VectorValued Functions Section 10 3 b Differentiation Rules

Vector-Valued Functions Section 10. 3 b

Differentiation Rules for Vector Functions Let u and v be differentiable functions of t, and C a constant vector. 1. Constant Function Rule: 2. Scalar Multiple Rules: f any differentiable scalar function 3. Sum Rule: c any scalar

Differentiation Rules for Vector Functions Let u and v be differentiable functions of t, and C a constant vector. 4. Difference Rule: 5. Dot Product Rule: 6. Chain Rule: r a differentiable function of t, t a differentiable function of s

Definition: Indefinite Integral The indefinite integral of r with respect to t is the set of all antiderivatives of r, denoted by. If R is any antiderivative of r, then Quick Example – Evaluate:

Definition: Definite Integral If the components of r(t) = f(t)i + g(t)j are integrable on [a, b], then so is r, and the definite integral of r from a to b is Quick Example – Evaluate:

Guided Practice The velocity vector of a particle moving in the plane (scaled in meters) is (a) Find the particle’s position as a vector function of t if when Initial Condition:

Guided Practice The velocity vector of a particle moving in the plane (scaled in meters) is (b) Find the distance the particle travels from t = 0 to t = 2. Graph the parametrization in [ – 1, 2] by [– 2, 4]: This is the path traveled by the particle, which is smooth, and the path is traversed exactly once on the interval… m

Guided Practice Solve the initial value problem for r as a vector function of t.

Guided Practice Solve the initial value problem for r as a vector function of t.

Guided Practice Solve the initial value problem for r as a vector function of t. Solution:

Guided Practice r(t) is the position vector of a particle in the plane at time t. Find the time, or times, in the given time interval when the velocity and acceleration vectors are perpendicular. We need to find when This is true for : , k any nonnegative integer