2.1 rates of change and tangent lines to curves

how do you find the eqn of the tangent line at a specified point on the curve

1) see if the given point is even on the graph by plugging in the x to see if it yields the y provided 2) calculate slope of tangent line 3)plug into slope intercept form to create eqn w given point

how to find slope of a secant line when given an eqn and an interval

1) you are only given two x's bc it's an internal, t so you only have xsub1 and xsub2, so plug the given x's in to get your ysub1 and ysub2 2)do change in y divided by change in x, so ysub2 - ysub1 all divided by xsub2 - xsub1

how do you find the slope of the tangent line?

Method: 1)use ∆y/∆x = (f(x2)−f(x1)) / (x2−x1) = (f(x1+h)−f(x1)) / h 2)simplify whichever way you can to get h out of the denominator because you are going to do the limit as h approaches 0 to get the slope of the tangent line. (i.e you can factor) 3) once you get h out of the denominator, plug 0 in to evaluate the limit as h approaches 0. the value you get is the slope of the tangent line Idealogical process: -secant lines become tangent lines when x approaches 0, so you take a secant line, use the above formula and do the limit as h approaches 0 because that secant line then becomes a tangent line.

what is the formula to expand (2+h)^3 or basically sum of cubes

a^3 + 3a^2b + 3ab^2 + b^3

what is the formula formula for the difference of cubes?

a^3 - 3a^2b +3ab^2 - b^3

how do you answer, "find the slope of the curve by finding the limiting value of the slope of the secants through p" given y=x^3 + 1 and P(2,9)

secant lines become tangent lines when x approaches 0, therefore find the slope of the tangent line