Preparing for University Calculus - Math and Computer Science

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3.8 Linear GraphsYou should be able to graph linear functions and inequalities, determine the slopeand intercept of a line from its equation and vice versa, determine where two linesmeet, use the negative-reciprocal rule for orthogonal lines, and find the distancebetween two points.Many of these ideas will be conceptually important in calculus, which deals alot with slopes, tangent lines, secant lines, etc.Examples:1. Find the point of intersection of the lines x = 2 and x + y =5.Solution: This does not require a graph to be drawn. Remember that theline (or curve) corresponding to an equation is the set of points (x, y) forwhich the equation is true; and the intersection of two lines or curves is theset of pairs (x, y) for which both equations are true.So we need to find a pair (x, y) such that x = 2 and x + y = 5. The firstequation gives us the value of x immediately; plugging that into the otherequation we get 2 + y =5,y = 3, so the answer is (x, y) =(2, 3).2. Find the slope of the line through the points (−5, 0) and (3, 2).Solution: The “rise” is 2 − 0 = 2 and the “run” is 3 − (−5) = 8.Slope = rise/run = 2/8 =1/4.3. Find the equation of the line through (5, 0) and orthogonal to x +2y =3.Solution: First we find the slope of the given line x +2y = 3. To find theslope we isolate y in the equation, rewriting it first as 2y =3− x and then asy = 3 2 − 1 2 x. Thus the given line has slope m = − 1 2 .The line we are looking for is orthogonal to the given line, so its slope is thenegative reciprocal of the first line’s slope. −1/(− 1 2) = 2, so we are lookingfor a line with slope 2, hence with eqation y =2x + b.Every point on that line satisfies that equation, so 0 = 2 × 5+b. Solving, wefind that b = −10, and our equation isy =2x − 10 .24
Problems: (answers on page 40)1. Find the slope of the line 2y = x − 22. Give the equation of a line with slope 3, through (0, 0).3. If a line has slope 3/2 and passes through (2, 2), give its equation.4. Find the distance from the point (0, 3) to the point (3, 0)5. Find the slope of the line through the points (0, 3) and (3, 0).6. Find the point of intersection of the lines y =5− x and y =2x +27. Find the equation of the line with slope −1/2 and y-intercept 38. Find the slope of a line orthogonal to the line y =3x +5.9. Find the equation of the line through (3, 2) and (1, 0)10. Which inequality is this the graph of?10-1-11(a) y>2 (b) x>y+ 2 (c) x2 (e) x + y>211. Which of these lines is parallel to 2y =6x − 1?(a) 2y =5x − 1 (b) y =6x + 1 (c) y =3x +1(a) 2y =3x + 2 (a) 2y = −6x +112. Which of these lines is orthogonal to y =3x − 1?(a) y = −3x − 1 (b) y = −x/3 + 1 (c) y = x/3+1(d) 3x = y + 2 (e) y =1− 3x25
Page 1 and 2: Preparing for University CalculusPr
Page 3 and 4: 1 About First-Year Calculus1.1 What
Page 5 and 6: 1.5 What is taking university calcu
Page 7 and 8: • Your university or student unio
Page 9 and 10: 2 Ordering informationGetting a cop
Page 11 and 12: Problems: (answers on page 40)1. Fi
Page 13 and 14: Problems: (answers on page 40)1. Si
Page 15 and 16: Problems: (answers on page 40)1. If
Page 17 and 18: Problems: (answers on page 40)1. If
Page 19 and 20: Problems: (answers on page 40)1. So
Page 21 and 22: Problems: (answers on page 40)11. S
Page 23: Problems: (answers on page 40)1. Ra
Page 28 and 29: 3.10 Exponents and rootsYou should
Page 30 and 31: 3.11 LogarithmsLogarithms are extre
Page 32 and 33: 3.12 Geometry and basic trigonometr
Page 34 and 35: 3.13 Trigonometric identitiesThere
Page 36 and 37: 3.14 Problem solvingSolving applied
Page 38 and 39: 3.14159265358979... Harder Problems
Page 40 and 41: Answers to problems:Section 3.1 (1)
Page 42 and 43: Some useful facts:Algebraic identit

Preparing for University Calculus - Math and Computer Science

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