Find the midpoint of the segment with the given endpoints (5,-9) and (-2,-2)

Another way to solve this is to use the Midpoint Formula.  The midpoint of a segment joining points and is



So the midpoint of your segment is



Perhaps it helps to see that the x-coordinate of the midpoint is just the average of the x-coordinates of the points.  Ditto for the y-coordinate of the midpoint; just average the y's.

C.

yes, part c is completely independent of parts a and b.

they give you information about a point and the slope of a tangent line at that particular point. then they tell you to find the values of a and b.

since the derivative gives the tangent line slope, we can differentiate  with respect to , with   and   as constants.

my reasoning is this: notice how the formula for the derivative from part b was in terms of x and y. we do know that   and   and the value of the derivative (the slope) being , so if we differentiate, we might be able to solve for one of our constants.

note that  because   is a constant, not a function of . so if we differentiated, we could end up solving for .



our derivative has an unknown constant of a. but we can find this constant with our information.

the tangent line at point   has to have a slope of . substitute 1 for , 3 for , and   for .



to get the value of , use this value of a along with the given coordinates on the equation of the curve.



i confirmed my answer using a graphing calculator like desmos, attached. it seems like a reasonable tangent line.