Calculus 1

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The main topics of Calculus I include limits, continuity, diffferentiation, and integration. Although this course is not considered an AP course, it should fully prepare students for the AP AB Calculus exam in the spring. Emphasis will be placed on demonstrating the connection between calculus and other subject areas, including physics, finance, and medicine.

 

Assessments will include group and individual projects, quizzes, and tests. Resources used throughout the course include textbooks, graphing calculators, and investigative labs.

Units

Unit Essential Questions Content Skills and Processes
Unit 1: Limits and Continuity
  • What is a limit?
  • How do we find limits graphically, numerically, and analytically?
  • What is continuity?
  • Definition of a limit
  • Definition of continuity
  • Squeeze Theorem, Extreme Value Theorem, and Intermediate Value Theorem
  • Finding limits graphically, numerically, and analytically
  • Determining the continuity of a function
  • Classifying points of discontinuity, including removable and nonremovable points and infinite limits
  • Using the squeeze theorem, extreme value theorem, and intermediate value theorem
Unit 2: Differentiation
  • What is a tangent line?
  • What is a derivative?
  • What are basic differentiation rules?
  • What is implicit differentiation?
  • What do the first and second derivatives tell us?
  • What are the important theorems involving differentiation?
  • How do we solve problems involving optimization?
  • The tangent line
  • The definition of a derivative
  • Basic differentiation rules
  • Implicit differentiation
  • Related rates
  • First and Second Derivative tests
  • Rolle's Theorem and Mean Value Theorem
  • Limits at infinity
  • Optimization
  • Differentials
  • Finding tangent lines and derivatives
  • Using basic differentiation rules, including the power rule, product rule, quotient rule, and chain rule.
  • Performing implicit differentiation
  • Using the first and second derivative tests to classify extrema
  • Using Rolle's Theorem and the Mean Value Theorem
  • Solving optimization problems using the derivative
  • Finding limits at infinity
  • Using differentials to approximate functions
Unit 3: Integration
  • What is a Riemann Sum?
  • What is an integral and how do we integrate?
  • What are the first and second Fundamental Theorems of Calculus?
  • How do we find the average value of a function?
  • How do we use substitution to integrate?
  • How do we approximate integrals using left hand, right hand, midpoint, and trapezoidal sums?
  • Riemann Sums
  • Basic Integration
  • Fundamental Theorems of Calculus
  • The Average Value formula
  • Integration by substitution
  • The left hand, right hand, midpoint, and trapezoidal approximations
  • Taking the limit of Riemann sums to find the integral
  • Understanding how the integral relates to the area under a curve
  • Integrating using basic rules as well as substitution
  • Using the First and Second Fundamental Theorems of Calculus
  • Finding the average value of a function
  • Approximating an integral using left hand, right hand, midpoint, and trapezoidal approximations
Unit 4: Exponential, logarithmic, and inverse functions
  • How does the natural logarithm relate to the integral of the inverse function?
  • How do we differentiate and integrate the exponential function?
  • How do we find the derivative of a function's inverse?
  • How do we differentiate inverse trigonometric functions?
  • The natural logarithm
  • The exponential function, including those with bases other than e
  • Inverse functions
  • Inverse trigonometric functions
 
  • Integration using the natural logarithm
  • Differentiating and integrating the exponential function, including those with bases other than e
  • Finding the derivative of a function's inverse
  • Differentiating inverse trigonometric functions
Unit 5: Differential equations and slope fields
  • What is a differential equation?
  • How do we solve differential equations using separation of variables?
  • What is Euler's Method?
  • What are slope fields and what do they tell us?
  • Differential equations
  • Separation of variables
  • Euler's method
  • Slope fields
  • How do we solve differential equations using separation of variables?
  • How can we approximate the solution to differential equations using Euler's method?
  • How can we investigate the behavior of differential equations using slope fields?
Unit 6: Area and volume
  • How can we find the area between two curves?
  • How can we find the volume of solids of revolution using the disk, washer, and shell methods?
  • How can we find the volume of solids with known cross sections?
  • Area between curves
  • Solids of revolution
  • Volumes of solids with known cross sections
  • Finding the area between curves
  • Finding the volume of solids of revolution through the disk, washer, and shell methods
  • Finding the volumes of solids with known cross sections