Question

How to find the equation of the Tangent line to f(x) at point (2,8) :
The expression dy/dx= x cuberoot (y) gives the slope at any point on the graph of the function f(x) where f(2)= 8

Answer 1

How don't even know how to begin

Answer 2

cuberoot(y)

Answer 3

4?

Answer 4

y-8 = 4(x-2)

Answer 5

y-8 = 4x - 8

Answer 6

for y or x?

Answer 7

y= 4x - 16

Answer 8

I see how would I write an expression for f(x) in terms of x

Answer 9

but hmmm y= 4x is the equation of the tangent line at 2,8 right

Answer 10

but I also wanted to know for the function

Answer 11

calc - differential equations

Answer 12

But does that give me f(x)....

Answer 13

yeah .so it would be y = 4x and f(x)= 4x

Answer 14

why y^2?

Answer 15

Okay this is very confusing thanks though

Answer 16

@Loser66 no I don't want you to do my homework (which this isn't homework I am studying for my placement test) I just need it to make sense to me

Answer 17

How to find the equation of the Tangent line to f(x) at point (2,8) :
The expression dy/dx= x cuberoot (y) gives the slope at any point on the graph of the function f(x) where f(2)= 8

The tangent line to \(f(x)\) at some point \((x,y)\) will have slope determined by \(\dfrac{dy}{dx}\) evaluated at this value of \(x\).

In this case, since \(x=2\), \(y=8\), and \(\dfrac{dy}{dx}=x\sqrt[3]y\), the slope is
\[\frac{dy}{dx}=2\sqrt[3]8=2\cdot2=4\]
The tangent line can be derived from the point-slope formula for a line:
\[y-y_0=m(x-x_0)\]
where you plug in a known point \((x_0,y_0)\) and slope \(m\). Using the given information, you have
\[y-8=4(x-2)~~\iff~~y=4x\]