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x. 3. 0. t. 20. 5,000. 10,000. M. Time (min). 2. 4. -1. 1. 2. 3. 1. y.

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  1. x 3 0 t 20 5,000 10,000 M Time (min) 2 4 -1 1 2 3 1 y Differential Calculus is the study of the derivative, and differentiation is the process of computing derivatives. What is a derivative? There are three equally important answers: A derivative is a rate of change, it is the slope of a tangent line, and (more formally), it is the limit of a difference quotient, as we will explain shortly. In this chapter, we explore all three facets of the derivative and develop the basic rules of differentiation. When you master these techniques, you will possess one of the most useful and flexible tools that mathematics has to offer. Calculus is the foundation for all of our understanding of motion, including the aerodynamic principles that made supersonic flight possible.

  2. t 20 5,000 10,000 M Time (min) 0 y x -1 1 2 3 1 2 3 4

  3. t 20 5,000 10,000 M Time (min) 0 y x -1 1 2 3 1 2 3 4

  4. t x 20 5,000 10,000 M Time (min) y 0 -1 2 3 1 2 3 4 1 Definition The Derivative Definition Tangent Line

  5. t 20 5,000 10,000 M Time (min) 0 y x -1 1 2 3 1 2 3 4

  6. t 20 5,000 10,000 M Time (min) 0 y x -1 1 2 3 1 2 3 4

  7. t 20 5,000 10,000 M Time (min) 0 y x -1 1 2 3 1 2 3 4

  8. t 20 5,000 10,000 M Time (min) 0 y x -1 1 2 3 1 2 3 4

  9. t x 20 5,000 10,000 M Time (min) 0 y 1 2 3 1 2 3 4 -1 Theorem 1 The Derivative of Linear and Constant Functions When h is small, the secant line has nearly the same slope as the tangent line.

  10. t 20 5,000 10,000 M Time (min) 0 y x -1 1 2 3 1 2 3 4

  11. t 20 5,000 10,000 M Time (min) 0 y x -1 1 2 3 1 2 3 4

  12. t 20 5,000 10,000 M Time (min) 0 y x -1 1 2 3 1 2 3 4

  13. Further Insights and Challenges 20 5,000 10,000 M Time (min) 0 y x -1 1 2 3 1 2 3 4 t 72. The SDQ usually approximates the derivative much more closely than does the ordinary difference quotient. Let Compute the SDQ with h = 0.001 and the ordinary difference quotients with h = ±0.001. Compare with the actual value, which is

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