Finding Tangent Line of f(x) at (4,(2/5))

In summary, the conversation is about finding an equation for the tangent line to the graph of a function at a specified point. The function is f(x) = Sqrt(x)/5 and the point is (4,(2/5)). The process involves finding the derivative of the function at the given point and using point-slope form to represent the tangent line. The final equation for the tangent line requires some algebraic manipulation.
  • #1
Victor Frankenstein
29
0
I need to find an equation for the tangent line to the graph of the function at the specified point, I had some trouble simplifing f(x) so that I can take the lim as x->0 which is the slope.

f(x) = Sqrt(x)/5 at (4,(2/5))
 
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  • #2
*Simply find the derivative of the curve at that point, and use point-slope to find the line.
[tex] f\left( x \right) = \frac{\sqrt x}{5} \Rightarrow f\,'\left( x \right) = \frac{1}{{10\sqrt x }} [/tex]

*To find the slope of the tangent line, simply calculate [itex] f\,' ( 4 ) [/itex]:
[tex] f\,'\left( 4 \right) = \frac{1}{{20}} [/tex]

*Next, just use point-slope to represent the tangent line, which I'll call [itex] y [/itex]:
[tex] \frac{{x - 4}}{{20}} = y - \frac{2}{5} [/tex]

And it's algebra from there :smile:
 
Last edited:
  • #3
Victor Frankenstein said:
I need to find an equation for the tangent line to the graph of the function at the specified point, I had some trouble simplifing f(x) so that I can take the lim as x->0 which is the slope.

f(x) = Sqrt(x)/5 at (4,(2/5))


Are you saying that you are required to use the basic formula for the derivative: [itex] lim_{h->0}\frac{f(a+h)-f(a)}{h}[/itex] rather than the more specific formula (derived from that) that bomba923 used?

If so try multiplying both numerator and denominator of
[tex]\frac{\sqrt{a+h}-\sqrt{a}}{h}[/tex]
by
[tex]\sqrt{a+h}+ \sqrt{a}[/tex].
 

What is a tangent line?

A tangent line is a straight line that touches a curve at a single point, without crossing through it. It represents the instant rate of change of the curve at that point.

How do you find the tangent line of a function at a given point?

To find the tangent line of a function at a given point, you need to find the slope of the function at that point. This can be done using the derivative of the function. Once you have the slope, you can use the point-slope formula to find the equation of the tangent line.

What is the point-slope formula?

The point-slope formula is used to find the equation of a line passing through a given point with a known slope. It is written as y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.

What is the derivative of a function?

The derivative of a function represents the rate of change of the function at a given point. It is the slope of the tangent line at that point and can be found by calculating the limit of the function as the change in x approaches 0.

How do you use the derivative to find the slope of a function at a given point?

To find the slope of a function at a given point, you need to take the derivative of the function and then plug in the x-coordinate of the given point. The resulting value is the slope of the function at that point.

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