Curriculum for the high school grades (9-12) for the state of North Carolina (US) : Curriculum for the high school grades (9-12) for the state of North Carolina (US)
Objectives : Objectives This module explains the curriculum for the high school grades (9-12) for the state of North Carolina (US) Country: United States
State: North Carolina
Time Zone: Eastern Standard Time (GMT-5) till 11th March 2007
Subject: Mathematics
Topic: Advanced Placement Calculus
Language: English
Grade: Grade 9-12
Introduction : Students will be expected to
Understanding of the concepts of calculus (functions, graphs, limits, derivatives and integrals)
Provides experience with its methods and applications
The course encourages the geometric, numerical, analytical, and verbal expression of concepts, results, and problems
Appropriate technology, from manipulative to calculators and application software, should be used regularly for instruction and assessment Introduction
Contents : Advanced Placement Calculus includes:
Understanding of the behavior of functions, graphs, limits, derivatives and integrals
Use of derivatives to solve problems
Use of integrals to solve problems Contents
Pre-requisites : What your student must already know?
Use of circle, trigonometric, and inverse trigonometric functions to solve problems.
Use of the trigonometric ratios and the laws of sines and cosines to solve problems.
Describe graphically, algebraically and verbally phenomena as functions; identifying independent and dependent quantities, domain, and range.
Translate among graphic, algebraic, tabular, and verbal representations of relations.
Pre-requisites
Pre-requisites : What your student must already know? (contd.)
Use of functions (linear, polynomial, exponential, logarithmic, rational, power, piecewise) to model and solve problems.
Use of the composition and inverse of functions to model and solve problems.
Transform relations in two and three dimensions; describe algebraically and/or geometrically the results.
Use of the conic relations to model and solve problems.
Write equivalent forms of algebraic expressions.
Find special points (zeros, intercepts, asymptotes, local maximum, local minimum, etc.) of relations and describe in the context of the problem. Pre-requisites
Competency Goal 1 : Competency Goal 1 1.01 Demonstrate an understanding of limits both local and global.
a) Calculate limits, including one-sided, using algebra.
b) Estimate limits from graphs or tables of data. Sample Problem: Solution: I f f(x) = 5x – 4 when x < 2 = 2 (x 2 – 1) when x > or = 2 Then find, LH ) x ( f 2 x Lt ® = ) h 2 ( f 0 h Lt - ® = 6 4 h) 5(2 0 h Lt = - - ® RH ) x ( f 2 x Lt ® = ) h 2 ( f 0 h Lt + ® = 6 1 2 ) h 2 ( 2 0 h Lt = - + ® Again, f(2) = 2 (2 2 – 1) = 6 As we find LHL = RHL = f(2) We have ) x ( f 2 x Lt ® = 6 Answer
Competency Goal 1 : 1.02 Recognize and describe the nature of aberrant behavior caused by asymptotes and unboundedness.
a) Understand asymptotes in terms of graphical behavior.
b) Describe asymptotic behavior in terms of limits involving infinity.
c) Compare relative magnitudes of functions and their rates of change. Competency Goal 1 Sample Problem: Solution: A straight line, at a finite distance from origin, is said to be an asymptote of the curve y = f(x) if the perpendicular distance of the point P on the curve from the line tends to zero whenx or y both tends to infinity. Y O P M Asymptote of the curve X
Competency Goal 1 : 1.03 Identify and demonstrate an understanding of continuity of functions.
a) Develop an intuitive understanding of continuity. (Close values of the domain lead to close values of the range.)
b) Understand continuity in terms of limits.
c) Develop a geometric understanding of graphs of continuous functions. (Intermediate Value Theorem and Extreme Value Theorem). Competency Goal 1 Sample Problem:
Show that the function f(x) = + Cos x, x = 2, x = 0 i s continuous at x = 0.
Competency Goal 1 : Competency Goal 1 Solution: We have LHL = f(x) = f(0 - h) = f( - h) = + Cos( - h) = + Cos h = 1 + 1 = 2
Competency Goal 1 : Competency Goal 1 Solution: We have RHL = f(x) = f(0+h) = f(h) = + Cos h = 1 + 1 = 2 Also we have, = 2 Therefore, f(x) = f(x) = f(0) Hence, f(x) is continuous at x = 0.
Competency Goal 2 : Competency Goal 2 Sample Problem: 2.01 Explore and interpret the concept of the derivative graphically, numerically, analytically and verbally.
a) Interpret derivative as an instantaneous rate of change.
b) Define derivative as the limit of the difference quotient.
c) Identify the relationship between differentiability and continuity. A balloon, which always remains spherical, has a variable radius. Find the rate at which its volume is increasing with respect to its radius when the radius is 7 cm.
Competency Goal 2 : Competency Goal 2 Solution: Let x be the radius and y be the volume of the balloon. Then y = = 196 p cm 2 Hence the volume is increasing w.r.t. its radius at the rate of 196 p cm 2 , when the radius is 7 cm.
Competency Goal 2 : Competency Goal 2 Sample Problem: 2.02 Apply the concept of the derivative at a point.
a) Find the slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.
b) Find the tangent line to a curve at a point and local linear approximation.
c) Find the instantaneous rate of change as the limit of average rate of
change.
d) Approximate a rate of change from graphs and tables of values. Find the slope of the curve y = 7x – x 3 at x = 2
Competency Goal 2 : Competency Goal 2 Solution: Slope of the curve = = 7 – 12 = - 5
Competency Goal 2 : Competency Goal 2 Sample Problem: Solution: Find the equation of the tangent of the curve y = 2x 2 – 3x – 1 at (1, 2) Given curve is y = 2x 2 – 3x – 1 = 1
Competency Goal 2 : Competency Goal 2 Solution (contd.): We know the equation of the tangent at p(x 1 , y 1 ) to the curve y = f(x) is y – y 1 = (x – x 1 ) Hence, the required tangent is y – ( - 2) = 1 ( x – 1) or, y + 2 = x – 1 or, x – y = 3
Competency Goal 2 : Competency Goal 2 Sample Problem 2.03 Interpret the derivative as a function.
a) Identify corresponding characteristics of graphs of ƒ and ƒ'.
b) Identify relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ'.
c) Investigate the Mean Value Theorem and its geometric consequences.
d) Translate between verbal and algebraic descriptions of equations involving derivatives. Show that the function f(x) = 2x + 3 is strictly increasing function on R.
Competency Goal 2 : Competency Goal 2 Solution: Let x 1 , x 2 Î R and next let x 1 < x 2 Then 2x 1 < 2x 2 or, 2x 1 +3 < 2x 2 +3 or, f(x 1 ) < f(x 2 ) for all x 1 , x 2 Î R so f(x) is strictly increasing function on R.
Competency Goal 2 : Competency Goal 2 Solution: Sample Problem Prove that f(x) = x 3 – 3x 2 + 3x – 100 is increasing on R. We have f(x) = x 3 – 3x 2 + 3x – 100 T herefore, f ’ (x) = 3x 2 – 6x + 3 or, f ’ (x) = 3(x 2 – 2x + 1) or, f ’ (x) = 3(x – 1) 2 Now x Î R Þ (x – 1) 2 ³ 0 Þ f ’ (x) ³ 0 Thus f(x) is increasing on R.
Competency Goal 2 : Competency Goal 2 Solution: Sample Problem: Using Lagrange’s (first) MVT, find a point on the curve defined on the interval [2, 3] where the tangent in parallel to the chord joining the end points of the curve. Let f(x) = . Since for each x Î [2, 3], the function f(x) attains a unique definite value. So, f(x) is continuous on [2, 3]. Also, f’(x) = exists for all x Î [2, 3]. So, f(x) is differentiable on (2, 3).
Competency Goal 2 : Competency Goal 2 Solution: (contd.) Thus, both conditions of first MVT are satisfied. Consequently there must exists some C Î [2, 3] such that f’(c) = Now, f(3) = 1 and f(2) = 0 or, 4(x - 2) = 1 ( squaring both sides) or, x = 9/4 thus, c = 9/4 Î [2, 3] such that f’(c) = 1 now, f(c) = ½ thus, (c, f(c)) i.e. (9/4, ½) is a point on the curve such that the tangent at it is parallel to the chord joining th e end points of the curve.
Competency Goal 2 : Competency Goal 2 Sample Problem: Solution: Find the derivative of 5 x y = 5 x = 5 x log 5.
Competency Goal 2 : Competency Goal 2 Sample Problem: Solution: Sample Problem: Solution: Find the derivative of x 4 y = x 4 = 4x 3 Find = = = e x cos x + sin x. e x + x n ( - sin x) + nx (n - 1) cos x = e x (cos x + sin x) - x n sin x + nx (n - 1) cos x
Competency Goal 2 : Competency Goal 2 Sample Problem: Solution: Find the Second derivative of x 4 y = x 4 y’ = = 4x 3 Now, y” = = 4. 3x 2 = 12 x 2
Competency Goal 2 : Competency Goal 2 Sample Problem: If log(x2+y2) = 2 tan-1(y/x), Show that,
Competency Goal 2 : Competency Goal 2 Hence proved ( ) ÷ ø ö ç è æ ÷ ø ö ç è æ + = + + Þ x y dx d . x y 1 1 . 2 y x dx d y x 1 2 2 2 2 2 ( ) ( ) ( ) ( ) y x y x dx dy y x x y dx dy y dx dy x 2 dx dy y x 2 - + = Þ + - = - Þ ÷ ø ö ç è æ - = ÷ ø ö ç è æ + Þ
Competency Goal 3 : Competency Goal 3 Sample Problem Solution: Evaluate: = = = = Answer
Competency Goal 3 : Competency Goal 3 Sample Problem Solution: Evaluate: Let I = -------------------- (1) Then, I = Or, I = ---------------------- (2) A d din g (1) & (2), we get 2I = = = = Therefore , I = 0
Competency Goal 3 : Competency Goal 3 Sample Problem Solution: Evaluate: Let, I = [put, x = a tan q ; Therefore, dx = a sec q ] Hence, dx = a sec 2 q d q , for x = 1, q = and for x = 0, q = 0 = =
Competency Goal 3 : Competency Goal 3 Sample Problem Solution: Evaluate: = = = = (1 – 0) = 1 Answer
Competency Goal 3 : Competency Goal 3 Sample Problem: 3.04 Define and use appropriate integrals in a variety of applications.
a) Interpret the integral of a rate of change to give accumulated change.
b) Find specific anti-derivatives using initial conditions.
c) Set up and use an approximating Riemann sum or trapezoidal sum and represent its limit as a definite integral.
d) Find the area of a region.
e) Find the volume of a solid with known cross sections.
f) Find the average value of a function.
g) Find the distance traveled by a particle along a line.
h) Solve separable differential equations and use them in modeling. In particular, study the equation y' = ky and exponential growth. Using integration find the area of the region bounded by the line x = 4 & the parabola y2 = 16 x.
Competency Goal 3 : Competency Goal 3 Solution: O Y X A B C(0, 4) Draw the line x = 4 & the curve y 2 = 16 x. As y 2 = 16 x is symmetric about the x - axis Therefore the required area = 2 (Area OCAO). Now the dimensions of the small vertical rectangle are Length = y , width = ?x and area = y?x The approximating area can m ove between x = 0 and x = 4. Therefore required area = 2 (Area OCAO) = 2 = 2 [ Since, y 2 = 16 x therefore, y = ] = 8 = 8 = = = sq units