Calculus Course Outline

 

 

Objectives:

 

Lesson Topic

Overall Objectives

Specific Objectives

 

A. PreCalculus Review

 

Lesson Topic

Overall Objectives

Specific Objectives

A1. Properties of Functions

1. Review characteristics of functions (R)

2. Review/Extend application of linear models (R/E)

3. Introduce new function concepts pertinent to Calculus (N)

 

1s. Fcn char like domain, range, max, min,

2s. Calculus concepts like intervals of increase, decrease, concavity, end behaviour, rate of change

 

A2. Linear Functions

1. Review linear fcns from an algebraic, graphic, numeric perspective (R)

2. Review/Extend application of linear models (R/E)

3. Incorporate new calculus concepts to linear fcns (N)

 

1s. Calculate slope

1s. Determine linear equations from given info (i. point-slope, ii. 2 points)

1s. Recognize and utilize slope relationships in parallel and perpendicular lines

1s. Solve linear equations

2s. model with linear relations

3s. Determine rates of change of functions using slope calcs.

 

A3. Quadratic Functions

1. Review Quadratics from an algebraic, graphic, numeric perspective (R)

2. Review/Extend application of quadratic models (R/E)

3. Incorporate new calculus concepts to quadratic fcns (N)

 

1s. Factor quadratic expressions

1s. Solve quadratic equations

1s. Find max/min on quadratic fcns

1s. sketch and graph quadratic fcns

2s. quadratic models in physics (motion), business (profit, cost), etc for finding roots, max/min, points

3s. Using technology, determine where a quadratic function in/decreases and its concavity

 

A4. Polynomial Functions

1. Review cubics and quartics from an algebraic, graphic, numeric perspective (R)

2. Review/Extend application of polynomial models (R/E)

3. Incorporate new calculus concepts to polynomial fcns (N)

1s. Factor polynomial expressions

1s. Solve polynomial equations

1s. Find max/min on polynomial fcns

1s. Sketch and graph polynomial fcns

2s. polynomial models in biology (populations), business (profit, cost, revenue) etc for finding roots, max/min, points

3s. Using technology, determine where a polynomial function in/decreases and its concavity

 

A5. Rational Functions

1. Review Rational Fcns from an algebraic, graphic, numeric perspective (R)

2. Review/Extend application of rational fcn models (R/E)

3. Incorporate new calculus concepts to rational fcns (N)

 

1s. Simplify rational expressions

1s. Solve rational expressions

1s. Find asymptotes on rational fcns

1s. Sketch and graph rational fcns

2s. rational fcn models in science, economics

3s. Using technology, determine where a quadratic function in/decreases and its concavity

 

A6. Inequalities (chap R5)

1. Review inequalities from an algebraic, graphic, numeric perspective (R)

1s. Solve linear equalities algebraically and graphically

1s. Solve quadratic equalities algebraically and graphically

1s. Solve polynomial equalities algebraically and graphically

 

A7. Exponential Functions

1. Review exponential fcns from an algebraic, graphic, numeric perspective (R)

2. Review/Extend application of exponential models (R/E)

3. Incorporate new calculus concepts to linear fcns (N)

 

1s. Simplify exponential expressions

1s. Solve exponential equations

1s. Find max/min on exponential fcns

1s. Sketch and graph exponential fcns

2s. Exponential models in biology (populations), business (profit, cost, revenue)

3s. Using technology, determine where an exponential function in/decreases and its concavity

 

A8. Logarithmic Functions

1. Review Quadratics from an algebraic, graphic, numeric perspective (R)

2. Review/Extend application of logarthmic models (R/E)

3. Incorporate new calculus concepts to logarithmic fcns (N)

 

1s. Simplify logarithmic expressions

1s. Solve logarithmic equations

1s. Find max/min on logarithmic fcns

1s. Sketch and graph logarithmic fcns

2s. Logarithmic models in science,  economics.

3s. Using technology, determine where a logarithmic function in/decreases and its concavity

 

 

B. Differentiation

 

 

Overall Objectives

Specific Objectives

B.3.1 - Limits

1. Define a limit

2. Determine if a limit exists

3. Determine the value of a limit, if it exists

 

1. Use algebraic, graphic and numeric (AGN) methods to determine if a limit exists

2. Use algebraic, graphic and numeric methods to determine the value of a limit, if it exists

3. Use algebraic, graphic and numeric methods to determine the value of a limit at infinity, if it exists

4. Be able to state and then work with the various properties of limits

5. Apply continuity to application/real world problems

 

B.3.2 - Continuity

1. Define continuity

2. Understand one sided limits

3. Understand the conditions under which a function is NOT continuous on both open and closed intervals

1. Use AGN methods to determine continuity or points of discontinuity

2. Sketch graphs having various limit and continuity conditions

3. Apply continuity to application/real world problems

 

B.3.3 - Rates of Change

1. Understand the geometrical interpretation of a derivative (slope of a tangent)

2. Use difference quotients and limits to find an instantaneous rate of change of a function at a point

 

1. Calculate an average rate of change

2. Calculate the value of a derivative at a point

3. Calculate instantaneous rates of change in real world problems

B.3.4 - Definition of a Derivative

1. Find the equation of a derivative function from the limit definition of a derivative

 

1. Calculate the derivative of simple polynomial and rational functions from first principles

2. Determine the conditions for existence of a derivative

3. Calculate derivatives and apply to real world scenarios

 

B.3.5 - Graphical Differentiation

1. Given the equation of a function, graph it and then make conjectures about the relationship between the derivative function and the original function

2. From a function, sketch its derivative and from a derivative, graph an original function

1. Relate “increasing function” to a positive derivative and vice versa

2. Relate “decreasing function” to a negative derivative and vice versa

3. Relate “max/min” to a zero derivative

4. Relate “concave up” to an increasing derivative and vice versa

5. Relate “concave down” to a decreasing derivative and vice versa

6. Relate function features of graphs of functions and derivatives modeling real world applications

 

 

 

B.4.1 - Techniques for finding the derivative

1. Derive the power rule

2. Derive the constant rule

3. Extend the power rule to the sum or difference rules

4. Determine the antiderivatives of power functions

 

1. Use first principles to develop the power rule

2. Use graphic differentiation to verify the power rule

3. Use graphic evidence to verify antiderivative functions

4. Apply the power rule to real world problems

B.4.2 - Derivatives of Products

1. Derive the product rule

2. Differentiate equations using the product rule

 

1. Use first principles to develop the product rule

2. Use graphic differentiation to verify the product rule

3. Apply the product rule to real world problems

 

B.4.3 - Derivatives of Rational Functions – The Quotient Rule

1. Derive the quotient rule

2. Differentiate equations using the quotient rule

 

1. Use first principles to develop the quotient rule

2. Use graphic differentiation to verify the quotient rule

3. Apply the quotient rule to real world problems

 

B.4.4 – Derivatives of Composite Functions – The Chain Rule

1. Derive the chain rule

2. Differentiate equations using the chain rule

3. Apply the chain rule to real world problems

 

B.4.5 - Derivatives of Exponential Functions

1. Derive the derivatives of exponential functions algebraically and graphically

2. Differentiate equations involving exponential functions

3. Apply exponential functions and their derivatives to real world problems

 

B.4.6 - Derivatives of Logarithmic functions

1. Derive the derivatives of logarithmic functions algebraically and graphically

2. Differentiate equations involving logarithmic functions

3. Apply logarithmic functions and their derivatives to real world problems

 

B.4.7 - Derivatives of Trigonometric functions

1. Derive the derivatives of trigonometric functions algebraically and graphically

2. Differentiate equations involving trigonometric functions

3. Apply sinusoidal functions and their derivatives to real world problems

 

 

 

 

B.5.1 - Increasing and Decreasing Functions

1. Apply knowledge of the first derivative to functions in a mathematical context and in the context of a real world problem

1. Define the terms increasing and decreasing

2. Use calculus methods to determine the intervals in which a function increases or decreases

3. Use calculus methods to determine critical numbers of a function

4. Apply the concepts of increase, decrease and critical numbers to a real world problem

 

B.5.2 - Relative Extrema

1. Calculate all extrema of a function in a mathematical context and in the context of a real world problem

 

1. Define the term relative (or local) extrema of a function

2. Apply definitions of extrema to functions on open and on closed intervals

3. Apply the first derivative test to classifying extremas

4. Predict the appearance of graphs based upon intervals of increase/decrease and extremas

5. Apply concepts of increase, decrease and critical numbers to a real world problem

 

B.5.3 - Higher Derivatives, Concavity

1. Calculate higher order derivatives of functions

2. Determine the intervals of concavity of a function in a mathematical context and in the context of a real world problem

 

1. Calculate second and third derivatives of functions

2. Define concavity and inflection point

3. Test for concavity in a function using the second derivative

4. Perform the second derivative test to determine the nature of relative extrema

5. Apply concepts of concavity, second derivatives, inflection points to a real world problem

 

B.5.4 - Curve Sketching

1. Apply the concepts of test for concavity, test for increasing and decreasing functions and the concepts of limits at infinity to help us sketch graphs and describe the behaviour of functions

 

 

 

B.6.1 - Absolute Extrema

1. Determine the absolute extrema of a given function in a mathematical context and in the context of a real world problem

 

1. Use Calculus methods to determine the absolute extrema of a function

2. State the absolute value theorem

3. Apply concepts of increase, decrease and critical numbers and absolute extrema to a real world problem

 

B.6.2 - Applications of Extrema

1. Solve optimization problems using a variety of calculus and non-calculus based strategies

2. Solve optimization problems with and without the use of the TI 89 GDC

3. Work with problems set in mathematical contexts or in the context of a real world problem

 

B.6.3 – Further Business Applications

1. Solve problems related to marginal costs, revenues and profits and average costs

2. Solve problems related to economic lot size, economic order quantity, and elasticity of demand

 

B.6.4 - Implicit Differentiation

1. Use the implicit differentiation method to find the derivatives of implicitly defined equations

 

1. Define the terms explicit and implicit equations

2. Implicitly differentiate implicitly defined equations

3. Determine the equation of tangents and normals of implicitly defined equations

4. Apply implicit differentiation to real world problems

 

B.6.5 - Related Rates

1. Given a situation in which several quantities vary, predict the rate at which one of them is changing when you know the other related rates

 

B.6.6 - Differentials: Linear Approximations

1. Given the equation of a function and a fixed point on its graph, find an equation for a linear function that best fits the given function

2. Use the linear equation to find approximate values for f(x) and values for the differentials dx and dy

1. Define differentials

2. Use differentials to find approximate values of Dy

3. Write equations of linear functions that best fit a function for values of x close to a given point x = c

 

C. Integration

 

 

Overall Objective

Specific Objective

C.7.1 - Antiderivatives

1. Determine the antiderivative of basic functions

2. Understand the connection between antiderivatives as indefinite integrals

1. Define an antiderivative

2. Recognize the role of and determine the value of a constant of integration

3. Understand the notation of ė f(x)dx

4. Antidifferentiate or integrate basic functions like power, exponential, simple trigonometric functions

5. Apply concepts of antiderivatives and indefinite integrals to a real world problems

 

C.7.2 – Integration by Substitution

1. Use the method of substitution to integrate  simple power, exponential, and trigonometric functions both in a mathematical context and in a real world problem context

 

C.7.3 - Area & the Definite Integral

1. Determine methods for approximating the area under a given function

2. Summarize the process of estimating the area under a curve by the use of the definite integral

 

1. Use a varying number of rectangles constructed at various “endpoints” (i.e. left endpoint, right endpoint, midpoints) to approximate total area under a curve

2. Use a varying number of trapezoids to approximate total area under a curve

3. Understand that the definite integral is simply the sum of an infinite number of rectangles

4. Understand that a definite integral can represent (i) area under a curve, (ii) total change in the function

5. Apply definite integrals to a real world problems

 

C.7.4 - The Fundamental Theorem of Calculus

1. Understand that the connection between antiderivatives and definite integrals is summarized by the Fundamental Theorem of Calculus

 

1. Calculate simple definite integrals using the FTC

2. Calculate definite integrals using the properties of definite integrals

3. Determine total areas under curves using the FTC

4. Differentiate integral functions with variable (i) upper limits of integration and (ii) lower limits of integration

5. Apply definite integrals to a real world problems

 

C.7.5 - The Area Between Two Curves

1. Given two functions, determine the area between the two functions using definite integrals

2. Apply definite integrals and area between curves to a real world problems

 

C.7.6 - Numerical Integration

1. Use numerical methods like the trapezoidal rule and Simpson’s rule to estimate integrals

2. Understand when numerical methods of integration are more useful than algebraic methods

3. Apply definite integrals to a real world problems

 

 

 

C.8.1 - Integration by parts

1. Use the method of integration by parts to integrate simple power, exponential, and trigonometric functions both in a mathematical context and in a real world problem context

 

C.8.2 - Volume

1. Determine the volume of revolution of an object rotated about the x-axis

2. Determine by slicing (disk and washer method) or cylindrical shells to calculate volumes of solids

3. Determine the average value of a function

4. Apply volumes and average values to a real world problems

 

C.8.4 - Improper Integrals

1. Calculate and interpret the values of improper integrals

2. Apply improper integrals to a real world problems

 

 

 

C.9.1 - Differential Equations

1. Solve differential equations with and without initial conditions in a variety of real world applications

 

C.9.2 – Slope Fields

1. Sketch a slope field for a given differential equation and use the given boundary conditions to identify a specific solution curve on their slope field.

2. Provide a geometric interpretation of differential equations using slope fields.

3. Explain the relationship between slope fields and solution curves for differential equations.

 

C.9.3 - Exponential Growth and Decay

1. Solve problems involving exponential decay in a variety of application (Radioactivity, Air resistance is proportional to velocity, Continuously compounding interest, Population growth)

 

C.9.4 - Populations

1. Solve problems involving population growth using exponential modeling and the logistic equation