Find the linear approximation of the function
f(x) = sqrt of (1 − x) at a = 0. use l(x) to approximate the numbers sqrt of 0.9 and sqrt of 0.99.

1−x−−−−√≈−12x+1

0.9−−−√=1−0.1−−−−−−√≈−12(0.1)+1

To simplify:

−1/2⋅1/100+1=−1/2⋅1/100+200/200=200−1/200=199/200 or =.995

So yes for sqrt (.99), that is a good approximation.

To write linear approximations:

Our curve should be  y=f(x) and substitute x=a for the numbers m

 Our formula for the linear equation is  y=mx+b

So subtsitute

y′=f′(x)

y=f′(a)x+b

Then find the  y-intercept, which is  b.

Given:  (a, f(a))= point of the line

Then substitute, that would be

f(a)=f′(a)⋅a+b

f(a)−f′(a)⋅a=b

So the tangent line to y=f(x) at x=a

Or the equation we will use for linear approximations is:

y=f′(a)x+f(a)−f′(a)⋅a

y=f′(a)⋅(x−a)+f(a)

Each time the number of hours (on the x axis) goes up by 1, the cost (on the y axis) goes up by up by $2. So each hour costs $2.

You can pick two points on the graph, such as (1,2) and (2,4) to compute the slope using the slope formula below

m = (y2 - y1)/(x2 - x1)

m = (4-2)/(2-1)

m = 2/1

m = 2 we get a slope of 2

The slope is the rise/run

rise = change in cost = change in y

run = change in time = change in x