Find the unit vector tangent to the curve y = x^4 at the point(x,y) = (1,1).
dy/dx = 4x^3
at (1,1), dy/dx = 4
equation of tangent : y-1 = 4(x-1)
y = 4x -3
we already have one point on this, (1,1)
another point: let x = 2, y = 5
direction vector of that tangent = < 2-1, 5-1> = < 1,4>
so a unit vector would be <1/√17 , 4/√17>
To find the unit vector tangent to the curve y = x^4 at the point (x,y) = (1,1), we need to find the derivative of the curve at that point and then normalize it.
Step 1: Find the derivative of the curve y = x^4.
To find the derivative, we can use the power rule: d/dx (x^n) = n*x^(n-1).
For this curve, when n = 4, we get: dy/dx = d/dx (x^4) = 4*x^(4-1) = 4*x^3.
Step 2: Evaluate the derivative at the point x = 1.
Substitute x = 1 into the derivative equation: dy/dx = 4*(1)^3 = 4*1 = 4.
So, the slope of the curve at the point (x,y) = (1,1) is 4.
Step 3: Convert the slope into a vector.
Since the slope is 4, we can write it as a vector <1, 4>.
Step 4: Normalize the vector.
The unit vector tangent to the curve is obtained by dividing the vector <1, 4> by its magnitude.
Magnitude of <1, 4> = sqrt(1^2 + 4^2) = sqrt(1+16) = sqrt(17).
Therefore, the unit vector tangent to the curve y = x^4 at the point (x,y) = (1,1) is (<1, 4>)/(sqrt(17)) = <1/sqrt(17), 4/sqrt(17)>, or approximately <0.24, 0.97>.
To find the unit vector tangent to the curve y = x^4 at the point (x, y) = (1, 1), we need to find the slope of the curve at that point and then normalize it to obtain the unit vector. Here's how you can do that:
Step 1: Calculate the derivative of the curve.
- Differentiate y = x^4 with respect to x.
- Applying the power rule, we find dy/dx = 4x^3.
Step 2: Substitute x = 1 into the derivative.
- Plug x = 1 into the derivative dy/dx = 4x^3.
- dy/dx = 4(1)^3 = 4.
Step 3: Determine the slope of the tangent line.
- The slope of the tangent line to the curve at the point (x, y) = (1, 1) is given by dy/dx evaluated at x = 1.
- Therefore, the slope of the tangent line is 4.
Step 4: Form the tangent vector.
- The tangent vector is a vector that points in the direction of the tangent line.
- The tangent vector will have the same slope as the tangent line; thus, its components can be represented as (1, 4).
Step 5: Normalize the tangent vector to get the unit tangent vector.
- To obtain the unit vector, divide the components of the tangent vector by its magnitude.
- The magnitude of the tangent vector is given by √(1^2 + 4^2) = √(1 + 16) = √17.
- Therefore, the unit vector tangent to the curve y = x^4 at the point (x, y) = (1, 1) is (1/√17, 4/√17).
So, the unit vector tangent to the curve y = x^4 at the point (x, y) = (1, 1) is (1/√17, 4/√17).