Question

An equation of the line normal to the graph of y = sqrt(3x^2+2x) at (2,4) is ?

Answer 1

\[y = \sqrt{(3x^2+2x)} at (2,4)\]

Answer 2

what does this mean? an equation of the line normal to the graph????

Answer 3

~ find the derivative.

Answer 4

what will the derivative of the function be?

Answer 5

the normal is the line that is perpendicular to the tangent at the given point

so find the 1st derivative, then the slope of the tangent...

Answer 6

yes.

Answer 7

\[y'= \frac{ 1 }{ 2 }(3x^2+2x)^\frac{ -1 }{ 2 } * 6x+2\]

Answer 8

you mean (6x+2) like this?

Answer 9

then it is correct.

Answer 10

you can simplify it a bit though.

Answer 11

\(\large\color{black}{ f'(x)=\frac{\LARGE 1 }{\LARGE 2}\cdot (\sqrt{3x^2+2x})^{-1/2}\cdot(6x+2) }\) please re-write this for me, with a positive exponent, and most reduced terms.

Answer 12

lost?

Answer 13

\[\frac{ 1 }{ \sqrt{3x^2+2x} }\times3x+1\]

Answer 14

Is it like that?

Answer 15

\[f'(x)=\frac{3x+1}{\sqrt{3x^2+2x}}\]

Answer 16

looks good

Answer 17

:D

Answer 18

now what happens?

Answer 19

well do you know how to find the slope of the tangent line to f at (2,4)

Answer 20

I know how to find slope between two points, but not of a tangent line (or don;t remember :( )

Answer 21

well f' gives you the slope number for each point (x,y) on f

Answer 22

evaluate f' at (2,4) to get the slope of the tangent line to f at (2,4)

Answer 23

we will then find the slope of line that runs perpendicular to that tangent line
by flipping your number and taking the opposite of it

Answer 24

OK lol Yeaahhh the derivative is the rate of change ok so I'll plug in

Answer 25

yea

Answer 26

I get 1.75

Answer 27

your normal line will be in this form
\[y-y_1=\frac{-1}{m}(x-x_1)\]
where m was the slope of the tangent line

Answer 28

\[y-4=-\frac{ 4 }{ 7 }(x-2)\]

Answer 29

\[f'(2)=\frac{3(2)+1}{\sqrt{3(2)^2+2(2)}}=\frac{7}{4}\]

Answer 30

looks great

Answer 31

you are done unless you want the line in slope-intercept form

Answer 32

y=-4/7x+36/7

Answer 33

\[\text{checking...} \\ y=\frac{-4}{7}(x-2)+4 \\ y=\frac{-4}{7}x+\frac{8}{7}+4 \\ y=\frac{-4}{7}x+\frac{8+28}{7}\]
yep looks great also

Answer 34

on your previous question
on the calculator I'm kinda thinking you entered in 100/20 sqrt(3)
instead of 100/(20 sqrt(3)) into calculator
and tht is why you got 5sqrt(3) instead of 5/sqrt(3)

Answer 35

yeaaahh that is what I did lol

Answer 36

VERIFICATION: https://www.desmos.com/calculator/os0v8fyubn

Answer 37

you can use ~ for a space, btw.

Answer 38

more space? ~~ Even more ~~~ and on....

Answer 39

thanks @freckles and @SolomonZelman