Question

conider the graph defined by
y=x-2/x-3
(a) Use the definition of the derivative to find the slope of the tangent line to the graph at the point (4, 2).
slope =

Answer 1

use below to find the slope of tangent to y = f(x), at x = a :

slope = \(\large \lim \limits_{x\to a}~ \dfrac{f(x) -f(a)}{x-a}\)

Answer 2

since you want the slope of tangent line at point (4, 2),
start by finding f(4)

Answer 3

plugin x = 4 in the given function

Answer 4

f(4)=2

Answer 5

whats f(a)-?

Answer 6

good, here \(a = 4\)

Answer 7

plug that value in and simplify

Answer 8

slope = \(\large \lim \limits_{x\to 4}~ \dfrac{f(x) -f(4)}{x-4}\)


= \(\large \lim \limits_{x\to 4}~ \frac{\dfrac{x-2}{x-3} -2}{x-4}\)

Answer 9

see if u can simplify^

Answer 10

-x^2 -x +20

Answer 11

no, try again

Answer 12

\(\large \lim \limits_{x\to 4}~ \frac{\dfrac{x-2}{x-3} -2}{x-4} \)


\(\large \lim \limits_{x\to 4}~ \frac{\dfrac{x-2-2x+6}{x-3} }{x-4} \)


\(\large \lim \limits_{x\to 4}~ \dfrac{-x+4}{(x-3)(x-4)} \)


\(\large \lim \limits_{x\to 4}~ \dfrac{-1}{x-3} \)


\(\large \ \dfrac{-1}{4-3} \)


\(\large -1\)

Answer 13

see if above makes sense...

Answer 14

oh yes, sorry i messed up the top part

Answer 15

so the slope is -1?

Answer 16

yes !