Unit 1 – Algebra Review

     Objectives:  The student will: 

          1.  a)  classify a number as Real, Complex, Rational, Irrational, Integer, Whole, and/or Natural.
               b)  understand the concept of denseness as related to the number line.
               c)  use a calculator to approximate irrational numbers and convert rational decimals into fractions.

          2.  a)  solve inequalities using a number line with boundaries and determine “true” or “false” regions
               b)  write solutions in interval notation and understand inclusion [bracket] and exclusion (parenthesis)

          3.  solve absolute value inequality problems and square root problems using a graphing calculator

          4.  a)  plot points in the Cartesian plane and calculate distance and midpoint.
               b)  write the equation of a line using slope-intercept form and point-slope form.
               c)  write the equation of a circle given the radius and center or points on the circle.

          5.  a)  use a graphing calculator to explore the graphs of basic polynomial functions
               b)  use a graphing calculator to find roots (zeros) of a function.
               c)  use a graphing calculator to solve equations.

DAY	DATE	Unit One Topics	ASSIGNMENT
1		Real Number System,  Denseness   Simplifying Expressions, Calculator Apprx	Section 1.1       3,  7,  9,  15,  19,  21,  25,  29,  31,  33,  34,  35,  37,  47 Section 1.2       23,  25,  27,  29,  30,  31,  41
2		Solving Inequalities, Interval Notation	Section 1.3       1,  2,  3,  13,  15,  19,  21,  25,  27,  31,  33
3		Solving Absolute Value, Square Roots	Section 1.4       1,  7,  11,  15,  27,  29
4		Coordinate System, Linear Equations Equation of a Circle, Midpoint, Distance	Section 1.5       1,  5,  11,  13,  15,  17,  21,  25,  Section 1.6       1,  7,  9,  11,  13,  15,  17,  19,  21,  23
5		Graphs of polynomials,  Roots of functions	Section 1.7       1,  3,  5,  7,  9,  11,  13,  15,  40
6		* * * TEST * * *

  




















Unit 2 – Limits
     Objectives:  The student will: 

          1.  a)  determine the domain and range of a function numerically, algebraically, and graphically.
               b)  describe symmetry of odd and even functions and explore symmetry with graphing calculator.
              
          2.  a)  examine function values through addition, subtraction, multiplication, and composition of f(x).
               b)  verbally describe (or write) what happens to the graph of a function under various translations.

          3.  a)  write a formula for piecewise functions by interpreting graphs.
               b)  use the greatest common integer function to develop an intuitive understanding of limit.

          4.  a)  use a graphing calculator graph feature to further increase an understanding of limit (graphically)
               b)  use a graphing calculator table feature to further increase an understanding of limit (numerically)

          5.  learn basic limit theorems to further increase an understanding limits (algebraically)

          6.  demonstrate a knowledge of continuity and discontinuity from a graphical and numerical view.

          7.  a)  demonstrate a knowledge of continuity and discontinuity at a point from an algebraic view
                          i.e.  Does the limit exist?   Is the function defined?  Are they equal?
               b)  learn and use the Intermediate Value Theorem

          8.  solve graphically and confirm analytically the limit of a function as x approaches ± infinity

          9)  a)  use a graphing calculator table feature to evaluate limits numerically as x approaches ± infinity
               b)  write the equation of vertical and horizontal asymptotes discovered by evaluation of limits

        10)  a)  use secant lines to develop a rudimentary understanding of average rate of change
               b)  use short secant line segments to approximate slope of tangent line (instantaneous rate of change)

        11)  a)  recognize that change in position yields velocity (a rate).
               b)  calculate approximate velocity using secant lines and instantaneous velocity using limits

DAY	DATE	Unit Two Topics	ASSIGNMENT
1		Functions, Domain & Range	Section 1.2  (1,  3,  5,  7,  10,  16,  18,  37–40,  56,  58
2		Operations on Functions          Translations	Section 1.2  (49,  51,  53,  66)
3		Piecewise f(x),   Limit concepts       Greatest Common Integer	Section 1.2  (29,  31,  34,  41,  44)     Section 2.1  (1–6)
4		Intuitive Limits (continued)	Section 2.1  (21,  23,  25,  31,  37–42,  46)
5		Limit Theorems	Section 2.1  (7,  9,  10,  11,  18,  19,  43)
6		Continuity & Discontinuity	Section 2.3  (1,  3,  5,  7,  11,  12)
7		Algebraic Continuity,    Intermediate Value Theorem	Section 2.3  (19,  20,  26,  35,  36,  37,  38)
8		Limits near infinity	Section 2.2  (4,  5,  7,  23,  24,  29–32)
9		Limits near infinity  & asymptotes	Section 2.2  (9,  10,  11,  13,  14,  17,  19,  35,  36)
10		Avg. rate of change,  Tangent Line	Section 2.4  (2,  7,  8,  9,  10,  11,  12)
11		Avg. & Instantaneous Velocity	Section 2.4  (23,  25,  26,  29)
12		Review for Test	Review for Test
13		* * * TEST * * *

Unit 3 – Derivatives of Algebraic Functions

     Objectives:  The student will: 

          1.  a)  algebraically determine a derivative by applying the limit definition.
               b)  use a graphing calculator to explore relationship of the graphs of f(x) and f '(x).
              
          2.  a)  analyze derivatives using the alternate definition of derivative (algebraically)
               b)  describe where and why derivatives fail to exist (graphically and numerically)

          3.  a)  learn and apply the basic rules of derivatives: power, product, quotient.
               b)  use a graphing calculator to calculate numeric derivatives.

          5.  learn and apply the basic rules of derivatives: chain rule

          6.  a)  explore higher order derivatives.
               b)  realize that the 2nd derivative of position is acceleration.

          7.  practice the process of implicit differentiation and solve for dy/dx.

          8.  a)  use differentials to approximate the amount of change on a function when ∆x is small.
               b)  use linear approximations to make statements about change.

          9.  learn and apply the Mean Value Theorem.

DAY	DATE	Unit Three Topics	ASSIGNMENT
1		Definition of Derivative	Section 3.1      (1,  2,  7–10,  11,  12,  15,  16)
2		Alternate def. of Derivative	Section 3.1      (3, 6, 18, 19)       Section 3.2 (1, 2, 5, 7, 8, 9)
3		Rules of Derivative	Section 3.3      (1 – 19)
4		Rules of Derivative (cont)	Section 3.3      (21, 22, 23, 25, 27, 33)  &  Worksheet
5		Chain Rule	Section 3.6      (9, 12, 15, 16, 33, 34, 37, 38, 45, 46)
6		Higher Order Derivatives	Section 3.3      (32, 34)     Section 3.4  (2, 3, 6, 8, 9, 18, 20)
7		Implicit Differentiation	Section 3.7      (9, 10, 12, 23, 25, 27, 29, 38)
8		Differentials & Approximation	Section 4.5      (19, 20, 27, 31, 33, 35, 39, 42)
9		Mean Value Theorem	Section 4.2      (15, 16, 19, 20, 21, 39, 40)
10		Review for Test	STUDY
11		* * * TEST * * *

Unit 4 – Properties and Applications of the Derivative

     Objectives:  The student will: 

          1.  a)  understand what a critical point is and how it is related to the first derivative.
               b)  develop an understanding of extreme values and where they occur.
               c)  use a graphing calculator to explore extrema from a graph and from a table
              
          2.  a)  analyze and describe where a function is increasing/decreasing and how that relates to f '(x).
               b)  give oral and written justification for monotonicity of a function on open and closed intervals.
               c)  analyze and describe where a function is concave up/down and how that relates to f ''(x)
               d)  give oral and written justification for concavity of a function (not just a number line.)

          3.  review and discuss symmetry of odd and even functions.

          4.  a)  learn and apply the 1st derivative test to monotonicity.
               b)  learn and apply the 2nd derivative test for concavity.
               c)  learn and apply the 2nd derivative test for local extrema.

          5.  a)  model real world problems and use calculus to optimize desired values.
               b)  use a graphing calculator to solve graphically and confirm analytically various optimizations.

          6.  model real world problems of related rates and use calculus to solve for desired value.

DAY	DATE	Unit Four Topics	ASSIGNMENT
1		Criticals & Extrema	Section 4.1   (1,  3,  5,  7,  8,  17,  18,  19,  21,  23,  29)
2		Monotonicity & Concavity	Section 4.2   (1, 7, 11)   Section 4.3 (1, 3, 5, 7, 9, 29, 31, 33)
3		Odd & Even functions & Review	Section 1.2   (19 – 24)   Section 4.3 (2,  8,  16,  34,  41,  42)
4		Derivative Tests	Section 4.3   (4,  13,  14,  28,  37)
5		Max – Min Optimization	Section 4.4   (1,  3,  5,  9,  11,  20)
6		More Max Min Optimization	Section 4.4   (2,  6,  10,  32,  36)
7		Related Rates	Section 4.6   (1,  3,  8,  11,  22)
8		More Related Rates	Section 4.6   (12,  14,  16,  17,  20)
9		Review for Test	STUDY
10		* * * TEST * * *

Unit 5 – Antiderivatives, Integrals, and Area

     Objectives:  The student will: 

          1.  a)  understand that the derivative of a function was produced from a function (primitive).
               b)  use basic models of integration to find antiderivatives and solve for initial values.
               c)  show that the derivative of the antiderivative is the original function (algebraically)
              
          2.  a)  learn to fit models of integration by multiplying by constant reciprocals (go to the store).
               b)  further develop integration skills by practicing more complex problems (algebraically).

          3.  a)  model word problems with a differential equation and solve for general solution.
               b)  describe a “family of functions” and select the specific solution from initial conditions.

          4.  practice motion problems starting from acceleration and working to position through integration.

          5.  review sum and sigma notation from pre-calculus.

          6.  a)  use geometry to approximate area under curves.
               b)  partition region into various sizes and approximate area by rectangles.

          7.  a)  calculate Riemann Sums by approximating areas (positives and negatives).
               b)  understand the relationship of the Riemann Sum with infinite partitions and integration.

          8.  a)  use the definition of definite integral to calculate exact area under various curves (analytical).
               b)  use a calculator program to view Riemann Sums (LRAM, MRAM, RRAM). (graphical)

          9.  a)  appreciate the beauty of the F.T.C. and its relationship to definite integration.
               b)  apply the F.T.C. from previous Riemann Sum problems to appreciate simplicity.

        10.  use the graphing calculator’s built in feature to find definite integrals of basic functions.

        11.  learn the properties of the definite integral and the Mean Value Theorem.

        12.  gain knowledge by practice with taking the derivative of a function defined by an integral.

        13.  Use the definite integral to calculate area of plane regions bound by two functions (vertical slice).

        14.  Use the definite integral to calculate area of plane regions bound by two functions (horizontal slice).

DAY	DATE	Unit Five Topics	ASSIGNMENT
1		Antiderivatives	Section 6.1   (1,  2,  3,  7,  8,  10,  11,  12,  13)
2		Fitting the Chain Model for Antiderivative	Section 6.2   (4,  6,  8,  9,  28,  45a )
3		Differential Equations	Section 6.1   (27,  28,  30,  31)       Section 6.2  (43)
4		Diff. Eq & Motion (velocity, acceleration, position)	Section 6.1   (36,  41,  53,  55,  56)
5		Sums & Sigma Notation	Quick Review 5.2  #1 - 10
6		Estimating Area under Curve	Section 5.1   (5,  6,  10,  11,  13)  & Worksheet
7		Riemann Sum & Definite Integral	Section 5.2   (1 - 7)       p. 298  #12
8		Riemann Sum & Definite Integral	Worksheet
9		Fundamental Theorem of Calculus	Section 5.4   (3 – 6,  13,  25,  26)
10		Fundamental Theorem of Calculus continued	Section 5.4  (15–18, 52)     pp.298-9  (15, 16, 18-22, 25, 26)
11		Integral Properties & Mean Value Theorem	Section 5.3   (1 – 11 odd,  25,  27,  29)
12		Derivative of Upper Variable Integral	Section 5.4   (19,  35,  37 – 40)
13		Plane Area – Vertical Slicing	Section 7.2   (3, 5, 6, 11, 13,  22,  31)
14		Plane Area – Horizontal Slicing	Section 7.2  (4,  7,  8,  9,  18,  21)
15		Review for Test	Study for Test
16		* * * TEST * * *

Unit 6 – The Calculus of Trigonometry

     Objectives:  The student will: 

          1.  a)  review basic understanding of trigonometry developed in Pre-Calculus.
               b)  convert angle measures from degrees to radians and radians to degrees.
               c)  develop the trig values for special angles on the unit circle and right triangles.
              
          2.  a)  review the trig identities developed in Pre-Calculus that will be used in Calculus.
               b)  use the graphing calculator to gain a conceptual knowledge of basic trig graphs.

          3.  a)  perform basic unit conversions to develop understanding of angular momentum.
               b)  solve arc length and rotation problems.

          4.  evaluate limits of trig functions graphically (calculator),  algebraically (rules), and numerically (table).

          5.  learn rules and properties of derivatives for trig functions.

          6.  extend derivative rules using chain rule containing trig functions.

          7.  a) perform implicit differentiation with equations containing trig functions.
               b) solve related rate problems of angular velocity and trigonometry

          8.  a)  discuss and justify monotonicity, critical points, and extrema of trig graphs.
               b)  discuss and justify concavity and inflection points of trig graphs.

          9.  solve optimization problems from a graphical and algebraic perspective.

        10.  learn the basic integral rules for trigonometric functions.

        11.  continue to develop understanding of area graphically and algebraically using F.T.C.

DAY	DATE	Unit Six Topics	ASSIGNMENT
1		Trigonometry & Unit Circle	Homework Worksheet Day 1
2		Trigonometric. Identities & Graphing	Homework Worksheet Day 2
3		Conversion & Arc Length	Homework Worksheet Day 3
4		Limits	Section 2.1     (26 – 30,  32,  49,  59)
5		Derivatives	Section 3.5     (1 – 15 odd,  16,  19,  31,  32)
6		Chain Rule	Section 3.6     (1,  3,  5,  7,  11,  13,  17,  29,  31)
7		Implicit Differentiation.  &  Related Rates	Section 3.7     (18, 19, 20, 33, 34, 35, 36) &      Section 4.6  (13, 21, 29, 35)
8		Monotonicity & Concavity	Section 4.2     (13,  14)        Section 4.3  (15,  32,  45,  46,  47)
9		Max/Min Optimization	Section 4.4     (15,  21,  27,  31,  34,  35)
10		Integration of Trigonometry.	Section 6.1     (14, 15, 21, 22, 32, 35, 38, 42)   &   Section 5.4  (7, 8, 9)
11		Area	Section 5.4     (27,  28,  51)  &  Section 7.2  (24,  25,  26,  27)
12		Review for Test	Study for Test
13		* * * TEST * * *

Unit 7 – Average Value and Volumes

     Objectives:  The student will: 

          1.  a)  gain an understanding of average value by discovering the relationship to graphical properties.
               b)  understand the average velocity algebraically and graphically.

          2.  use the method of disks to calculate the volume of a solid of rotation about an axis.

          3.  use the method of washers to calculate the volume of a solid of rotation about an axis.

          4.  use the method of disks and washers to calculate volume about arbitrary lines.

          5.  use the method of shells to calculate volume of solid about line or axis. (non AP)

          6.  visualize the solid created by method of slabs with known cross section and compute volume.

          7.  a) work with a group to visualize, describe, and build a model of a solid with known cross sections.
               b) work with a group to calculate volume of the solid the group built.

          8.  a)  read the problem and verbally describe to the class the solid the group created. (Reader)
               b)  design and draw 2D and 3D representations of the solid. (Artist)
               c)  present the model to the class for better understanding of method of slabs (Engineer)
               d)  teach the class how to calculate the volume using the method of slabs (Teacher)

DAY	DATE	Unit Seven Topics	ASSIGNMENT
1		Average Value of Function	pp.  275-276         (26,   28,   30,   40)
2		Volume by Disks around axis	p.  392             (13,   14,   15,   16,   17,   18,   20)
3		Volumes by Washers around axis	p.  392             (21,   22,   23,   26,   34)
4		Volumes by Disks & Washers around line	p.  392             (28,   35,   36,   37)
5		Volumes by Shells	p.  392             (13,   14,   39,   41)
6		Volumes of Known Cross Sections – Slabs	pp.  390-391         (1a,   1b,   1c,   1d,   2b,   7b,   10)
7		Group Design & Construction
8		Group Presentation
9		Review for Test
10		* * * TEST * * *

Unit 8 – The Calculus of the Transcendentals

     Objectives:  The student will: 

          1.  review properties of logarithmic and exponential functions.

          2.  learn and apply the derivative rule of the natural log function.

          3.  examine the relationship of the derivative of a function and the derivative of the inverse function.

          4.  learn and apply the derivative rule and integral models of the natural exponential function.

          5.  learn and apply the derivative rule and the integral models of the general exponential functions.

          6. a) solve differential equations where growth is proportional to amount of substance present.
              b) solve initial value problems of exponential growth and decay.
              c) use a table (numerically) and a graph (graphically) to examine simple and compound interest.

          7. continue to solve growth and decay problems and evaluate long term patterns (limit)

          8.  learn and apply the derivatives and integrals of inverse trig functions.

DAY	DATE	Unit Eight Topics	ASSIGNMENT
1		Review of Logarithm & Exponent	Worksheet C8 – 1
2		Derivative of Natural Log	p. 170 ( 21,  22,  23,  25,  27,  28 )      p. 268 ( 39 )      p. 312 ( 17,  34,  37 )
3		Inverse Functions & Deriv.	Worksheet C8 – 3
4		Natural Exponential Function	p. 170 ( 1, 2, 3, 4, 7, 8, 9, 41, 42 )    p. 275  (15)     p. 313 (33)    p. 358  (7)
5		General Exponential. Functions	p. 170     ( 15,  16,  17,  18,  40 )         Worksheet C8 – 5
6		Exponential Growth & Decay	p. 338     ( 1,  3,  5,  6,  7,  8,  11,  12,  13 )
7		Exponential Growth & Decay (continued)	p. 338-339     ( 15,  16,  20,  22,  28,  29 )
8		Inverse Trig Functions	p. 48-49     ( 7,  8,  9,  10,  21,  31 )     p. 162 Quick Review Exercises 1 – 5
9		Derivative of Inverse Trig	p. 162     (1,  2,  3,  19,  22,  23 )       Worksheet C8 – 9
10		Review for Test	Study for test
11		* * * TEST * * *

Unit 9 – Focus Topics of Calculus

     Objectives:  The student will: 

          1.  a) recognize indeterminate forms of limits that allow l’Hopital’s rule to be applied.
               b) apply l’Hopital’s rule to calculate limits of indeterminate form.

          2.  learn to integrate more complex integrals by applying method of integration by parts. (non AP)

          3.  a) use Newton’s Method of Iteration to find roots of a function.
               b) understand, from a calculus perspective, the process of Newton’s method

          4.  a) graphically approximate a definite integral using Trapezoidal rule.
               b) gain an understanding of how the rule is developed.

          5.  learn about slope fields by manual graphing approximate slopes and guessing the antiderivative.

          6.  use a graphing calculator program to graph slope fields and match with the antiderivative.

          7.  review linearization techniques gained in previous section to approximate change.

DAY	DATE	Unit Nine Topics	ASSIGNMENT
1		L’Hopital’s Rule	p. 423  (1,  2,  4,  5,  6,  8,  9,  15,  17 )
2		Integration by Parts	p. 328  ( 1,  2,  4,  10,  11)
3		Newton’s Method of Iteration	p. 229  (15 – 18)
4		Trapezoidal Rule	p. 295  (2,  3,  4,  5,  8 )
5		Slope Fields	Slope Field Worksheet
6		Slope Fields (cont)	Worksheet #2
7		Linearization & Approximation	p. 229  (1,  3,  5,  7,  28 )
8		Review for Test	Study for test
9		* * * TEST * * *