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 nDeriv function on the calculator to find numerical derivatives.
  • Use the equation of a tangent line to approximate function values.
  • Find the Derivative of a product or quotient of two functions.
  • Find higher order derivatives.
 
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       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 IV       Applications of the Derivative I
 
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.
  • 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 V        Applications of the Derivative II
 
By the end of this unit, the student will be able to:
  • Use the derivative to make conclusions about motion.
  • Relate the position, velocity, and acceleration functions.
  • Take derivatives of relations implicitly.
  • Use implicit differentiation to find higher order derivatives.
  • Solve related rates problems.
  • Identify key information from a logistic growth equation
  • Solve separable differential equations that arise from logistic or exponential growth
 
 
Unit VI       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 VII     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.
  • Evaluate Limits at infinity.
  • Interpret Limits at infinity in terms of end behavior of the graph.