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

In summary, the conversation is about finding an equation for the tangent line to the graph of the function f(x) = sqrt(x)/5 at the point (4, 2/5). The person has trouble simplifying f(x) in order to take the limit as h approaches 0, which is the slope. They discuss using the basic formula for derivative and multiplying both the numerator and denominator of the fraction by sqrt(a+h)+sqrt(a). The person got a slope of 1/4, but the correct answer is y=(1/20)*x+(1/5). They also ask if the factor mentioned is called a conjugate.
  • #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 h->0 which is the slope.

f(x) = Sqrt(x)/5 at (4,(2/5))
 
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  • #2
Do you mean that you have to use the basic formula for derivative
[tex]lim_{h->0}\frac{f(a+h)-f(a)}{h}[/tex] rather than derivative formulas?

Try multiplying both numerator and denominator of
[tex]\frac{\sqrt{a+h}-\sqrt{a}}{h}[/tex]
by [tex]\sqrt{a+h}+\sqrt{a}[/tex].
 
  • #3
I did that and I got something like 1/2*Sqrt(a) by taking the lim as h->0, giving a slope of 1/4, but the answer is sopposed to be y=(1/20)*x+(1/5) can you please show me how they got this ?

By the way is that factor you posted called a conjugate ?
 
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1. What is a tangent line?

A tangent line is a straight line that touches a curve at a specific point, without crossing over or intersecting it. It represents the slope of the curve at that particular point.

2. How do you find the equation of a tangent line?

To find the equation of a tangent line, you need to first determine the slope of the curve at the given point. This can be done by taking the derivative of the function at that point. Then, using the point-slope form of a line, you can plug in the slope and the coordinates of the point to find the equation of the tangent line.

3. What is the process for finding the tangent line to f(x) at (4, (2/5))?

To find the tangent line to f(x) at (4, (2/5)), you need to first take the derivative of the function f(x). Then, plug in the x-coordinate (4) into the derivative to find the slope of the curve at that point. Finally, using the point-slope form of a line, plug in the slope and the coordinates of the given point to find the equation of the tangent line.

4. Can there be more than one tangent line to a curve at a specific point?

Yes, there can be more than one tangent line to a curve at a specific point. This is because a curve can have multiple slopes at a single point, depending on the shape of the curve. However, there will always be one unique tangent line that represents the slope of the curve at that point.

5. What is the significance of finding the tangent line to a curve at a specific point?

Finding the tangent line to a curve at a specific point allows us to determine the slope of the curve at that point. This can be useful in many applications, such as optimization problems, where we need to find the maximum or minimum value of a function, or in physics, where the slope represents the rate of change of a variable.

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