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Unit 1: Limits and Continuity |
- What is a limit?
- How do we find limits graphically, numerically, and analytically?
- What is continuity?
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- Definition of a limit
- Definition of continuity
- Squeeze Theorem, Extreme Value Theorem, and Intermediate Value Theorem
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- 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
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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?
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- 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
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- 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
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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?
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- Riemann Sums
- Basic Integration
- Fundamental Theorems of Calculus
- The Average Value formula
- Integration by substitution
- The left hand, right hand, midpoint, and trapezoidal approximations
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- 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
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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?
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- 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
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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?
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- Differential equations
- Separation of variables
- Euler's method
- Slope fields
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- 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?
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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?
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- Area between curves
- Solids of revolution
- Volumes of solids with known cross sections
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- 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
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