Difference between revisions of "MAT1214"

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*Domain of function.
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*Interval notation.
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*Evaluate limits.
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* Continuity at a point.
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* Describe three kinds of discontinuities.
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* Define continuity on an interval.
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* State the theorem for limits of composite functions and use the theorem to evaluate limits.
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* Provide an example of the intermediate value theorem.
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|[[Limits_at_infinity|Limits at infinity and asymptotes]]
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* Horizontal asymptote for the graph of a function
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* Calculate the limit of a function that is unbounded.
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* Identify a horizontal asymptote for the graph of a function.
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|[[Derivative_definition|Defining the Derivative]]
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* Evaluation of a function at a value or variable expression.
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* Find equation of a line given a point on the line and its slope.
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* Evaluate limits.
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* Recognize the meaning of the tangent to a curve at a point.
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* Calculate the slope of a secant line (average rate of change of a function over an interval).
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* Calculate the slope of a tangent line.
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* Find the equation of the line tangent to a curve at a point.
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* Identify the derivative as the limit of a difference quotient.
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* Calculate the derivative of a given function at a point.
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|[[Derivative_function|The Derivative as a Function]]
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* Graphing functions.
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* The definition of continuity of a function at a point.
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* Understanding that derivative of a function at a point represents the slope of the curve at a point.
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* Understanding when a limit fails to exist.
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* Define the derivative function of a given function.
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* Graph a derivative function from the graph of a given function.
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* State the connection between derivatives and continuity.
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* Describe three conditions for when a function does not have a derivative.
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* Explain the meaning of and compute a higher-order derivative.
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|[[Differentiation_rules|Differentiation Rules]]
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* Radical and exponential notation.
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* Convert between radical and rational exponents.
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* Use properties of exponents to re-write with or without negative exponents.
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* State the constant, constant multiple, and power rules.
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* Apply the sum and difference rules to combine derivatives.
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* Use the product rule for finding the derivative of a product of functions.
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* Use the quotient rule for finding the derivative of a quotient of functions.
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* Extend the power rule to functions with negative exponents.
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* Combine the differentiation rules to find the derivative of a polynomial or rational function.
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|[[Derivatives_Rates_of_Change|Derivatives as Rates of Change]]
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* Function evaluation at a value or variable expression.
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* Solving an algebraic equation.
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* Find derivatives of functions using the derivative rules.
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* Determine a new value of a quantity from the old value and the amount of change.
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* Calculate the average rate of change and explain how it differs from the instantaneous rate of change.
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* Apply rates of change to displacement, velocity, and acceleration of an object moving along a straight line.
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* Predict the future population from the present value and the population growth rate.
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* Use derivatives to calculate marginal cost and revenue in a business situation.
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Edit
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|-
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|[[Derivatives_Trigonometric_Functions|Derivatives of the Trigonometric Functions]]
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* State and use trigonometric identities.
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* Graphs of the six trigonometric functions.
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* Power, Product, and Quotient Rules for finding derivatives.
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* Find the derivatives of the sine and cosine function.
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* Find the derivatives of the standard trigonometric functions.
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* Calculate the higher-order derivatives of the sine and cosine.
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Edit
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|-
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|[[Chain_Rule|The Chain Rule]]
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* Composition of functions.
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* Solve trigonometric equations.
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* Power, Product, and Quotient Rules for finding derivatives.
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 +
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 +
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* State the chain rule for the composition of two functions.
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* Apply the chain rule together with the power rule.
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* Apply the chain rule and the product/quotient rules correctly in combination when both are necessary.
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* Recognize and apply the chain rule for a composition of three or more functions.
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* Use interchangeably the Newton and Leibniz Notation for the Chain Rule.
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Edit
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|-
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|[[Derivatives_Inverse_Functions|Derivatives of Inverse Functions]]
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* Determine if a function is 1-1.
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* The relationship between a 1-1 function and its inverse.
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* Knowing customary domain restrictions for trigonometric functions to define their inverses.
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* Rules for differentiating functions.
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* State the Inverse Function Theorem for Derivatives.
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* Apply the Inverse Function Theorem to find the derivative of a function at a point given its inverse and a point on its graph.
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* Derivatives of the inverse trigonometric functions.
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|[[Implicit_Differentiation|Implicit Differentiation]]

Revision as of 13:45, 11 June 2020

Topics List

Topic Pre-requisite Objective Examples
The Limit of a Function
  • Evaluation of a function including the absolute value, rational, and piecewise functions
  • Domain and Range of a Function
  • Describe the limit of a function using correct notation.
  • Use a table of values to estimate the limit of a function or to identify when the limit does not exist.
  • Use a graph to estimate the limit of a function or to identify when the limit does not exist.
  • Define one-sided limits and provide examples.
  • Explain the relationship between one-sided and two-sided limits.
  • Describe an infinite limit using correct notation.
  • Define a vertical asymptote.

Edit soon


The Limit Laws


  • Simplifying algebraic expressions.
  • Factoring polynomials
  • Identifying conjugate radical expressions.
  • Evaluating expressions at a value.
  • Simplifying complex rational expressions by obtaining common denominators.
  • Evaluating piecewise functions.
  • The trigonometric functions and right triangle trigonometry.
  • Recognize the basic limit laws.
  • Use the limit laws to evaluate the limit of a function.
  • Evaluate the limit of a function by factoring.
  • Use the limit laws to evaluate the limit of a polynomial or rational function.
  • Evaluate the limit of a function by factoring or by using conjugates.
  • Evaluate the limit of a function by using the squeeze theorem.
  • Evaluate left, right, and two sided limits of piecewise defined functions.
  • Evaluate limits of the form K/0, K≠0.
  • Establish and use this to evaluate other limits involving trigonometric functions.

edit soon


Continuity


  • Domain of function.
  • Interval notation.
  • Evaluate limits.
  • Continuity at a point.
  • Describe three kinds of discontinuities.
  • Define continuity on an interval.
  • State the theorem for limits of composite functions and use the theorem to evaluate limits.
  • Provide an example of the intermediate value theorem.

Edit


Limits at infinity and asymptotes
  • Horizontal asymptote for the graph of a function
  • Calculate the limit of a function that is unbounded.
  • Identify a horizontal asymptote for the graph of a function.

Edit


Defining the Derivative
  • Evaluation of a function at a value or variable expression.
  • Find equation of a line given a point on the line and its slope.
  • Evaluate limits.
  • Recognize the meaning of the tangent to a curve at a point.
  • Calculate the slope of a secant line (average rate of change of a function over an interval).
  • Calculate the slope of a tangent line.
  • Find the equation of the line tangent to a curve at a point.
  • Identify the derivative as the limit of a difference quotient.
  • Calculate the derivative of a given function at a point.

Edit


The Derivative as a Function
  • Graphing functions.
  • The definition of continuity of a function at a point.
  • Understanding that derivative of a function at a point represents the slope of the curve at a point.
  • Understanding when a limit fails to exist.
  • Define the derivative function of a given function.
  • Graph a derivative function from the graph of a given function.
  • State the connection between derivatives and continuity.
  • Describe three conditions for when a function does not have a derivative.
  • Explain the meaning of and compute a higher-order derivative.

Edit


Differentiation Rules
  • Radical and exponential notation.
  • Convert between radical and rational exponents.
  • Use properties of exponents to re-write with or without negative exponents.
  • State the constant, constant multiple, and power rules.
  • Apply the sum and difference rules to combine derivatives.
  • Use the product rule for finding the derivative of a product of functions.
  • Use the quotient rule for finding the derivative of a quotient of functions.
  • Extend the power rule to functions with negative exponents.
  • Combine the differentiation rules to find the derivative of a polynomial or rational function.

Edit


Derivatives as Rates of Change
  • Function evaluation at a value or variable expression.
  • Solving an algebraic equation.
  • Find derivatives of functions using the derivative rules.
  • Determine a new value of a quantity from the old value and the amount of change.
  • Calculate the average rate of change and explain how it differs from the instantaneous rate of change.
  • Apply rates of change to displacement, velocity, and acceleration of an object moving along a straight line.
  • Predict the future population from the present value and the population growth rate.
  • Use derivatives to calculate marginal cost and revenue in a business situation.

Edit


Derivatives of the Trigonometric Functions
  • State and use trigonometric identities.
  • Graphs of the six trigonometric functions.
  • Power, Product, and Quotient Rules for finding derivatives.
  • Find the derivatives of the sine and cosine function.
  • Find the derivatives of the standard trigonometric functions.
  • Calculate the higher-order derivatives of the sine and cosine.

Edit


The Chain Rule
  • Composition of functions.
  • Solve trigonometric equations.
  • Power, Product, and Quotient Rules for finding derivatives.
  • State the chain rule for the composition of two functions.
  • Apply the chain rule together with the power rule.
  • Apply the chain rule and the product/quotient rules correctly in combination when both are necessary.
  • Recognize and apply the chain rule for a composition of three or more functions.
  • Use interchangeably the Newton and Leibniz Notation for the Chain Rule.

Edit


Derivatives of Inverse Functions
  • Determine if a function is 1-1.
  • The relationship between a 1-1 function and its inverse.
  • Knowing customary domain restrictions for trigonometric functions to define their inverses.
  • Rules for differentiating functions.
  • State the Inverse Function Theorem for Derivatives.
  • Apply the Inverse Function Theorem to find the derivative of a function at a point given its inverse and a point on its graph.
  • Derivatives of the inverse trigonometric functions.

Edit


Implicit Differentiation
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