1. Chapter 13 – Vector Functions 13.2 Derivatives and Integrals of Vector Functions 1 Objectives:  Develop Calculus of vector functions.  Find vector, parametric, and general forms of equations of lines and planes.  Find distances and angles between lines and planes

2. Definition – Derivatives of Vector Functions The derivative r’ of a vector function is defined in much the same way as for real-valued functions: if the limit exists. 13.2 Derivatives and Integrals of Vector Functions2

3. Definition – Tangent Vector The vector r’(t) is called the tangent vector to the curve defined by r at the point P, provided: ◦ r’(t) exists ◦ r’(t) ≠ 0 13.2 Derivatives and Integrals of Vector Functions3

4. Visualization Secant and Tangent Vectors 13.2 Derivatives and Integrals of Vector Functions4

5. Definition – Unit Tangent Vector We will also have occasion to consider the unit tangent vector which is defined as: 13.2 Derivatives and Integrals of Vector Functions5

6. Theorem The following theorem gives us a convenient method for computing the derivative of a vector function r: ◦ Just differentiate each component of r. 13.2 Derivatives and Integrals of Vector Functions6

7. Second Derivative Just as for real-valued functions, the second derivative of a vector function r is the derivative of r’, that is, r” = (r’)’. 13.2 Derivatives and Integrals of Vector Functions7

8. Differentiation Rules 13.2 Derivatives and Integrals of Vector Functions8

9. Integrals The definite integral of a continuous vector function r(t) can be defined in much the same way as for real- valued functions—except that the integral is a vector. 13.2 Derivatives and Integrals of Vector Functions9

10. Integral Notation However, then, we can express the integral of r in terms of the integrals of its component functions f, g, and h as follows using the notation of Chapter 5. 13.2 Derivatives and Integrals of Vector Functions10

11. Integral Notation - Continued Thus, ◦ This means that we can evaluate an integral of a vector function by integrating each component function. 13.2 Derivatives and Integrals of Vector Functions11

12. Fundamental Theorem of Calculus We can extend the Fundamental Theorem of Calculus to continuous vector functions: ◦ Here, R is an antiderivative of r, that is, R’(t) = r(t). ◦ We use the notation ∫ r(t) dt for indefinite integrals (antiderivatives). 13.2 Derivatives and Integrals of Vector Functions12

13. Example 1- pg. 852 #8 Sketch the plane curve with the given vector equation. Find r’(t). Sketch the position vector r(t) and the tangent vector r’(t) for the given value of t. 13.2 Derivatives and Integrals of Vector Functions13

14. Example 2- pg. 852 #9 Find the derivative of the vector function. 13.2 Derivatives and Integrals of Vector Functions14

15. Example 3 Find the unit tangent vector T(t) at the point with the given value of the parameter t. 13.2 Derivatives and Integrals of Vector Functions15

16. Example 4- pg. 852 #24 Find the parametric equations for the tangent line to the curve with the given parametric equations at the specified point. 13.2 Derivatives and Integrals of Vector Functions16

17. Example 5- pg. 852 #31 Find the parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Illustrate by graphing both the curve and the tangent line on a common screen. 13.2 Derivatives and Integrals of Vector Functions17

18. Example 6- pg. 856 #36 Evaluate the integral. 13.2 Derivatives and Integrals of Vector Functions18

19. Example 7 Evaluate the integral. 13.2 Derivatives and Integrals of Vector Functions19

20. More Examples The video examples below are from section 13.2 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦ Example 1 Example 1 ◦ Example 3 Example 3 ◦ Example 4 Example 4 13.2 Derivatives and Integrals of Vector Functions20