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Term 2 Grade 11 Core Project

Term 2 Grade 11 Core Project. Student Name: ATEEQ SAEED Section : 11.04 Student Name: ALI SAEED Section : 11.04. Approximation by Means of Polynomials.

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Term 2 Grade 11 Core Project

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  1. Term 2 Grade 11 Core Project Student Name: ATEEQ SAEED Section: 11.04 Student Name: ALI SAEED Section: 11.04

  2. Approximation by Means of Polynomials • Many scientific experiments produce pairs of numbers [x, p(x)] that can be related by a formula. If the pairs form a function, you can fit a polynomial to the pairs in exactly one way. • For 2 pairs of numbers you can write a polynomial of degree 1. px=A+Bx-x0 • For 3 pairs of numbers you can write a polynomial of degree 2. px=A+Bx-x0+Cx-x0x-x1 • For 4 pairs of numbers you can write a polynomial of degree 3. px=A+Bx-x0+Cx-x0x-x1+Dx-x0x-x1(x-x2) And so on.

  3. Example: For the following 4 pairs we can find a polynomial of degree 3.

  4. To find the polynomial p(x): A. Substitute the given values into this expression. And find the values of A, B, C, and D px=A+Bx-x0+Cx-x0x-x1+Dx-x0x-x1(x-x2) -5=A 13=A+B4-1 51=A+B5-1+C5-1(5-4) 211=A+B7-1+C7-17-4+D7-17-4(7-5) Or -5=A 13=A+3B 51=A+4B+4C 211=A+6B+18C+36D

  5. B. We solve the above system by substitution to get: A=-5, B=6, C=8, and D=1 C. To find the polynomial we substitute the values of A, B, C, D, x0 , x1 , and x2 in px=A+Bx-x0+Cx-x0x-x1+Dx-x0x-x1(x-x2) and simplify: px=-5+6x-1+8x-1x-4+1x-1x-4(x-5) px=x3-2x2-5x+1

  6. Task 1: Find the polynomial that gives the following values px=A+Bx-x0+Cx-x0x-x1+Dx-x0x-x1(x-x2)

  7. a. Write the system of equations in A, B, C, andD that you can use to find the desired polynomial. 10 =A -6 =A+B (X1-X0) (1-(-1)) -17=A+B (X2 –X0)+C (X2-X0)(X2-X1) (2-(-1)) + C (2-(-1)(2-1) 82 =A+B (X2 –X0)+C (X2-X0)(X3-X1)+D (X3-X0)(X3 –X1)(X3-X2) (5-(-1)+C (5-(-1))(5-1)+D (5-(-1))(5-1) (5-2) 10 = A -6 = A + 2B -17 = A + 3B + 3C 82 = A + 6B + 24C + 72D b. Solve the system obtained from part a. A = 10 -6=10+B (1-(-1)  2B=-16  B=-8 B = -8 -17=10+-8(2-(-1)+C (2-(-1)(2-1)  3C=-3C=-1 C = -1 82=10+-8(5-(-1)+-1(5-(-1)(5-1)+D (5-(-1)(5-1)(5-2) D = 2 72D=144 D=2

  8. c. Find the polynomial that represents the four ordered pairs. px=10+-8x+(-1+-1x+(-1x-1+2x+(-1x-1(x-2) px=2x3-5x2-10x+7 =10-8X-8-X2+1+(X3-2X2-X+2) =3-X2-8X+2X3-4X2-2X+4 =2X3-5X2-10X+7 d. Write the general form of the polynomial of degree 4 for 5 pairs of numbers.   px=A+Bx-x0+Cx-x0x-x1+Dx-x0x-x1(x-x2)+Ex-x0x-x1(x-x2)(x-x3) =EX4+(2-7E) X3+(9E-5) X2+(7E-10) X+(7-10E)

  9. The Bisection Method for Approximating Real Zeros The bisection method can be used to approximate zeros of polynomial functions like fx=x3+x2-3x-3 (To the nearest tenth) • Since f (1) = -4 and f (2) = 3, there is at least one real zero between 1 and 2. • The midpoint of this interval is 1.5 • Since f(1.5) = -1.875, the zero is between 1.5 and 2. • The midpoint of this interval is 1.75. • Since f(1.75) is about 0.172, the zero is between 1.5 and 1.75. • The midpoint of this interval is 1.625 • Since f(1.625) is about -0.94. The zero is between 1.625 and 1.75. • The midpoint of this interval is 1.6875. • Since f(1.6875) is about -0.41, the zero is between 1.6875 and 1.75. • Therefore, the zero is 1.7 to the nearest tenth.

  10. The diagram below summarizes the results obtained by the bisection method.

  11. Task 2: Find the zeros of the polynomial found in task 1. • a. Show that the 3 zeros of the polynomial found in task 1 are: First zero lies between f (-2)=1, f (-1)=10  midpoint = -1.5 Second zero lies between f (0)=7, f (1) =-6  = 0.5 Third zero lies between f (3)=-14, f (4)=15  = 3.5 b. Find to the nearest tenth the third zero using the Bisection Method for Approximating Real Zeros. px=2x3-5x2-12x+7 • Since f (3) = -20 and f (4) = 7, there is at least one real zero between 3 and 4. • The midpoint of this interval is 3.5. • Since f (3.5) = -10.5, the zero is between 3.5 and 4. • The midpoint of this interval is 3.75 • Since f (3.75) is about -2.843, the zero is between 3.5 and 3.75. • The midpoint of this interval is 3.62 • Since f (3.62) is about -7.086. The zero is between 3.62 and 3.75. • The midpoint of this interval is 3.68 • Since f (3.68) is about -5.199, the zero is between 3.68 and 3.75. • Therefore, the zero is 3.7 to the nearest tenth.

  12. Task 3: Real World Construction You are planning a rectangular garden. Its length is twice its width. You want a walkway w feet wide around the garden. Let x be the width of the garden. a. any value for the width of the walkway w that is less than 6 ft. W = 1 ft. B . Write an expression for the area of the garden and walk. width=x , Length=2x Garden Area =W×L= x × (2x)= 2x2 All Area=x+82x+8 All Area=fx=2x2+16x+64

  13. c. Write an expression for the area of the walkway only. =2(f + w ) =2(2w+w) =2(3w) =(3x) =6x Ft d. You have enough gravel to cover 1000ft2 and want to use it all on the walk. How big should you make the garden? 1000 = 2w2 1000=2x2 500=x2 22.36 Ft=X .:F= 2X =2X22.36 F=44.72 FT

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