- 3.7 Continuity And Differentiablityap Calculus Solver
- 3.7 Continuity And Differentiablityap Calculus Algebra
- 3.7 Continuity And Differentiablityap Calculus 14th Edition
Introduction
Math AP®︎/College Calculus AB Differentiation: definition and basic derivative rules Connecting differentiability and continuity: determining when derivatives do and do not exist Differentiability at a point: algebraic. To state whether the above function is differentiable and continuous, we will study the continuity of the function first. This is because if a function is not continuous then it is also not differentiable. To study the continuity of the function, we need to substitute x = 2 in both the functions. Limits, Continuity and Differentiability can in fact be termed as the building blocks of Calculus as they form the basis of entire Calculus. The basic concept of limit of a function lays the groundwork for the concepts of continuity and differentiability. These concepts can in fact be called the natural extensions of the concept of limit.
In Section 1.2, we learned about how the concept of limits can be used to study the trend of a function near a fixed input value. As we study such trends, we are fundamentally interested in knowing how well-behaved the function is at the given point, say (x = a). In this present section, we aim to expand our perspective and develop language and understanding to quantify how the function acts and how its value changes near a particular point. Beyond thinking about whether or not the function has a limit (L) at (x = a), we will also consider the value of the function (f (a)) and how this value is related to (lim_{x→a} f (x)), as well as whether or not the function has a derivative (f '(a)) at the point of interest. Throughout, we will build on and formalize ideas that we have encountered in several settings.
We begin to consider these issues through the following preview activity that asks you to consider the graph of a function with a variety of interesting behaviors.
Preview Activity (PageIndex{1})
3.7 Continuity And Differentiablityap Calculus Solver
A function (f) defined on (−4 < x < 4) is given by the graph in Figure 1.7.1. Use the graph to answer each of the following questions. Note: to the right of (x = 2), the graph of (f) is exhibiting infinite oscillatory behavior similar to the function (sin( frac{π}{ x })) that we encountered in the key example early in Section 1.2.
(a) For each of the values (a) = −3, −2, −1, 0, 1, 2, 3, determine whether or not (lim_{x→a} f (x)) exists. If the function has a limit (L) at a given point, state the value of the limit using the notation (lim_{x→a} f (x)= L). If the function does not have a limit at a given point, write a sentence to explain why.
3.7 Continuity And Differentiablityap Calculus Algebra
3.7 Continuity And Differentiablityap Calculus 14th Edition
Figure (PageIndex{1}): The graph of (y = f (x)).
(b) For each of the values of a from part (a) where (f) has a limit, determine the value of (f (a)) at each such point. In addition, for each such a value, does (f (a)) have the same value as (lim_{x→a} f (x)) ?
(c) For each of the values (a) = −3, −2, −1, 0, 1, 2, 3, determine whether or not (f '(a)) exists. In particular, based on the given graph, ask yourself if it is reasonable to say that f has a tangent line at ((a, f (a))) for each of the given (a)-values. If so, visually estimate the slope of the tangent line to find the value of (f '(a)).