Unit Knowledge and Skills (Performance Objectives)
Unit I. Derivatives
By the end of this unit, the student will be able to:
· Use the Power Rule and Exponential Rules to find Derivatives.
· Find the Derivative of Composite Functions.
· Find Derivatives involving Trig, Trig Inverse, and Logarithmic Functions.
· Use the equation of a tangent line to approximate function values.
· Use Euler’s Method to approximate a numerical solution to a differential equation at a given point.
· Find the Derivative of a product or quotient of two functions.
· Find higher order derivatives.
· Take derivatives of relations implicitly.
· Use implicit differentiation to find higher order derivatives.
· Determine when it is appropriate to use logarithmic differentiation.
· Use logarithmic differentiation to take the derivatives of complicated functions.
· Solve related rates problems.
Unit II Anti-Derivatives
By the end of this unit, the student will be able to:
· Find the anti-derivative of a polynomial.
· Integrate functions involving Transcendental operations.
· Use Integration to solve rectilinear motion problems.
· Use the integration by substitution to integrate composite expressions.
· Use the Integration by Substitution to integrate integrands involving Sine and Cosine.
· Use the Integration by Substitution to integrate integrands involving Secant and Tangent or Cosecant and Cotangent.
· Given a separable differential equation, find the general solution.
· Given a separable differential equation and an initial condition, find a particular solution.
· Given a differential equation, sketch its slope field.
· Given a slope field, sketch a particular solution curve.
· Given a slope field, determine the family of functions to which the solution curves belong.
· Given a slope field, determine the differential equation that it represents.
Unit III Applications of the Derivative
By the end of this unit, the student will be able to:
· Find critical values and extreme values for functions.
· Use the 1st and 2nd derivative tests to identify maxima vs. minima.
· Find Points of Inflection and Intervals of Concavity.
· Sketch the graph of a function using information from its first and/or second derivatives.
· Sketch the graph of a first and/or second derivative from the graph of a function.
· Solve optimization problems.
· Use the derivative to make conclusions about motion.
· Relate the position, velocity, and acceleration functions.
· Sketch the graphs of parametric equations.
· Eliminate the parameter of parametric equations.
· Interpret information in the graph of a derivative in terms of the graph of the “original” function.
· Use the graph of a function to answer questions concerning maximums, minimums, and intervals of increasing and decreasing
· Use the graph of a function to answer questions concerning points of inflection and intervals of concavity.
· Use the graph of a function to answer questions concerning the area under a curve.
Unit IV Integrals
By the end of this unit, the student will be able to:
· Find approximations of integrals using different rectangles.
· Use proper notation when dealing with integral approximation.
· Differentiate integral expressions with the variable in the boundary
· Evaluate Definite Integrals
· Find the average value of a continuous function over a given interval
· Evaluate definite integrals using the Fundamental Theorem of Calculus.
· Evaluate definite integrals applying the Substitution Rule, when appropriate.
· Use proper notation when evaluating these integrals.
· Relate definite integrals to area under a curve.
· Understand the difference between displacement and total distance.
· Extend that idea to understanding the difference between the two concepts in other contexts.
· Analyze the interplay between rates and accumulation in context.
Unit V Limits
By the end of this unit, the student will be able to:
· Evaluate one-sided limits graphically, numerically, and algebraically.
· Evaluate two-sided limits in terms of one-sided limits.
· Prove continuity or discontinuity of a given function.
· Interpret Vertical Asymptotes in terms of one-sided limits.
· Determine if a function is differentiable or not.
· Demonstrate understanding of the connections and differences between differentiability and continuity.
· Evaluate Limits algebraically.
· Evaluate Limits using L’Hopital’s Rule.
· Recognize and evaluate Limits which are derivatives.
· Use the nDeriv function on the calculator to find numerical derivatives.
· Evaluate Limits at infinity.
· Interpret Limits at infinity in terms of end behavior of the graph.
· Evaluate Type I Improper Integrals.
· Determine convergence or divergence of a Type II Improper Integral.
Unit VI Numerical Sequences and Series
By the end of this unit, the student will be able to:
· Identify Sequences and Series
· Find Partial Sums of a given Series.
· Find the terms, partial sums, infinite sums, or n in a geometric sequence.
· Determine the convergence or divergence of a sequence.
· Determine the divergence of a series.
· Use the Comparison Tests to check for convergence or divergence.
· Use the Integral Test to check for convergence or divergence.
· Use the Ratio and Nth Root Tests to check for convergence or divergence.
· Use the Alternating Series Test to check for convergence or divergence.
Unit VII Applications of the Integral
By the end of this unit, the student will be able to:
· Find the area of the region between two curves.
· Find the volume of a solid rotated when a region is rotated about a given axis
· Find the volume of a solid rotated when a region is rotated about a given line
· Find the volume of a solid with given cross sections.
· Find the arc length of a function in Cartesian mode between to points.
Unit VIII Techniques of Integration
By the end of this unit, the student will be able to:
· Identify integrals where Integration by Parts is appropriate.
· Apply the Integration by Parts method.
· Integrate radical integrands using trig substitution.
· Determine the appropriate technique to apply to a rational integral.
· Determine the appropriate technique to apply to a rational integral.
· Apply the Partial Fractions technique.
· Recognize the carrying capacity in a logistic growth setting.
· Determine when the maximum growth rate in a logistic growth setting.
· Know the solution to a logistic differential equation.
· Apply Partial Fractions to the proper type of integral.
· Apply Partial Fractions to integrals with Quadratic factors.
· Determine the correct technique to use and perform the integration.
Unit IX Parametric and Polar Coordinates
By the end of this unit, the student will be able to:
· Graph relations in parametric mode.
· Eliminate the parameter to identify the function form of a parametric.
· Find the slope of a tangent line to a curve in parametric mode.
· Find the concavity of a curve in parametric mode.
· Find the arc length of a curve in parametric mode.
· Find the position of an object in motion in two dimensions from its velocity.
· Find the arc length of a curve expressed in parametric mode.
· Graph curves in polar form.
· Recognize certain polar equations as having particular graphs.
· Determine and interpret intervals of increasing or decreasing of a polar curve.
· Find slopes of lines tangent to polar curves.
· Find the arc length of a shape describe in polar coordinates.
· Find the area of a shape described in polar coordinates.
Unit X Power Series
By the end of this unit, the student will be able to:
· Create a Taylor polynomial from give numerical derivatives.
· Identify numerical derivatives from a given Taylor or Maclaurin polynomial.
· Use a Taylor or Maclaurin polynomial to approximate function values.
· Create new series from a Taylor or Maclaurin polynomial.
· Show that the error involved in an approximation of a function value is below a given amount.
· Create a new series from a known series.
· Find whether a given numerical series converges or diverges.
· Find the Radius of Convergence for a given series.
· Find the Interval of Convergence for a given series.
Unit I. Derivatives
By the end of this unit, the student will be able to:
· Use the Power Rule and Exponential Rules to find Derivatives.
· Find the Derivative of Composite Functions.
· Find Derivatives involving Trig, Trig Inverse, and Logarithmic Functions.
· Use the equation of a tangent line to approximate function values.
· Use Euler’s Method to approximate a numerical solution to a differential equation at a given point.
· Find the Derivative of a product or quotient of two functions.
· Find higher order derivatives.
· Take derivatives of relations implicitly.
· Use implicit differentiation to find higher order derivatives.
· Determine when it is appropriate to use logarithmic differentiation.
· Use logarithmic differentiation to take the derivatives of complicated functions.
· Solve related rates problems.
Unit II Anti-Derivatives
By the end of this unit, the student will be able to:
· Find the anti-derivative of a polynomial.
· Integrate functions involving Transcendental operations.
· Use Integration to solve rectilinear motion problems.
· Use the integration by substitution to integrate composite expressions.
· Use the Integration by Substitution to integrate integrands involving Sine and Cosine.
· Use the Integration by Substitution to integrate integrands involving Secant and Tangent or Cosecant and Cotangent.
· Given a separable differential equation, find the general solution.
· Given a separable differential equation and an initial condition, find a particular solution.
· Given a differential equation, sketch its slope field.
· Given a slope field, sketch a particular solution curve.
· Given a slope field, determine the family of functions to which the solution curves belong.
· Given a slope field, determine the differential equation that it represents.
Unit III Applications of the Derivative
By the end of this unit, the student will be able to:
· Find critical values and extreme values for functions.
· Use the 1st and 2nd derivative tests to identify maxima vs. minima.
· Find Points of Inflection and Intervals of Concavity.
· Sketch the graph of a function using information from its first and/or second derivatives.
· Sketch the graph of a first and/or second derivative from the graph of a function.
· Solve optimization problems.
· Use the derivative to make conclusions about motion.
· Relate the position, velocity, and acceleration functions.
· Sketch the graphs of parametric equations.
· Eliminate the parameter of parametric equations.
· Interpret information in the graph of a derivative in terms of the graph of the “original” function.
· Use the graph of a function to answer questions concerning maximums, minimums, and intervals of increasing and decreasing
· Use the graph of a function to answer questions concerning points of inflection and intervals of concavity.
· Use the graph of a function to answer questions concerning the area under a curve.
Unit IV Integrals
By the end of this unit, the student will be able to:
· Find approximations of integrals using different rectangles.
· Use proper notation when dealing with integral approximation.
· Differentiate integral expressions with the variable in the boundary
· Evaluate Definite Integrals
· Find the average value of a continuous function over a given interval
· Evaluate definite integrals using the Fundamental Theorem of Calculus.
· Evaluate definite integrals applying the Substitution Rule, when appropriate.
· Use proper notation when evaluating these integrals.
· Relate definite integrals to area under a curve.
· Understand the difference between displacement and total distance.
· Extend that idea to understanding the difference between the two concepts in other contexts.
· Analyze the interplay between rates and accumulation in context.
Unit V Limits
By the end of this unit, the student will be able to:
· Evaluate one-sided limits graphically, numerically, and algebraically.
· Evaluate two-sided limits in terms of one-sided limits.
· Prove continuity or discontinuity of a given function.
· Interpret Vertical Asymptotes in terms of one-sided limits.
· Determine if a function is differentiable or not.
· Demonstrate understanding of the connections and differences between differentiability and continuity.
· Evaluate Limits algebraically.
· Evaluate Limits using L’Hopital’s Rule.
· Recognize and evaluate Limits which are derivatives.
· Use the nDeriv function on the calculator to find numerical derivatives.
· Evaluate Limits at infinity.
· Interpret Limits at infinity in terms of end behavior of the graph.
· Evaluate Type I Improper Integrals.
· Determine convergence or divergence of a Type II Improper Integral.
Unit VI Numerical Sequences and Series
By the end of this unit, the student will be able to:
· Identify Sequences and Series
· Find Partial Sums of a given Series.
· Find the terms, partial sums, infinite sums, or n in a geometric sequence.
· Determine the convergence or divergence of a sequence.
· Determine the divergence of a series.
· Use the Comparison Tests to check for convergence or divergence.
· Use the Integral Test to check for convergence or divergence.
· Use the Ratio and Nth Root Tests to check for convergence or divergence.
· Use the Alternating Series Test to check for convergence or divergence.
Unit VII Applications of the Integral
By the end of this unit, the student will be able to:
· Find the area of the region between two curves.
· Find the volume of a solid rotated when a region is rotated about a given axis
· Find the volume of a solid rotated when a region is rotated about a given line
· Find the volume of a solid with given cross sections.
· Find the arc length of a function in Cartesian mode between to points.
Unit VIII Techniques of Integration
By the end of this unit, the student will be able to:
· Identify integrals where Integration by Parts is appropriate.
· Apply the Integration by Parts method.
· Integrate radical integrands using trig substitution.
· Determine the appropriate technique to apply to a rational integral.
· Determine the appropriate technique to apply to a rational integral.
· Apply the Partial Fractions technique.
· Recognize the carrying capacity in a logistic growth setting.
· Determine when the maximum growth rate in a logistic growth setting.
· Know the solution to a logistic differential equation.
· Apply Partial Fractions to the proper type of integral.
· Apply Partial Fractions to integrals with Quadratic factors.
· Determine the correct technique to use and perform the integration.
Unit IX Parametric and Polar Coordinates
By the end of this unit, the student will be able to:
· Graph relations in parametric mode.
· Eliminate the parameter to identify the function form of a parametric.
· Find the slope of a tangent line to a curve in parametric mode.
· Find the concavity of a curve in parametric mode.
· Find the arc length of a curve in parametric mode.
· Find the position of an object in motion in two dimensions from its velocity.
· Find the arc length of a curve expressed in parametric mode.
· Graph curves in polar form.
· Recognize certain polar equations as having particular graphs.
· Determine and interpret intervals of increasing or decreasing of a polar curve.
· Find slopes of lines tangent to polar curves.
· Find the arc length of a shape describe in polar coordinates.
· Find the area of a shape described in polar coordinates.
Unit X Power Series
By the end of this unit, the student will be able to:
· Create a Taylor polynomial from give numerical derivatives.
· Identify numerical derivatives from a given Taylor or Maclaurin polynomial.
· Use a Taylor or Maclaurin polynomial to approximate function values.
· Create new series from a Taylor or Maclaurin polynomial.
· Show that the error involved in an approximation of a function value is below a given amount.
· Create a new series from a known series.
· Find whether a given numerical series converges or diverges.
· Find the Radius of Convergence for a given series.
· Find the Interval of Convergence for a given series.