Date |
Sections |
Topics |
Prerequisite Skills |
Student Learning Outcomes
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Week 1
|
2.2
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The Limit of a Function
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- 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.
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Week 1/2
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2.3
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The Limit Laws
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- 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.
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Week 2/3
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2.4
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Continuity
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- 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.
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Week 3
|
4.6
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Limits at Infinity and Asymptotes
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- Calculate the limit of a function that is unbounded.
- Identify a horizontal asymptote for the graph of a function.
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Week 3/4
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3.1
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Defining the Derivative
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- 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.
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Week 4
|
3.2
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The Derivative as a Function
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- 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.
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Week 4/5
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3.3
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Differentiation Rules
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- 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.
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Week 5
|
3.4
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Derivatives as Rates of Change
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- 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.
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Week 5
|
3.5
|
Derivatives of the Trigonometric Functions
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- 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.
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Week 6
|
3.6
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The Chain Rule
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- 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.
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Week 6
|
3.7
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Derivatives of Inverse Functions
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- 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.
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Week 6/7
|
3.8
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Implicit Differentiation
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- Assuming, for example, y is implicitly a function of x, find the derivative of y with respect to x.
- Assuming, for example, y is implicitly a function of x, and given an equation relating y to x, find the derivative of y with respect to x.
- Find the equation of a line tangent to an implicitly defined curve at a point.
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Week 7
|
3.9
|
Derivatives of Exponential and Logarithmic Functions
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- Find the derivative of functions that involve exponential functions.
- Find the derivative of functions that involve logarithmic functions.
- Use logarithmic differentiation to find the derivative of functions containing combinations of powers, products, and quotients.
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Week 7/8
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4.1
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Related Rates
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- Express changing quantities in terms of derivatives.
- Find relationships among the derivatives in a given problem.
- Use the chain rule to find the rate of change of one quantity that depends on the rate of change of other quantities.
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Week 8
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4.2
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Linear Approximations and Differentials
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- Approximate the function value close to the center of the linear approximation using the linearization.
- Given an expression to be evaluated/approximated, come up with the function and its linearization
- Understand the formula for the differential; how it can be used to estimate the change in the dependent variable quantity, given the small change in the independent variable quantity.
- Use the information above to estimate potential relative (and percentage) error
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Week 8/9
|
4.3
|
Maxima and Minima
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- Know the definitions of absolute and local extrema.
- Know what a critical point is and locate it (them).
- Use the Extreme Value Theorem to find the absolute extrema of a continuous function on a closed interval.
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Week 9
|
4.4
|
Mean Value Theorem
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- Determine if the MVT applies given a function on an interval.
- Find c in the conclusion of the MVT (if algebraically feasible)
- Know the first 3 Corollaries of MVT (especially the 3rd)
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Week 9
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4.5
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Derivatives and the Shape of a Graph
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- Use the First Derivative Test to find intervals on which the function is increasing and decreasing and the local extrema and their type
- Use the Concavity Test (aka the Second Derivative Test for Concavity) to find the intervals on which the function is concave up and down, and point(s) of inflection
- Understand the shape of the graph, given the signs of the first and second derivatives.
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Week 10
|
4.7
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Applied Optimization Problems
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- Set up a function to be optimized and find the value(s) of the independent variable which provide the optimal solution.
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Week 10
|
4.8
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L’Hôpital’s Rule
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- Identify indeterminate forms produced by quotients, products, subtractions, and powers, and apply L’Hôpital’s rule in each case.
- Recognize when to apply L’Hôpital’s rule.
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Week 11
|
4.10
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Antiderivatives
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- Find the general antiderivative of a given function.
- Explain the terms and notation used for an indefinite integral.
- State the power rule for integrals.
- Use anti-differentiation to solve simple initial-value problems.
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Week 11/12
|
5.1
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Approximating Areas
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- Calculate sums and powers of integers.
- Use the sum of rectangular areas to approximate the area under a curve.
- Use Riemann sums to approximate area.
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Week 12
|
5.2
|
The Definite Integral
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- State the definition of the definite integral.
- Explain the terms integrand, limits of integration, and variable of integration.
- Explain when a function is integrable.
- Rules for the Definite Integral.
- Describe the relationship between the definite integral and net area.
- Use geometry and the properties of definite integrals to evaluate them.
- Calculate the average value of a function.
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Week 12/13
|
5.3
|
The Fundamental Theorem of Calculus
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- Describe the meaning of the Mean Value Theorem for Integrals.
- State the meaning of the Fundamental Theorem of Calculus, Part 1.
- Use the Fundamental Theorem of Calculus, Part 1, to evaluate derivatives of integrals.
- State the meaning of the Fundamental Theorem of Calculus, Part 2.
- Use the Fundamental Theorem of Calculus, Part 2, to evaluate definite integrals.
- Explain the relationship between differentiation and integration.
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Week 13
|
5.4
|
Integration Formulas and the Net Change Theorem
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- Apply the basic integration formulas.
- Explain the significance of the net change theorem.
- Use the net change theorem to solve applied problems.
- Apply the integrals of odd and even functions.
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Week 14
|
5.5
|
Integration by Substitution
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- Use substitution to evaluate indefinite integrals.
- Use substitution to evaluate definite integrals.
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Week 14/15
|
5.6
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Integrals Involving Exponential and Logarithmic Functions
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- Integrate functions involving exponential functions.
- Integrate functions involving logarithmic functions.
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Week 15
|
5.7
|
Integrals Resulting in Inverse Trigonometric Functions
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- Integrate functions resulting in inverse trigonometric functions.
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