Question

Find the slope of the curve yequals=x2minus−33xminus−55 at the point ​p(22​,negative 7−7​) by finding the limit of the secant slopes through point p.

Answer 1

`(dy)/(dx) = \lim_(h \to 0) (f(x + h) - f(x))/(h)`

`= \lim_(h \to 0) ((x + h)^(2) - 33(x + h) - 55 - x^(2) + 33x + 55)/(h)`

`= \lim_(h \to 0) (x^(2) + 2xh + h^(2) - 33x - 33h - 55 - x^(2) + 33x + 55)/(h)`

`= \lim_(h \to 0) (2xh + h^(2) - 33h)/(h)`

`= \lim_(h \to 0) 2x + h - 33`

`= 2x - 33`


`\text{At x = 22, } (dy)/(dx) = 2(22) - 33 = 44 - 33 = 11`

Thus, the slope at P is 11.