2. THEOREM 2 Vector-Valued Derivatives Are Computed Componentwise A vector-valued function r(t) = x (t), y (t) is differentiable iff each component is differentiable. In this case, Calculate r”(3), where r(t) = ln t, t.

3. THEOREM 3 Product Rule for Dot Product Assume that r 1 (t) and r 2 (t) are differentiable. Then

4. The Derivative as a Tangent Vector The derivative vector r’(t 0 ) has an important geometric property: It points in the direction tangent to the path traced by r(t) at t = t 0. To understand why, consider the difference quotient, where Δr = r (t 0 + h) − r (t 0 ) and Δt = h with h 0:

5. The Derivative as a Tangent Vector The vector Δr points from the head of r(t 0 ) to the head of r(t 0 + h). The difference quotient Δr/Δt is a scalar multiple of Δr and therefore points in the same direction.

6. The Derivative as a Tangent Vector The tangent vector r’(t 0 ) (if it is nonzero) is a direction vector for the tangent line to the curve. Therefore, the tangent line has vector parametrization: Although it has been our convention to regard all vectors as based at the origin, the tangent vector r(t) is an exception; we visualize it as a vector based at the terminal point of r (t). This makes sense because r(t) then appears as a vector tangent to the curve. As h = Δt tends to zero, Δr also tends to zero but the quotient Δr/Δt approaches a vector r’(t 0 ), which, if nonzero, points in the direction tangent to the curve. The figure illustrates the limiting process. We refer to r’(t 0 ) as the tangent or velocity vector at r(t 0 ).

7. The Derivative as a Tangent Vector Plotting Tangent Vectors Plot r (t) = cos t, sin t together with its tangent vectors at and. Find a parametrization of the tangent line at. tangent line is parametrized by and thus the

8. The Derivative as a Tangent Vector There are some important differences between vector- and scalar-valued derivatives. The tangent line to a plane curve y = f (x) is horizontal at x 0 if f’(x 0 ) = 0. But in a vector parametrization, the tangent vector is horizontal and nonzero if y’(t 0 ) = 0 but x’(t 0 ) 0.

9. The Derivative as a Tangent Vector Horizontal Tangent Vectors on the Cycloid The function traces a cycloid. Find the points where: (a) r’(t) is horizontal and nonzero. (b) r ’(t) is the zero vector. The tangent vector is By periodicity, we conclude that r’(t) is nonzero and horizontal for t = π, 3π, 5π,…and r’(t) = 0 for t = 0, 2π, 4π,…

10. The Derivative as a Tangent Vector CONCEPTUAL INSIGHT The cycloid below has sharp points called cusps at points where x = 0, 2π, 4π,…. If we represent the cycloid as the graph of a function y = f (x), then f’(x) does not exist at these points. By contrast, the vector derivative r’(t) = 1 − cos t, sin t exists for all t, but r’(t) = 0 at the cusps. In general, r’(t) is a direction vector for the tangent line whenever it exists, but we get no information about the tangent line (which may or may not exist) at points where r’(t) = 0.

11. The Derivative as a Tangent Vector GRAPHICAL INSIGHT A vector parametrization r(t) consisting of vectors of constant length R traces a curve on the circle of radius R with center at the origin. Thus r’(t) is tangent to this circle. But any line that is tangent to a circle at a point P is orthogonal to the radial vector through P, and thus r(t) is orthogonal to r’(t).

12. Vector-Valued Integration The integral of a vector-valued function can be defined via componentwise integration. The integral exists if each of the components x (t), y (t) is integrable. For example,

13. Vector-Valued Integration An antiderivative of r(t) is a vector-valued function R(t) such that R’(t) = r(t). In the single-variable case, two functions f 1 (x) and f 2 (x) with the same derivative differ by a constant. Similarly, two vector-valued functions with the same derivative differ by a constant vector (i.e., a vector that does not depend on t). THEOREM 4 If R 1 (t) and R 2 (t) are differentiable and for some constant vector c.

14. Vector-Valued Integration The general antiderivative of r(t) is written Fundamental Theorem of Calculus for Vector-Valued Functions If r(t) is continuous on [a,b], and R(t) is an antiderivative of r(t), then

15. Vector-Valued Integration Finding Position via Vector-Valued Differential Equations The path of a particle satisfies Find the particle’s location at The general solution is obtained by integration: The initial condition gives us