VISION OF THE GRADUATE

Collaboration, Communication, Knowledge, Resilience

LONG TERM GOALS

Students will be able to independently use their learning to analyze and evaluate various functions and apply their understanding to real world applications.

BIG IDEAS

• Limits of functions can be evaluated numerically, graphically, and algebraically.
• Limits allow us to analyze functions and talk about the rate of change at an exact point.
• There exists relationships between asymptotes and limits.
• Continuity is property of functions that is defined using limits.
• When limits fail to exist.
• The relationship between limits and rates of change.

ESSENTIAL QUESTIONS

• What is a limit?
• When is it appropriate to use each method to evaluate a limit?
• What are the different types of limits and the graphical representations of each?
• Why might a limit fail to exist?
• How can asymptotic behavior of a function be expressed using limits?
• What is continuity? What are the different types of discontinuity?
• How can properties of functions help to better understand a function's continuity?
• How is the average rate of change of a function related to the slope?
• How can limits be used to find the instantaneous rate of change of a function at a point?

LEARNING OUTCOMES Students will be skilled at . . .

• finding limits of functions graphically and numerically.
• understanding the definition of the limit of a function.
• using properties of limits to evaluate limits of functions.
• using different analytical techniques to evaluate limits of functions.
• evaluating one-sided limits.
• recognizing unbounded behavior of functions.
• determining continuity of functions.
• determining the continuity of functions on a closed interval.
• identifying tangent lines to a graph at a point.
• approximating the slopes of tangent lines to graphs at points.
• using the limit definition to find the slopes of graphs at points.
• using the limit definition to find derivatives of functions.
• describing the relationship between differentiability and continuity.

CONTENT STANDARDS

Limits And Continuity

• C.LC.1: Understand the concept of limit and estimate limits from graphs and tables of values.
• C.LC.2: Find limits by substitution.
• C.LC.3: Find limits of sums, differences, products, and quotients.
• C.LC.4: Find limits of rational functions that are undefined at a point.
• C.LC.5: Find limits at infinity.
• C.LC.6: Decide when a limit is infinite and use limits involving infinity to describe asymptotic behavior. Find special limits
• C.LC.7: Find one-sided limits.
• C.LC.8: Understand continuity in terms of limits.
• C.LC.9: Decide if a function is continuous at a point.
• C.LC.10: Find the types of discontinuities of a function.

Differentiation

C.D.2: State, understand, and apply the definition of derivative.

CONTENT The students will know . . .

• the most appropriate technique for determining the limit of a function. Methods include algebraic techniques and graphical interpretation.
• how to evaluate and interpret limits, including one sided limits, limits at infinity and infinite limits.
• the three reasons why limits fail to exists: 1) left and right sided limits are not equal, 2) function oscillates as it approaches a specific x value, and 3) function increases without bound as it approaches a specific x value.
• properties of limits (sum, difference, product, quotient and composite) to evaluate limits algebraically.
• how to evaluate limits of the indeterminate form using factoring,rationalizing or other algebraic techniques.
• a function f is continuous at x = c given the following conditions: 1) f(c) exists, 2) the limit of f(x) as x approaches c exists, and 3) f(c) equals the limit of f(x) as x approaches c.
• functions are continuous at all points in their domains.
• there are three types of discontinuity: 1) removable, 2) jump, 3)infinite.
• if a function is differentiable at x = c, then f is continuous at x = c.
• common situations where a function will not be differentiable at a point (i.e: vertical tangent lines, discontinuity, cusp, nodes.
• the average rate of change of a function over an interval can be found using the difference quotient.
• the instantaneous rate of change of a function at a specific point can be found by finding the limit of the difference quotient as Δx approaches zero.
• the limit as Δx approaches zero of the difference quotient is the derivative of that function, assuming the limit exists.

VISION OF THE GRADUATE

Collaboration, Communication, Knowledge, Resilience

LONG TERM GOALS

Students will be able to independently use their learning to evaluate and understand rates of change.

BIG IDEAS

• The rate at which a function changes can be described using slope.
• How derivatives can be used to describe the rate of change of a function at a given point.
• How the limit process can be applied to find a function's rate of change.
• The various rules of differentation to find a function's rate of change.
• How to find higher order derivatives and understand their rates of change.

ESSENTIAL QUESTIONS

• What is the definition of a derivative?
• How can the slope of a curve be estimated at a single point using local linearity?
• How can the slope at an arbitrary x-value be determined?
• When is it appropriate to apply the various methods of differentiation to a function?
• When might a derivative fail to exist?
• What is the relationship between differentiability and continuity?
• How are higher order derivatives used to find velocity and acceleration?

LEARNING OUTCOMES Students will be skilled at . . .

• Identifying tangent lines to a graph at a point.
• approximating the slopes of tangent lines to graphs at a point.
• describing the relationship between differentiability and continuity.
• finding the derivatives of functions using the Constant, Power, Sum, Difference, Product, Quotient, and Chain rules.
• finding the average rates of change of functions over intervals.
• finding the instantaneous rates of change of functions at points.
• finding marginal revenues, marginal costs, and marginal profits for products.
• using derivatives to answer real-life situations.
• finding higher-order derivatives.
• identifying indeterminate forms of limits.
• applying L’Hospital’s Rule to evaluate limits involving indeterminate forms.

CONTENT STANDARDS

Differentiation

C.D.1: Understand the concept of derivative geometrically, numerically, and analytically, and interpret the derivative as a rate of change.

C.D.2: State, understand, and apply the definition of derivative.

C.D.3: Find the derivatives of functions, including algebraic, trigonometric, logarithmic, and exponential functions.

C.D.4: Find the derivatives of sums, products, and quotients.

C.D.5: Find the derivatives of composite functions, using the chain rule.

C.D.8: Find second derivatives and derivatives of higher order.

C.D.9: Find derivatives using logarithmic differentiation.

C.D.10: Understand and apply the relationship between differentiability and continuity.

Application Of Derivatives

C.AD.1: Find the slope of a curve at a point, including points at which there are vertical tangents and no tangents.

C.AD.2: Find a tangent line to a curve at a point and a local linear approximation.

C.AD.10: Find average and instantaneous rates of change. Understand the instantaneous rate of change as the limit of the average rate of change. Interpret a derivative as a rate of change in applications, including distance, velocity, and acceleration.

CONTENT The students will know . . .

• the relationship between average rates of change vs instantaneous rates of change.
• to evaluate derivatives of functions of sums, differences, products, quotients and composites.
• if the first derivative exists, differentiating the first derivative produces the second derivative; higher order derivatives can be found by repeating this process.
• for y = f(x) functions, notation for the derivative include: f’(x), dy/dx, and y’.
• the derivative can be represented graphically, numerically, analytically, and verbally.
• if a function is differentiable at x = c, then f is continuous at x = c.
• common situations where a function will not be differentiable at a point (i.e: vertical tangent lines, discontinuity, cusp, nodes).

*Functions include: polynomial, power, rational, logarithmic, e^x, trigonometric and compositions of these functions.

VISION OF THE GRADUATE

Collaboration, Innovation, Mindfulness, Communication, Knowledge, Resilience

LONG TERM GOALS

Students will be able to independently use their learning to evaluate and understand rates of change and apply their learning to model functions and real world applications.

BIG IDEAS

• A function’s behavior can be determined using derivatives.
• Conclusions can be drawn about original functions by analyzing graphs of their derivatives.
• Analyzing derivatives, along with knowledge of function properties, can be used to sketch the graph of a function without the use of a graphing calculator.
• The derivative has multiple interpretations and applications including those that involve instantaneous rates of change.

ESSENTIAL QUESTIONS

• How can the first derivative be used to describe the behavior of the original function and the shape of its graph?
• How can the second derivative be used to determine the behavior of the original function and the shape of its graph?
• When is it appropriate to use implicit differentiation vs. explicit differentiation?
• How can derivatives be used to describe relationships between quantities and their rates of change?
• How can a derivative be used to optimize a function over a given interval?

LEARNING OUTCOMES Students will be skilled at . . .

• finding derivatives explicitly.
• finding derivatives implicitly.
• solving related rate problems by creating mathematical models, identifying all given quantities and determining unknown quantities to be found.
• finding the critical numbers of functions.
• finding the open intervals on which functions are increasing and decreasing using the first derivative.
• using increasing and decreasing functions to model and solve real world examples.
• finding the open intervals on which functions are concave up and concave down.
• finding points of inflection.
• sketching the graph of a function using the first and second derivatives and the key concepts of graphing.
• solving optimization problems by creating mathematical models, identifying all given quantities and determining unknown quantities to be found.
• finding the differentials of functions using differentiation formulas.
• using differentials in economics to approximate changes in cost,revenue and profit.

CONTENT STANDARDS

Differentiation

C.D.1: Understand the concept of derivative geometrically, numerically, and analytically, and interpret the derivative as a rate of change.

C.D.3: Find the derivatives of functions, including algebraic, trigonometric, logarithmic, and exponential functions.

C.D.4: Find the derivatives of sums, products, and quotients.

C.D.5: Find the derivatives of composite functions, using the chain rule.

C.D.6: Find the derivatives of implicitly-defined functions.

C.D.8: Find second derivatives and derivatives of higher order.

C.D.9: Find derivatives using logarithmic differentiation.

Application Of Derivatives

C.AD.1: Find the slope of a curve at a point, including points at which there are vertical tangents and no tangents.

C.AD.2: Find a tangent line to a curve at a point and a local linear approximation.

C.AD.3: Decide where functions are decreasing and increasing. Understand the relationship between the increasing and decreasing behavior of f and the sign of f'.

C.AD.4: Solve real-world and other mathematical problems finding local and absolute maximum and minimum points with and without technology.

C.AD.5: Analyze real-world problems modeled by curves, including the notions of monotonicity and concavity with and without technology.

C.AD.6: Find points of inflection of functions. Understand the relationship between the concavity of f and the sign of f". Understand points of inflection as places where concavity changes.

C.AD.7: Use first and second derivatives to help sketch graphs modeling real-world and other mathematical problems with and without technology. Compare the corresponding characteristics of the graphs of f, f', and f".

C.AD.9: Solve optimization real-world problems with and without technology.

CONTENT The students will know . . .

• how to differentiate with respect to all variables in a given relation.
• the chain rule is the basis for implicit differentiation.
• if two or more variables that are changing with respect to time are related to each other, then their rates of change with respect to time are also related.
• how to create a mathematical model from a verbal description of a related rate.
• how to analyze the first derivative to determine critical numbers.
• how to analyze the first derivative to determine on what interval(s) a function is increasing and decreasing.
• how to apply the First Derivative Test to locate relative extrema.
• not all x-values at which the first derivative equals zero or the first derivative is undefined are relative extrema.
• how to test for absolute extrema on closed intervals.
• how to analyze the second derivative to determine concavity.
• how to analyze the second derivative to find points of inflection.
• not all x-values at which the second derivative equals zero or the second derivative is undefined are points of inflection.
• the Second Derivative Test (to find relative extrema using concavity).
• calculus concepts can be used to analyze and sketch functions without a graphing calculator.
• the derivative can be used to solve optimization problems.
• how to find a maximum or a minimum value of a function over a given interval in order to solve real world problems.
• the definition of differentials.
• how differentials are used in economics.
• how marginal analysis is used in economics.

*Functions include: polynomial, power, rational, logarithmic, e^x, trigonometric and compositions of these functions.

VISION OF THE GRADUATE

Collaboration, Mindfulness, Communication, Knowledge, Resilience

LONG TERM GOALS

Students will be able to independently use their learning to evaluate and understand integrals and apply their learning to model functions and real world applications.

BIG IDEAS

• The Fundamental Theorem of Calculus relate differentiation and integration.
• Antidifferentiation is the inverse process of differentiation.
• The definite integral of a function over an interval is the limit of a Riemann Sum over that interval and can be calculated using a variety of strategies.
• The definite integral of a function has many application involving accumulation.

ESSENTIAL QUESTIONS

• What is integration?
• How are the derivative and the integral related?
• What is the relationship between an integral and area?
• How is the anti-derivative related to the definite and indefinite integral?
• What geometric methods can be used to approximate the area under a curve?
• What is a definite integral?

LEARNING OUTCOMES Students will be skilled at . . .

• understanding the definition of antiderivatives.
• using indefinite integral notation for antiderivatives.
• using basic integration rules to find antiderivatives.
• using initial conditions to find particular solutions of indefinite integrals.
• using antiderivatives to solve real life applications.
• using the general power rule, exponential and log rules to find indefinite integrals.
• using substitution to find indefinite integrals.
• understanding the relationship between area and definite integrals.
• evaluating definite integrals using the Fundamental Theorem of Calculus.
• using definite integrals to solve marginal analysis examples.
• finding the average values of functions over closed intervals.
• using properties of even and odd functions to help evaluate definite integrals.
• finding the areas of regions bounded by two graphs.
• finding consumer and producer surpluses.
• using the areas of regions bounded by two graphs to model and solve real world examples.
• using the midpoint rule, the trapezoid rule, left Riemann sums and right Riemann sums to approximate definite integrals.
• understanding the definite integral as the limit of a sum.

CONTENT STANDARDS

Integrals

C.I.1: Use rectangle approximations to find approximate values of integrals.

C.I.2: Calculate the values of Riemann Sums over equal subdivisions using left, right, and midpoint evaluation points.

C.I.3: Interpret a definite integral as a limit of Riemann Sums.

C.I.4: Understand the Fundamental Theorem of Calculus: Interpret a definite integral of the rate of change of a quantity over an interval as the change of the quantity over the interval, that is

C.I.5: Use the Fundamental Theorem of Calculus to evaluate definite and indefinite integrals and to represent particular antiderivatives. Perform analytical and graphical analysis of functions so defined.

C.I.6:Understand and use these properties of definite integrals.

C.I.7: Understand and use integration by substitution (or change of variable) to find values of integrals.

C.I.8: Understand and use Riemann Sums, the Trapezoidal Rule, and technology to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values.

Applications of Integrals

C.AI.1: Find specific antiderivatives using initial conditions, including finding velocity functions from acceleration functions, finding position functions from velocity functions, and applications to motion along a line.

C.AI.4: Use definite integrals to find the area between a curve and the x-axis, or between two curves.

C.AI.5: Use definite integrals to find the average value of a function over a closed interval.

CONTENT The students will know . . .

• an antiderivative of a function, f, is a function g whose derivative is f.
• how to evaluate a definite integral.
• differentiation rules provide the foundation for finding antiderivatives.
• areas of certain regions in the plane can be calculated with definite integrals.
• the limit of a Riemann sum can be written as a definite integral.
• techniques for finding antiderivatives include algebraic manipulation such as substitution of variables.
• the area between two curves can be found using integration techniques.
• definite integrals can be approximated for functions that are represented graphically, numerically, and algebraically.