Schaum's Outline of Theory and Problems of Beginning Calculus

CHAP. 113THE SLOPE OF A TANGENT LINE89ALGEBRAa c ad bc ad-bc---=---=bd bd bd bdand---h-rO1= --= 1 _- 1lim (x + h) lim x x x x2h-0 h+OThus, the slope of the tangent line at (x, l/x) is - 1/x2.From part (a), the slope of the tangent line at (2, $) is1 1 1--=--=--x2 22 4Hence, the slope-intercept equation of the tangent line 9 has the form1y=--x+b4Since 9 passes through (2, i), substitution of 2 for x and 4 for y yieldsHence, the equation of 9’ is-- 1 1 1 12--4(2)+b or -=--+b 2 2 or b=l1y=--x+l4Find a formula for the slope of the tangent line at any point of the graph of the functionfsuch thatf(x) = 3x2 - 6x + 4.Find the slope-intercept equation of the tangent line at the point (0,4) of the graph.Draw the graph offand show the tangent line at (0,4).Computef(x + h) by replacing all occurrences of x in the formula forf(x) by x + h:f(x + h) = 3(x + h)2 - 6(x + h) + 4= 3(x2 + 2xh + h2) - 6x - 6h + 4= 3x2 + 6xh + 3h2 - 6~ - 6h + 4f(x) = 3x2 - 6~ + 4f(x + h) -f(x) = (3x2 + 6xh + 3h2 - 6x - 6h + 4) - (3x2 - 6x + 4)= 3x2 + 6xh + 3h2 - 6x - 6h + 4 - 3x2 + 6x - 4= 6xh + 3h2 - 6h = h(6x + 3h - 6)f(x + h) -f(x) - h(6x + 3h - 6) = 6x + 3h - 6hhThus, the slope of the tangent line at (x,f(x)) is 6x - 6.