What is the equation of the tangent at (3, f(3))?

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In summary, to find the equation of the tangent to a curve at a given point, you need to use the formula y - f(c) = f'(c)(x - c), where c is the x-coordinate of the point. This formula uses the derivative at that point, which represents the slope of the tangent line. This means that the derivative at the point (c, f(c)) is the orientation coefficient of the tangent line.
  • #1
Joza
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I have a function.

I am asked to find the equation of the tangent to the curve at (3, f(3))

I think I am just confused of notation, but what is it asking here? To find the points, don't you put in the x value into the first derivative and this gets the y value, and use the formula for an equation of a line? What are those coordinates for then?
 
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  • #2
Well, you're supposed to find the equation of the tangent at that very point.
 
  • #3
So I already have my x and y, so just use y-y1=m(x-x1)?
 
  • #4
If you meant y - f(c) = f'(c)(x - c), then yes. (c = 3)
 
  • #5
radou said:
If you meant y - f(c) = f'(c)(x - c), then yes. (c = 3)

As you might know f'(c)=tg@, so the derivative of the curve at the point (c,f(c)) is actually the orientation coeficient of the tagent drawn at that point.

So this is actually what radou said, just giving u some more hints.
 

FAQ: What is the equation of the tangent at (3, f(3))?

What is the equation of a tangent?

The equation of a tangent is a mathematical representation of a straight line that touches a curve at one point, known as the point of tangency. It is used to determine the slope of the curve at that point.

How is the equation of a tangent different from the slope?

The slope of a curve is a measure of its steepness at a given point, while the equation of a tangent is a specific line that touches the curve at that point. The equation of a tangent includes both the slope and the coordinates of the point of tangency.

How do you find the equation of a tangent?

To find the equation of a tangent, you need to know the coordinates of the point of tangency and the slope of the curve at that point. You can use the point-slope form of a line, y - y1 = m(x - x1), where m is the slope and (x1, y1) are the coordinates of the point of tangency, to find the equation of the tangent.

What is the significance of the equation of a tangent in calculus?

The equation of a tangent is a key concept in calculus because it allows us to calculate the instantaneous rate of change of a function at a specific point. This is important for understanding the behavior of a curve and solving optimization problems.

Can the equation of a tangent be used to find the equation of a curve?

Yes, the equation of a tangent can be used to find the equation of a curve. By finding the equation of multiple tangents at different points on a curve, we can approximate the overall shape of the curve and find its equation. This process is known as curve fitting.

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