1. Vector-Valued Functions Section 10.3b

2. 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: c any scalar f any differentiable scalar function 3. Sum Rule:

3. 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: r a differentiable function of t, t a differentiable function of s 6. Chain Rule:

4. 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:

5. 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:

6. 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:

7. 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. This is the path traveled by the particle, which is smooth, and the path is traversed exactly once on the interval… Graph the parametrization in [ –1, 2] by [–2, 4]: m

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

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

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

11. 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