# 3: Derivative Essentials


• 3.1: How do we Measure Velocity?
The average velocity on [a,b] can be viewed geometrically as the slope of the line between the points (a,s(a)) and (b,s(b)) on the graph of y=s(t). The instantaneous velocity of a moving object at a fixed time is estimated by considering average velocities on shorter and shorter time intervals that contain the instant of interest
• 3.2: The Notion of Limit
Limits enable us to examine trends in function behavior near a specific point. In particular, taking a limit at a given point asks if the function values nearby tend to approach a particular fixed value.
• 3.3: The Derivative of a Function at a Point
An idea that sits at the foundations of calculus is the instantaneous rate of change of a function. This rate of change is always considered with respect to change in the input variable, often at a particular fixed input value. This is a generalization of the notion of instantaneous velocity and essentially allows us to consider the question “how do we measure how fast a particular function is changing at a given point?”
• 3.4: The Derivative Function
The limit definition of the derivative produces a value for each x at which the derivative is defined, and this leads to a new function whose formula is y = f'(x). Hence we talk both about a given function f and its derivative f'. It is especially important to note that taking the derivative is a process that starts with a given function (f) and produces a new, related function (f').
• 3.5: Limits, Continuity, and Differentiability
A function f has limit as x → a if and only if f has a left-hand limit at x = a, has a right-hand limit at x = a, and the left- and right-hand limits are equal. A function f is continuous at x = a whenever f (a) is defined, f has a limit as x → a, and the value of the limit and the value of the function agree. This guarantees that there is not a hole or jump in the graph of f at x = a. A function f is differentiable at x = a whenever f' (a) exists.
• 3.6: The Mean Value Theorem
The Mean Value Theorem is one of the most important theorems in calculus. We look at some of its implications at the end of this section. First, let’s start with a special case of the Mean Value Theorem, called Rolle’s theorem.

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