# CHAPTER 8

## 8.1 INTRODUCTION

This chapter will discuss what a derivative is and why it is important in engineering. The concepts of maxima and minima along with the applications of derivatives to solve engineering problems in dynamics, electric circuits, and mechanics of materials are emphasized.

### 8.1.1 What Is a Derivative?

To explain what a derivative is, an engineering professor asks a student to drop a ball (shown in Fig. 8.1) from a height of y = 1.0 m to find the time when it impacts the ground. Using a high-resolution stopwatch, the student measures the time at impact as t = 0.452 s. The professor then poses the following questions:

(a) What is the average velocity of the ball?

(b) What is the speed of the ball at impact?

(c) How fast is the ball accelerating? Figure 8.1 A ball dropped from a height of 1 meter.

Using the given information, the student provides the following answers:

(a) Average Velocity, : The average velocity is the total distance traveled per unit time, i.e., Note that the negative sign means the ball is moving in the negative y-direction.

(b) Speed at Impact: The student finds that there is not enough information to find the ...

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