Question

I am not sure I understand the following instructions on my book (Calculus: The Classic Edition, by Swokowski):

"[a] Use Definition (3.1) to find the slope of the tangent line to the graph of the equation at the point of x-coordinate a. [b] Find an equation of the tangent line at P. [c] Sketch the graph and the tangent line at P."

Then, I am given:
y = square root of x; P(4, 2)

The Definition (3.1) refers to the following:

limit when h approaches 0 of
[ f(a + h) - f(a) ] / h

The answer book says that [a] is 1/(2*sqrt(a)) and [b] is y = (1/4)x + 1

Answer 1

What I need is to be sure I understand the instructions correctly. I was trying to use the function y = sqrt(x) in the Definition (3.1), like this:

limit when h approaches 0 of
[(sqrt(a + h)) - sqrt(a)] / h

Am I correct?

Answer 2

yes that is what you want, although in this case you can do it with numbers

Answer 3

the point is
\[(4,2)\]so you can compute
\[\lim_{h\to 0}\frac{\sqrt{4+h}-2}{h}\]

Answer 4

usual gimmick is to multiply by the conjugate on the top and bottom.
you got it from here ?

Answer 5

This is where I get confused.