Unit I. Derivatives**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.