Question

Find the unit tangent vector to the curve at the specified value of the parameter. The curve is defined by the equation r(t) = t³i + 5t²j, and the parameter value is t = 1.

Answer 1

To find the unit tangent vector at t = 1 for the curve defined by r(t) = t³i + 5t²j, calculate the derivative, evaluate it at t = 1 to get the tangent vector, find the magnitude of this vector, and then divide the tangent vector by its magnitude to get the unit vector.

Step-by-step explanation:

To find the unit tangent vector to the curve given by r(t) = t³i + 5t²j at the specified value of the parameter t = 1, we follow these steps:

1. First, we find the derivative of the curve, which gives us the tangent vector. The derivative of r(t) with respect to t is r'(t) = 3t²i + 10tj.
2. Then, we evaluate this derivative at t = 1, which yields r'(1) = 3i + 10j.
3. The magnitude of this vector is found by calculating the square root of the sum of the squares of its components, which is |r'(1)| = √(3² + 10²) = √109.
4. The unit tangent vector is then found by dividing the tangent vector by its magnitude, resulting in t(1) = (1/√109)(3i + 10j).