Lesson Tangent lines
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A function y = f(x) and an x-value are given. (a) Find a formula for the slope of the tangent line to the graph of f at a general point {{{x=x[0]}}} (b) Use the formula obtained in part (a) to find the slope of the tangent line for the given value of {{{x[0]}}} 16. a. {{{f(x)=x^2+3x+2; x[0]=2}}} {{{lim(x->x[0],(f(x)-f(x[0]))/(x-x[0]))}}} use this formula to evaluate the limit to find the slope of the tangent line. {{{lim(x->x[0],((x^2+3x+2)-(x[0]^2+3x[0]+2))/(x-x[0]))}}} distribute the negative {{{lim(x->x[0],(x^2+3x+2-x[0]^2-3x[0]-2)/(x-x[0]))}}} the 2s cancel {{{lim(x->x[0],(x^2+3x-x[0]^2-3x[0])/(x-x[0]))}}} rearrange the terms {{{lim(x->x[0],(x^2-x[0]^2+3x-3x[0])/(x-x[0]))}}} factor using difference of squares {{{lim(x->x[0],((x-x[0])(x+x[0])+3(x-x[0]))/(x-x[0]))}}} cancel all {{{x-x[0]}}} factors {{{lim(x->x[0],(x+x[0]+3))}}} now substitute {{{x[0]}}} for x {{{x[0]+x[0]+3}}} {{{2x[0]+3}}} this is the general formula for the slope of the tangent line at any {{{x[0]}}} b. at {{{x[0]=2}}} {{{mtan=2(2)+3=7}}} the slope of the tangent line at x=2 is 7.
