Academics at Champion

The purpose of this class is to enable students to become proficient in introductory analytic geometry, the theory of limits, differential calculus of algebraic and trigonometric functions, applications of differentiation, anti-derivatives, and the definite integral. This course provides an excellent opportunity for the student to experience a college level mathematics course in a high school setting. Students who are interested in math and who desire to enter the math or science field will find this course challenging and very helpful in their further studies.

PREREQUISITES:

Successful completion of Trigonometry

COURSE OBJECTIVES: During the year long course, students will be able to:

Trigonometry

1.  Recall and use the definitions of the sine, cosine, and tangent of an angle.

2.  2.  Recall and use the 30-60-90~ and 45-45-90~ triangles to find the sine, cosine, and tangent of 30, 45 and 60.

3.  3.  Recall the sign of the sine, cosine, and tangent in the four quadrants of a coordinate axis system.

4.  Know how to find the related angle of an angle greater than 900, and to use the related angle to find the sine, cosine, or tangent of the large angle.

5.  Recall and use the definitions of cosecant, secant, and cotangent as inverse functions of sine, cosine, and tangent.

6.  Understand that for any point (x,y) on the unit circle, x is the cosine, and y is the sine of the corresponding angle.

7.  Use the unit circle to determine the sine, cosine, and tangent of 00, 900, 1800, 2700, and 3600.

8.  Compute the sine, cosine, and tangent of angles greater than 3600.

9.  Know the definition of the radian as angular measure, and be able to from degrees to radians and vice versa.

10.  Sketch sine and cosine curves, given their equation.

11.  Compute powers of trigonometric functions.

12.  Solve trigonometric equations for the value of the ukknown angle(s).

13.  Recall and use the cofunction identities (i.e., sin[90-x] = cos x) and the trigonometric functions of (-x) to solve trigonometric identities.

14.  Derive the main Pythagorean identify (sin2x + cos2x = 1) from the sides of a right triangle and use the main identify to derive the other two Pythagorean identities. Use the identities in solving trigonometric identities.

15.  Use the sum and difference identities for sine and cosine to derive the sum and difference identities for the tangent, as well as the double angle identities for all three functions.

Logic

16.  Understand the parts of a syllogism and recognize valid and invalid arguments.

17.  Use a Venn diagram to show the equivalence of the conditional and the contrapositive, the inverse and the converse.

18.  Use the above equivalencies to evaluate validity of arguments and write valid arguments.

Limits and Continuity

19.  Understand the concept of the limit of a function as x approaches a number c.

20.  State the definition of a limit in terms of the existence and equality of the left and right hand limits.

21.  Manipulate functions in order to find the limit of the function as x approaches c.

22.  Define and determine continuity of a function at a point and on an interval

The derivative

23.  State the definition of the derivative f (x) of a function f(x) and use the definition to find the derivative of basic functions

24.  Understand the concept of the derivative in terms of the slope of the tangent line at a particular x value, and be able to label a graph of the different parts of the definition of the derivative.

25.  Use Pascal’s triangle or the binomial theorem to expand binomials.

26.  Use the definition of a derivative and the process of binomial expansion to find the derivative of xn, where n is a positive integer.

27.  Demonstrate graphically the derivative of ex and 1n½x½ by drawing tangent lines and estimating their slopes.

28.  Demonstrate graphically the derivatives of sin(x) and cos(x) by comparing the graphs of the two functions.

29.  Use the product rule and quotient rule to find the derivatives of fluictions.

30.  Use U substitution and the chain rule to find the derivative of functions.

31.  Find the derivative of trigonometric functions.

32.  Use the first derivative to find critical points of a function.

33.  Use the first derivative to determine whether a critical point is a maximum or minimum point.

34.  Use the second derivative to determine whether a critical point is a maximum or minimum point.

35.  Use the derivative to solve related rate problems, maximization and minimization problems, and falling body problems.

Logarithms/Exponential Functions

36.  Understand the exponential and logarithmic forms.

37.  Use exponential equations to solve exponential growth and decay problems.

38.  Graph exponential and logarithmic equations.

39 Use the three laws of logarithms (product, quotient, power) to solve exponential and logarithmic equations.

40.  Understand and calculate common logarithms and natural logarithms.

Conic Sections

41.  Use the locus definition of a line, a circle, and a parabola to derive the equations of each.

42.  Use the distance formula to write an equation that expresses the locus definition of an ellipse and a hyperbola.

43.  Complete the square as necessary to find the standard form of the equation of a parabola, an ellipse, and a hyperbola.

44.  Graph the equation of a line, a circle, a parabola, an ellipse, and a hyperbola.

Polynomial functions

45.  Use synthetic division and the rational root theorem to find the roots of a polynomial equation and to factor nth-degree polynomial equations into their n linear factors or

combination of linear and non-linear factors.

46.  Make very rough sketches of un-factored polynomial equations.

47.  Make rough sketches of rational polynomial equations with factored numerator and denominator, determining zeroes and asymptotes.

48.  Sketch reciprocal functions.

Integration

49.  Define the indefinite integral of a function f(x) in terms of all possible anti-derivatives of the function.

50.  Integrate by inspection, by the power rule (integral of xn), by guessing, by parts, and by u substitution with change of variable.

51.  Approximate the area under a curve by partitioning the interval and finding upper and lower sums.

52.  Find the exact area under a curve by using the fundamental theorem of calculus.

53.  Distinguish between the indefinite integral and the definite, or Riemann integral.

54.  Use the definite integral to find the area between two curves (functions of x or y).

55.  Use the definite integral to find the volume of a solid formed by rotating a curve about the x or y axis. Use the “disk,” “washer,” and “shell” methods.

56.  Apply the definite integral to find the work done when the force applied is not a constant.

MAJOR ACTIVITIES

1.  Written assignments from textbook

2.  Occasional projects.

MAJOR RESOURCE MATE~ALS:

1.  Calculus with Trigonometry and Analytic Geometry, by Saxon

MEANS OF STUDENT EVALUATION:

1.  Daily assignments.
2.  Quizzes and tests