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Explore functions and their derivatives
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
Explore the rate of change of a function
Define and compute its derivative from a given function by methods which include symbolic manipulation
Compute and interpret a velocity function given a position function
Make inferences and connections between the graph of a function and its derivative
Compute limits both intuitively and symbolically
Compute land interpret limits at infinity
Compute and interpret infinite limits
Determine the derivative as a limit
Explore continuous functions
Determine whether a given function is continuous both by intuitive and symbolic methods
Demonstrate the connection between the concept of continuity and the concept of a limit
Check if the answer to a derivative or limit problem is consistent with the information given
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Compute derivatives
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
Compute symbolically the derivatives of polynomials
Compute symbolically the derivatives of rational functions
Compute symbolically the derivatives of power functions with rational exponents
Compute symbolically the derivatives of trigonometric functions
Compute symbolically the derivatives of products of functions
Compute symbolically the derivatives of quotients of functions
Compute symbolically the derivatives of compositions of functions
Compute a derivative implicitly
Compute higher order derivatives
Compute the tangent (linear and quadratic approximations to a given function)
Demonstrate the connection of the linear approximation and differential notation to the concept of the derivative
Solve related rate problems in a variety of verbally stated applications
(Optional) Approximate the roots of a function using Newton's method
(Optional) Demonstrate the connection between Newton's method and the linear approximation
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Solve inverse functions and their derivatives
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
Demonstrate the inverse of a 1-1 function
Compute the derivative of the inverse in terms of the derivative of the original function
Demonstrate the exponential function including the significance of the natural base e
Compute symbolically the derivatives of exponential functions
Demonstrate the logarithmic functions
Demonstrate the significance of the natural logarithm
Compute symbolically the derivatives of logarithmic functions
Construct and interpret graphs of exponential functions
Construct and interpret graphs of logarithmic functions
Demonstrate the inverse trigonometric functions including computing their derivatives symbolically
Construct and interpret graphs of inverse trigonometric functions
(Optional) Demonstrate the hyperbolic trigonometric functions including computing their derivatives symbolically
Recognize an indeterminate limit and calculate the answer symbolically by a variety of methods including L'Hospitals Rule
Distinguish between indeterminate forms, L'Hospital's Rule and derivatives
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Use derivatives of functions
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
Examine the content and uses of Rolle's Theorem
Examine the content and uses of the Mean Value Theorem
Use the sign of the first derivative to determine where a function is increasing or decreasing
Use the sign of the second derivative to determine where a function is convex or concave down
Find the absolute and relative maximums and minimums of a given function on a given domain using analytic methods
Find points of inflection of a given function on a given domain using analytic methods
Sketch and interpret the graph of a function using analytical methods
Record properties about the function and its derivatives on a graph
Solve problems of maximization or minimization in a variety of verbally stated applications
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Use antiderivatives of functions
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
Explore the process of finding an antiderivative of a given function
Compute indefinite integrals of functions which are derivatives of elementary functions
Find both the position and velocity functions given the acceleration and the initial position and velocity
Explore the definition of the definite integral as the limit of a Riemann sum and use the properties of the definite integral for a continuous functions
Approximate definite integrals using different Riemann sums (left rectangles, right rectangles, mid points), Simpson’s Rule, and the Trapezoid Rule
Explore the content and uses of the Fundamental Theorems of Calculus
Compute definite integrals and analytically use the Fundamental Theorem
Demonstrate the connection between the two forms of the Fundamental Theorem
Recognize the potential for wrong answers using the Fundamental Theorem if the integrand is not continuous
Compute integrals involving substitutions
Recognize the connection between the substitution method and the chain rules
Handle the limits of integration when substitution is used to evaluate a definite integral
Check the result of an indefinite integral by differentiation
Check the reasonableness of the value of a definite integral by a numerical estimation
Check analytical integrations results using numerical/graphical methods
Relate the definition of a natural logarithm as a definite integral to its derivative
Find the error bounds of numerical integration using Simpson’s Rule or the Trapezoid Rule
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Use definite integrals
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
Set up and compute definite integrals which represent the areas between two curves
Set up and compute definite integrals which represent volumes by using a cross-sectional area
Set up and compute definite integrals which represent the volumes of revolution
Set up volumes of revolution using washers (including disks)
(Optional) Set up volumes of revolution using cylindrical shells
Set up and compute definite integrals which represent the average value of a function over an interval
(Optional) Set up and compute definite integrals which represent the location of the center of mass of an object
(Optional) Set up and compute definite integrals which represent mechanical work done by a force
(Optional) Set up and compute definite integrals which represent the force exerted against a surface by a fluid