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Fluency in Limits, Derivatives and Integrals
Maintain fluency and extend skills in evaluating limits, derivatives and integrals to solve corresponding applied problems.
- I can evaluate limits of functions
- Graphically
- Numerically and
- Analytically
- I can explain what a derivative represents graphically, and can give examples of functions that are
not differentiable
- I can differentiate functions using
- basic differentiation rules
- product and quotient rules, and the
- chain rule
- I can describe the relationship between position, velocity and acceleration both analytically and
graphically
- I can differentiate implicitly defined functions and evaluate these derivatives at particular coordinates
- I can model related rates problems using differentiable equations
- I can solve related rates problems and describe the meaning of the solution in the context of the
problem
- I can use integration techniques to
- Write general solutions to differential equations
- Evaluate definite integrals
- Represent area and volume
Advanced Integration
Apply advanced integration techniques to evaluate integrals to include integration by parts, partial fractions, and improper integrals.
- I can use integration techniques to
- Write general solutions to differential equations
- Evaluate definite integrals
- Represent area and volume
- I can evaluate an improper integral that has an infinite limit of integration.
- I can evaluate an improper integral that has an infinite discontinuity.
- I can explain the difference between a proper and improper integral graphically.
Parametric
Analyze curves given in parametric form, and use derivatives and integrals of parametric equations to solve applied problems including motion, vectors, and curve length.
- I can convert parametric equations to a rectangular equation and vice versa and describe the difference between them graphically
- I can differentiate parametric equations and use them to model real world applications.
Polar
Analyze curves given in polar form, evaluate derivatives of polar equations and use integrals to find arc lengths and areas of polar curves.
- I understand the relationship between polar coordinates and rectangular coordinates.
- I can convert polar coordinates and rectangular coordinates and vice versa.
- I can use a graph to explain how polar coordinates and rectangular coordinates are related
- I can differentiate polar equations
- I can model areas of polar graphs and calculate their value
Series Convergence
Understand the difference between convergent and divergent sequences and series and apply appropriate tests to determine convergence.
- I can list the terms of a sequence and determine whether the sequence converges or diverges.
- I can write a formula for the nth term of a sequence.
- I can explain the properties of monotonic and bounded sequences and give examples of each.
- I understand the difference between a convergent and divergent series.
- I can use properties of infinite geometric and telescoping series to test for convergence.
- I know the nth term, integral, p-series, comparison, limit comparison, ratio and root tests and can apply them to determine convergence of a series.
- I understand and can explain when to use each of the above-named tests for convergence.
- I can find the radius and interval of convergence of a power series
- I can differentiate and integrate a power series
- I can explain the relationship between the intervals of convergence for a function, its derivative and its anti-derivative
Taylor Polynomials
Represent and approximate functions using Taylor polynomials and series and determine the accuracy of such approximations by analyzing the remainder.
- I can find Taylor and Maclaurin polynomial approximations of elementary functions
- I can explain what the Lagrange form of the Remainder represents in Taylor's Theorem and can apply this concept to determine accuracy of my approximations
- I can explain how to obtain better approximations using these techniques
- I can find a Taylor or Maclaurin series for a function
- I can perform operations on known Taylor series to derive more complicated Taylor series approximations
Advanced Differential Equations
Apply a variety of techniques to solve differential equations, including separation of variables, Euler’s method and logistic growth models.
- I can use Euler's method to numerically approximate solutions to differential equations
- Given a function, I can estimate function values using Euler's method, series methods, and others and describe how to decrease the error in the approximations.
-I understand the logistic growth model and how it applies to real-world biological populations
-I can apply the logistic growth model to help find solutions to population related problems
Advanced Calculus Modeling
Apply advanced calculus skills to model real-life situations with a function, a differential equation, or an integral and to solve corresponding applied problems.
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