3: Derivative Essentials
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- 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.
