Ch 1 Intro to Calculus Notes - 1.a Chapter 1 Calculus...

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1 Chapter 1, Calculus Calculus A study of functions which can be approximated by functions which are linear. Factoring a 2 b 2 = (a b)(a + b) a 2 + 2ab + b 2 = (a + b)(a + b) = (a + b) 2 a 3 b 3 = (a b)(a 2 + ab + b 2 ) a 3 + b 3 = (a + b)(a 2 ab + b 2 ) a 4 b 4 = (a 2 b 2 )(a 2 + b 2 ) = (a b)(a + b)(a 2 + b 2 ) Factor Theorum If a polynomial, P(x), has (x b) as a factor, the P(b) = 0. Synthetic division 3 3 2 27 0 0 27 3 3 x x x x x x 3 1 0 0 27 3 9 27 1 3 9 0 i.e. since Dividend = Divisor Quotient + Remainder (x 3 27) = (x 3)(1x 2 + 3 x + 9) + 0 When factoring fractions: To get rid of the fractions in the final term factor out the least common multiple of the denominator. ( The new term is obtained by multiplying the original term by the reciprocal of this factor) 1. 1 2 1 (5 6 ) 3 5 15 x y x y Note 1 1 1 3 5 15 so we multiply through by 15. 2. 3 9 3 3 40 3 40 9 40 3 8 5 10 40 3 8 3 5 3 10 x y z x y z 3 5 24 4 40 x y z When taking out the common factor of the variable or a function as a common factor we take out the least power of the common factor. ( Least = farthest left on the number line) 3. x 5 3x 2 = x 2 (x 3 5) 4. x 2 2x 3 = x 3 (x 5 2) When factoring we can always check by multiplying (Remember we add exponents when multiplying) 1.a
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2 5. 2 3 4 4 7 2 7 1 3 6 6 6 6 3 6 2 1 1 x x x x x x x x 6. 1 3 1 2 3 2 2 4 5 1 5 0 2 4 2 4 4 4 4 4 4 2 4 2 1 4 3 1 1 6 4 3 6 4 3 3 2 6 6 6 6 x x x x x x x x x x x x x 7. 3 7 7 7 4 4 4 4 2 1 4 2 1 2 1 (2 1) 4 2 1 2 3 x x x x x x Slopes, Tangents to Curves Slope = m = 2 1 2 1 y y rise y run x x x If we use a variable point P(x,y) and the slope, m, is known the equation of the line comes from the point slope formula : 1 1 y y m x x For parallel lines m 1 = m 2; for perpendicular lines m 1 m 2 = 1 or 1 2 1 m m 1. What is the equation of a line perpendicular to y = 2x + 4 if this perpendicular line is on (6, 2)? y = 2x + 4 is of the form y = mx + b and has slope = m = 2 slope perpendicular to m = 2 is m = 1 2 m line on (6, 2) with this slope is 1 1 y y m x x 1 2 2 6 y x 1(x 6) = 2(y 2) 0 = x + 2y 10 or 1 5 2 y x 2. What is the equation of a line parallel to 4x + 2y 7 = 0 and having an x intercept at 3. 4x + 2y 7 = 0 2y = 4x + 7 y = 2x + 3.5 desired line has m = 2 and is on ( 3, 0) 1 1 y y m x x 0 2 3 y x y = 2x 6 1.b 5 5 0 -5 -5 5 0 new line
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3 Equation of a Tangent Line to a Curve For a curve the slope of the tangent to the curve is taken as the limiting value of the slopes of a sequence of secant lines for which the point of tangency is fixed as one end of all the secant lines. Tangent touches but does not cut a curve Secant Cuts a curve in 2 or more points Chord Line segment between secant line intercepts 1. What is the equation of the tangent to y = x 2 at the point P(2, 4)? We could use either side but we will find the slope of PQ n as Q n approaches P from the left.
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