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Fluency in Advanced Functions
Maintain fluency in working with functions represented in a variety of ways (graphical, numerical, analytical, and/or verbal) and understand the connections among these representations.
Limits
Understand the limit process and evaluate limits analytically, graphically, and numerically.
I can evaluate limits of functions
- Graphically
- Numerically and
- Analytically
Derivatives
Understand the meaning of the derivative in terms of a rate of change and find derivatives by applying differentiation rules to include implicit differentiation and higher-order derivatives.
- 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
Derivative Applications
Use derivatives to solve a variety of applied problems including related rates and optimization.
- 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 model optimization problems using differentiable equations
- I can solve optimization problems and describe the meaning of the solution in the context of the problem
Curve Analysis
Apply calculus techniques to analyze and graph functions and understand the relationship between the sign, direction and concavity of curves and their equations.
- I can locate absolute extrema on a closed interval
- I can explain Rolle's Theorem and the Mean Value Theorem and explain their relevance in real-life situations
-I can locate relative extrema on an open interval by using the first derivative test or the second derivative test
- I can determine intervals where a function is increasing/decreasing and concave up/concave down
- I can analyze and accurately sketch the graph of a function using calculus techniques
Antiderivatives/Integrals
Evaluate indefinite integrals and understand the meaning of the definite integral as a limit of Riemann sums, the net accumulation of change, and as an area.
- I can write the general solution to a differential equation
- I can find a particular solution to a differential equation
- I understand the relationship between general and particular solutions graphically
- I can represent a graphical region using an integral expression
- Given a function (such as velocity), I can explain the graphical representation of the derivative and integral and find those values
- I can approximate areas using Riemann sums and trapezoidal sums and can explain how to obtain better approximations using these techniques
Fundamental Theorem of Calculus
Describe the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus and use integrals to solve a variety of applied problems.
- I can evaluate a definite integral using the Fundamental Theorem (and u-substitution if necessary)
- I can explain the relationship between derivatives and integrals and use the second part of the Fundamental Theorem to differentiate integral functions
- I can apply the Fundamental Theorem to initial condition problems
Transcendentals
Apply differentiation and integration techniques to exponential, logarithmic, trigonometric and inverse functions.
- I can evaluate derivatives and integrals involving natural logs and exponential functions
- I can explain the relationship between inverse functions and their derivatives
- I can apply these derivative and anti-derivative rules to real-world examples including optimization and rate of change problems
Differential Equations
Understand the relationship between slope fields and solutions to differential equations and use separation of variables to solve differential equations.
I understand the relationship between slope fields and solutions to differential equations
- Given a differential equation and initial condition, I can sketch a slope field and draw a solution curve
-Given a differential equation and initial condition, I can find the general and particular solution
-I can explain the relationship between the slope field with solution curves and the differential equations and solutions
Integral Volume
Represent and calculate the volume of solid figures using integrals.
The volume of solid figures can be represented and calculated using integrals
- I can use the disk, washer and cross-section methods to represent and calculate the volume of solid figures
- I can distinguish between, and know when to use, the disk and washer methods
- I can explain the relationship between these methods of representing volume of solids, describe how they look, and how differences in the integral expressions represent the differences between the solids themselves.
Calculus Modeling
Model real-life situations with a function, a differential equation, or an integral and use calculus techniques to solve corresponding applied problems.
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