Question

Find an equation for the tangent to the curve at the given point: f(x)=4(sqrt(x))-x+3, (16,3)

Answer 1

do you know how to differentiate?

Answer 2

\[f(x)=4\sqrt{x}-x+3\]

Answer 3

Yes but not so well with radicals

Answer 4

remember that \(\sqrt{x}\) can be written as \(\displaystyle x^{\frac{1}{2}}\)

Answer 5

does that help?

Answer 6

so 2/sqrt(x)

Answer 7

that is the first term - yes

Answer 8

\[d/dx=2/\sqrt{x}-1\]

Answer 9

yup, although usually written out as:\[f'(x)=\frac{2}{\sqrt{x}}-1\]

Answer 10

so what do I do with coordinates (16,3)? Just substitute?

Answer 11

the differential will give you the value of the slope of the tangent at any point

Answer 12

so, since the differential involves terms only containing x, you would substitute x=16 to get the value of the slope of the tangent line

Answer 13

so 1/2 is the slope and 3 is the y intercept so the answer should be y=1/2x+3

Answer 14

that doesn't seem right to me

Answer 15

oohhhso 2/4x-16+3
the answer is 1/2x-11??

Answer 16

lets do this one step at a time to avoid mistakes.

first what do you get for the value of the slope at (16,3) ?

Answer 17

2/sqrt(16)x
=2/4x
=1/2x

Answer 18

Or (1/2)x

Answer 19

no - remember the equation that tells us what the slope of the tangent line will be at any point is given by:\[f'(x)=\frac{2}{\sqrt{x}}-1\]so substitute x=16 into this equation and what do you get for the value of f'(x)?

Answer 20

-1/2

Answer 21

correct, so now you know the equation of the tangent line is:\[y=-\frac{1}{2}x +c\]

Answer 22

next, we know that this line must also pass through the point (16,3). So use this to calculate the value for c.

Answer 23

so I need to solve -(1/2)(16)+c=0?

Answer 24

or =3?

Answer 25

not =0, but =3

Answer 26

ok give me one sec to get this

Answer 27

remember it passes through the point (16,3) => x=16, y=3

Answer 28

that cant be right...-(1/2)(16)=-8
c=3-8
=-5
That is not an option. It has to be 3, -11, or 11

Answer 29

nevermind 11 :)

Answer 30

you made a mistake. remember the equation is:\[y=-\frac{1}{2}x +c\]

Answer 31

I think you know what I did :D

Answer 32

:)

Answer 33

Thank you so much for you help

Answer 34

yw :)