20804231Calculus and Analytic Geometry 1
Course Information
Description
This course covers differential and integral calculus, plane analytic geometry, applications and the properties and uses of elementary transcendental functions. The course provides an introduction to the basic properties of limits, rate of change of functions, continuity, derivatives of algebraic and elementary transcendental functions, their products quotients and compositions, curve sketching, finding maxima and minima, and indefinite and definite integration with applications. This is the first course in a three-semester calculus sequence completed with Calculus III 20-804-233, which is normally required for all higher-level math courses and should be taken by those preparing for major study in mathematics, the physical sciences, computer sciences, or engineering. It is also recommended for students in the social and life sciences who may want a substantial introduction to calculus.
Total Credits
5
Prior Learning Assessment
  • Exam-Advanced Placement
Course Competencies
  1. 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

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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