculus AB Learning Targets

2016-2017

Chapter 0: Pre-Calculus Skills
• I can identify minimums, maximums, intercepts, intervals of positive/negative, and key values from a graphing calculator
• I can sketch all parent functions by hand and identify their domain and range
• I can write the equation of a line given a point and a slope
• I can rewrite an absolute value function into a piecewise function
• I can rewrite expressions using factoring
• I can rewrite expressions using rational operations
• I can rewrite expressions using long division
• I can rewrite expressions using completing the square
• I can solve equations and inequalities graphically and algebraically
• I can find the inverse of a function algebraically
• I can compose and decompose functions
• I can use properties of logarithms to rewrite expressions and solve equations
• I can identify exact trig values of important angles in the unit circle and use them to sole trigonometric equations

Chapter 1: Limits & Their Properties
• I can evaluate the limit using a table
• I can use a graph to evaluate the limit
• I can evaluate a limit using properties
• I can evaluate a limit by using direct substitution
• I can evaluate a limit algebraically
• I can write a simpler function to evaluate a limit
• I can evaluate a limit using two special trigonometric limits
• I can evaluate a limit using the squeeze theorem**
• I can evaluate a one-sided limit
• I can determine if a function is continuous (satisfy 3 conditions)
• I can discuss the continuity of a function on a closed interval
• I can identify the type of discontinuity by name
• I can use Intermediate Value Theorem (IVT) to analyze function behavior in an interval
• I can write the equation for a vertical asymptote
• I can evaluate limits with function values approaching ±∞
• I can evaluate limits as x approaches ±∞
• I can find the horizontal asymptotes of a function

Chapter 2: Differentiation
• I can explain how the slope of secant lines can approximate the slope of a tangent line
• I can use the average rate of change (slope formula) to approximate the derivative of a function
• I can identify derivative as an instantaneous rate of change
• I can find the equation of a tangent line at a specific point
• I can find the equation of a normal line at a specific point
• I can find the general derivative using the limit process
• I can find the derivative at a point using the limit provess
• I can explain the relationship between the limit definition formulas (at a point and general) and the slope formula from previous courses
• I can use a graphing utility to find the slope at a specific point, sketch a possible graph of the derivative of a function
• I can apply to alternative form of the derivative
• I can find where a function is differentiable
• I can differentiate using the power rule
• I can find where horizontal tangents occur
• I can find the derivative of sine and cosine
• I can understand how the derivative applies to Position/Velocity/Acceleration
• I can differentiate using product rule
• I can differentiate using quotient rule
• I can find the derivative of tangent, cotangent, secant, and cosecant
• I can differentiate using chain rule
• I can differentiate using more than one rule
• I can understand function notation to find derivatives, including differentiating from a table
• I can find derivatives implicitly
• I can find the second derivative of an equation implicitly
• I can find horizontal and vertical tangents of an implicitly defined function
• I can identify when L’Hopital’s rule applies to an indeterminant form (p.567)
• I can evaluate a limit of the form 00 or using L’Hopital’s rule (p.567)
• I can evaluate (f-1)'(a) (p.341)
• I can find derivatives of functions involving the natural logarithmic function
• I can find dy/dx using logarithmic differentiation(p.322) **
• I can find the derivative of the exponential function (p.350)
• I can find the derivative of a function involving a base other than e  (p.360)
• I can find the derivative of an inverse trig functions (arcsin and arctan must be memorized)

Chapter 3: Applications of Differentiation
• I can apply the extreme value theorem
• I can find critical values of a function
• I can find relative extrema of a function
• I can find absolute extrema of a function using the closed interval test
• I can use the First Derivative Test to find relative extrema of a function
• I can relate the First Derivative Test to the second derivative
• I can find points of inflection of a function
• I can find intervals of concavity of a function
• I can sketch the graph of f ′ and f ″ given the graph of f(x)
• I can sketch the graph of f(x) given the graph of f ′ and f ″
• I can use a tangent line to approximate function values
• I can use differentials and the graph of f to approximate values**
• I can verify that a the criteria for Mean Value Theorem apply
• I can apply the Mean Value Theorem
• I can write an argument to justify my use of Mean Value Theorem
• I can apply Rolle’s Theorem
• I can apply the Second Derivative Test to find extrema
• I can solve applied minimum and maximum problems
• I can identify important quantities and equations in a related rate problem
• I can solve related rate problems involving distance
• I can solve related rate problems involving area and volume

Chapter 4A: Integration (General Antiderivative)
• I can find the general
• anti-derivative of an algebraic function
• I can recognize why we need a constant of integration
• I can define the indefinite integral and its parts
• I can find the general anti-derivative of a trigonometric function
• I can find the original function from the graph of the derivative
• I can find a particular function given certain conditions
• I can find the anti-derivative of a natural logarithmic function (p. 332)
• I can find the anti-derivative of a function involving a base other than e
• I can find the anti-deriative of an exponential function
• I can integrate functions whose antiderivatives involve inverse trig functions
• I can use the method of completing the square to integrate a function
• I can integrate functions using u-substitution
• I can integrate functions using long division

Chapter 6: Differential Equations
• I can sketch the slope field to represent a differential equation
• I can sketch the solution curve to fit a given slope field
• I can choose a differential equation to fit a given slope field
• I can use separation of variables to solve a simple differential equation
• I can use exponential functions to model growth and decay in applied problems
• I can use exponential functions to model compounded continuously problems
• I can identify a problem as exponential (y=Cekt) when it discusses the rate being proportional to the amount present
• I can find the general solution of a differential equation
• I can find the particular solution with conditions

Chapter 4B: Definite Integration and the Fundamental Theorem of Calculus
• I can estimate the area under a curve using a Riemann Sum
• I can estimate the area under a curve using the Midpoint Rule
• I can approximate the area under a curve using the Trapezoidal Rule
• I can compare Left, Right, Midpoint, and Trapezoidal approximations of the area under a curve
• I can evaluate problems involving summations both with and without calculators
• I can identify the limit definition of the definite integral
• I can represent the area of a region using a definite integral
• Ican recognize that the value of a definite integral can be found using geometry
• I can use properties to help evaluate definite integrals
• I can use the Fundamental Theorem of Calculus to evaluate definite integrals
• I can accurately calculate a definite integral using u substitution and change of bounds
• I can evaluate a definite integral using a graphing calculator
• I can find the average value of a function using the Mean Value Theorem for integrals
• I can use the Second Fundamental Theorem of Calculus to find the derivative of a definite integral
• I can use the differential equation and a given point to find the function
• I can use an accumulation function to answer application questions

Chapter 7: Applications of Integration
• I can finding area between two curves
• I can use the disk method to find the volume of a solid of revolution
• I can use the washer method to find the volume of a solid of revolution
• I can use the shell method to find the volume of a solid of revolution**
• I can find the volume of solids whose cross-sections are known