foreverlost said:Homework Statement
Find an equation of the tangent line to the curve y = x√x that is parallel to the line
y = 1+3x.
Homework Equations
m = 3
The Attempt at a Solution
Here is my attempt: dy/dx(x) * dy/dx(x^(1/2)) = (1) * (1/2x^(-1/2)) = (1/2x^(-1/2))
(1/2x^(-1/2)) = 3 → x^(-1/2) = 6 → -√x = 6
x = -√36 = 6 → -6=6 -1=1
Thank you, I will not be able to respond for a bit as I am leaving for home now.
You don't take "dy/dx of x" or "dy/dx of x^(1/2)." You can take the derivative with respect to x of something, which you write as d/dx(x) + d/dx(x^(1/2)).foreverlost said:dy/dx(x) * dy/dx(x^(1/2))
A tangent line is a straight line that touches a curve at only one point, without crossing over the curve. It represents the instantaneous rate of change of the curve at that point.
To find the equation of a tangent line, you first need to find the slope of the tangent line at the desired point on the curve. This can be done by taking the derivative of the curve at that point. Then, you can use the point-slope form of a line to write the equation of the tangent line.
No, there can only be one tangent line to a curve at a given point. This is because the tangent line represents the instantaneous rate of change of the curve at that point, and there cannot be more than one instantaneous rate of change.
A parallel line has the same slope as the tangent line at a given point on the curve. This means that the parallel line will have the same instantaneous rate of change as the curve at that point.
Tangent lines and parallel lines can be used to approximate the behavior of a curve at a specific point. This can be helpful in predicting future trends or making informed decisions in fields such as economics, engineering, and physics.