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Do Now:

Do Now:. What do you know about the following: Slope x-intercept, y-intercept Linear equations Calculus. Lines. Increments. Calculus has proven to be useful for relating the rate of change of a quantity to the graph of the quantity.

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Do Now:

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  1. Do Now: • What do you know about the following: • Slope • x-intercept, y-intercept • Linear equations • Calculus

  2. Lines

  3. Increments • Calculus has proven to be useful for relating the rate of change of a quantity to the graph of the quantity. • In order to begin explaining this relationship we must begin with the slopes of lines.

  4. Increments • When a particle in the plane moves from one point to another we must use the starting point and stopping point to discuss change.

  5. Increments • If a particle moves from point (x1,y1) to the point (x2,y2), the increments in its coordinates are: x = x2 – x1 and y = y2 – y1

  6. Increments Let a particle move from P1(x1, y1) to P2(x2, y2). The increments in its coordinates are Dx = (x2 – x1) and Dy= (y2 – y1). Ex. The coordinate increments from (4, -3) to (2, 5) are: Solution Dx = (x2 – x1) = 2 – 4 = -2 and Dy = (y2 – y1) = 5- -3 = 8. Ex. The coordinate increments from (5, 6) to (5, 1) are: Solution Dx = (x2 – x1) = 5 – 5 = 0 and Dy = (y2 – y1) = 1- 6 = -5.

  7. Slope • Each nonvertical line has a slope, and we can use increments to calculate our slope.

  8. Slope • Let P1(x1, y1) and P2(x2, y2) be points on a nonvertical line, L. The slope of L is

  9. Parallel and Perpendicular lines m1 = m2 PERPENDICULAR PARALLEL

  10. Equations of vertical & horizontal lines Ex. Find the equations of the vertical and horizontal lines that pass through the point (2, 3) Solution (2, 3) Y = 3 x = 2

  11. Point-slope Equation The equation y - y1 = m(x –x1) is the point slope formula with m = slope and (x1, y1) is a pt on line • Ex. Find the equation of the line that passes through (-1, 2) and is • Parallel to y = 3x – 4 • Perpendicular to y = 3x - 4 Solution • y – 2 = 3(x - -1) => y = 3x + 5 b) Y – 2 = (-1/3)*(x - -1) => y = -x/3 + 5/3

  12. Slope-intercept Equation The equation y = mx + b is the slope-intercept formula with m = slope and b = y-intercept Ex. Find the equation of the line through (-1, 2) that passes through (0, 5). Solution m = (5 – 2)/(0 - –1) => m = 3. Since (0, 5) is the y-intercept, we get y = 3x + 5.

  13. 8 X X 5 General Linear Equation The equation Ax + By = C (A and B not both 0) is the general linear equation. Sketch the line 8x + 5y = 40 Solution Substitute x = 0 => y = 8; substitute y = 0 => x = 5. Y X

  14. Linear Regression Using the best fitting line to predict future trends Ex Use a linear model of the data in the table to predict the population in the year 2010.

  15. Method 1 Draw a scatter plot by hand and overlap the best fitting straight line. Best fit

  16. Method 1, continued Find the equation of the line and use its equation to predict population in 2010. • Eqn: Use point slope formula. Use 2 pts on best line. • (1986, 4936mil), (1990, 5300mil) • m = (5300-4936)/(1990-1986) = 364/4 = 91 (mil/yr) • y – 4936mil = 91mil(x – 1986) • y = 91mil*x - 175790mil. • In 2010, the population will be 91(2010) – 175790 • Pop = 7120 mil <= INACCURATE

  17. ENTER STO Method 2 Use the capabilities of the TI-89 calculator. To simplify, let x = 0 represent 1986, x = 1 represent 1987 etc. {0, 1, 2, 3, 4, 5} -> L1 ENTER 0, 1, 2, 3, 4, 5 2nd { 2nd } alpha L 1 (Upper case L used for clarity.)

  18. ENTER ENTER STO Method 2, continued {4936, 5023, 5111, 5201, 5329, 5422} -> L2 ENTER 4936, 5023, 5111, 5201, 5329, 5422 2nd { 2nd } alpha L 2 (Upper case L used for clarity.) LinReg L1, L2 ENTER 6 3 2 alpha L 1 , alpha L 2 2nd MATH The calculator should return: LinReg Done Statistics Regressions

  19. ENTER Method 2, continued 6 8 2nd MATH Statistics ShowStat The calculator gives you an equation and constants: y = ax + b a = 98.228571, b = 4924.761905, corr = 0.997826 R2 = 0.995658

  20. WINDOW ENTER ENTER Y= Method 2, continued Use calculator - plot new curve & the original points: x y1=regeq(x) ) regeq 2nd VAR-LINK Plot 1 Use alpha etc. Type in L1 and L2

  21. WINDOW GRAPH Method 2, continued Xmin = 0 Xmax = 5 Xsc = 1 Ymin = 4936 Ymax = 5422 Ysc = 1 Xres = 1 produces the graph

  22. Method 2, continued Go to the homescreen 2nd QUIT To get the population in 2010, note that 2010 is 24 years after 1986. So we enter y1(24) at homescreen and obtain 7282.25 (mil). Compare this value with the value obtained using the “by hand” method p

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