**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

- 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.