Question

Estimate the instantaneous rate of change of y with respect to x at x =9 for y=-square root X

Answer 1

Hint:

y = -sqrt(x)

y = -x^(1/2)

y ' = -(1/2)x^(-1/2)

y ' = -1/(2*sqrt(x))

Answer 2

were did the 1/2 at the from and back come from for y ' = -(1/2)x^(-1/2)

Answer 3

front*

Answer 4

in general, sqrt(x) = x^(1/2)

Answer 5

when you derive x^n you get n*x^(n-1)

this is by the power rule

Answer 6

ummm what does derive mean? I didn't learn anything about derivatives yet

Answer 7

so that's how I went from y = -x^(1/2) to y ' = -(1/2)x^(-1/2)

Answer 8

oh you haven't?

Answer 9

that's odd considering how they want you to find the instantaneous rate of change

Answer 10

no , we just learned how to get average and instant rate of change so far.

Answer 11

it says estimate, I tried using the difference quotient, didn't work, I tried picking values close to 9 so x=9.001 and finding it through delta y divided by delta x, it didn't work

Answer 12

well what you could do is find the average rate of change from x = 9 to x = 10

then you can find the average rate of change from x = 9 to x = 9.5

then you can find the average rate of change from x = 9 to x = 9.1

then you can find the average rate of change from x = 9 to x = 9.01


Notice the pattern? I'm getting the other x value closer and closer to 9 and this will estimate the instantaneous rate of change at x = 9

Answer 13

but when I picked x=9.001 it didn't work. For another simpler question were x=7 and y=2/x, I tried picking 7.001 for x , as well as 7 and finding the average rate of change, but I got the wrong answer
the question says to use technology, like a graphing calc

Answer 14

yeah i recommend a graphing calculator as well because you can type in the function and then go to the table to input any value you want

Answer 15

so you would type -sqrt(x) into y1

then go to the table and type in 9, 9.1, 9.01, 9.001 etc into the table to get your outputs

Answer 16

similar*

Answer 17

which you would use in the average rate of change formula given below

Average Rate of Change from x1 to x2 = (f(x2) - f(x1))/(x2 - x1)

Answer 18

Ok. I will do it and see.

x=9.001 x=9

(9.001, -3.00017) (9, -3)

y/x = -3.00017+3/9.001-9 =-0.17

The answer at the back of the book is -1/6.

I guess that could be because I rounded, Thanks for you help.

Answer 19

could you help with another question?

Answer 20

well you're estimating, so -0.17 or -0.16667 is very close to what the true value is (-1/6)

If you used the derivative, you would have found -1/6 exactly on the first try without any guesses or estimations, but you don't know about that yet

Answer 21

sure

Answer 22

two points P(1,1) and Q(x, 2x-x^2) lin on the curve y= 2x-x^2. Find the simplified expression for the slope of the secant PQ

Answer 23

Use the slope formula

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

Answer 24

In this case

x1 = 1
x2 = x

y1 = 1
y2 = 2x - x^2

Answer 25

I did, but then I didn't know what to do after since I had,

2x-x^2-1/x-1

Answer 26

now factor the numerator

Answer 27

(x+1)(x+1)/x-1 is that right?

Answer 28

2x-x^2-1

-x^2+2x-1

-(x^2-2x+1)

-(x-1)^2

Answer 29

so

(2x-x^2-1)/(x-1)

turns into

( -(x-1)^2 )/(x-1)

and that simplifies into

-(x-1)

-x + 1

Answer 30

so the slope of secant PQ is -x+1

Answer 31

oh ok. I was just factoring it wrong. ughh, so silly. Thank you so so so so so much.

Answer 32

you're welcome