Problem 14 - A Course in Calculus and Real Analysis

Question

Let \(a, b, c \in \mathbb{R}\) with \(a

Step-by-Step Solution

Answer

In summary, the antiderivatives \(F_j(x)\) for functions \(f_j(x)\) are given by: \(F_1(x) = \left\lbrace \begin{array}{ll}0 & \text{ if } x\leq c \\ x-c & \text{ if } c

Step 1: Finding the antiderivatives F_j(x) for each function f_j(x)#

First, let's find the antiderivative for each function: Function \(f_1(x)\): \(f_1(x) = 0, x\leq c\) \(f_1(x) = 1, c

Step 2: Analyzing continuity and differentiability#

Now, let's analyze the continuity and differentiability of the functions and their antiderivatives at point \(x=c\). Function \(f_1(x)\): As \(f_1(x) = 0\) for \(x\leq c\) and \(f_1(x) = 1\) for \(c

Key Concepts

Limit of a Function

When learning calculus, understanding the limit of a function is crucial. It is the value that a function approaches as the input approaches some value. Limits are essential when dealing with piecewise defined functions where abrupt changes occur, such as in our exercise with function f_1(x).

In the provided exercise, the limit of f_1(x) at x = c does not exist because the left-hand limit (0) does not equal the right-hand limit (1). This is called a jump discontinuity. The concept of limits also helps us understand the behavior of functions at points of discontinuity and is the backbone of concepts like continuity and differentiability.

Integrals and Antiderivatives

Integrals and antiderivatives are a fundamental part of calculus, closely tied to the concept of area under a curve. An antiderivative of a function is another function whose derivative gives back the original function. The process of finding antiderivatives is known as integration.

In our textbook exercise, we calculated the antiderivatives F_j(x) for the piecewise functions f_j(x). For instance, integrating f_1(x), which equals 1 for c < x and 0 otherwise, resulted in an antiderivative F_1(x) that is a simple linear function for c < x and constant elsewhere.

Continuity and Differentiability

Continuity refers to the property of a function to have no breaks, jumps, or holes at a certain point. A function f(x) is continuous at a point if lim f(x) as x approaches the point equals the function's value at that point. Differentiability, on the other hand, implies continuity and adds that the function has a defined derivative at that point.

In our exercise, f_3(x) is a highlight because it showcases continuous behavior (no abrupt changes) at x = c, yet it is not differentiable there, indicating a 'corner' or 'kink' in the graph. A function that is not differentiable at a point may still be continuous there, but not vice versa.

Piecewise Defined Functions

Piecewise defined functions are functions that have different expressions or rules for different intervals of the input variable. They are quite common in real-world scenarios where different conditions yield different outcomes. In calculus, these functions often present interesting challenges regarding continuity and differentiability.

The functions f_1(x), f_2(x), and f_3(x) from our exercise are all piecewise functions with different behaviors at the point x = c. Piecewise functions require us to consider limits, integrals, and derivatives piece by piece and challenge us to stitch these different pieces together to analyze the overall behavior of the function.

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Problem 13

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Problem 15

Other exercises in this chapter

Problem 12

Evaluate the following limits. (i) \(\lim _{h \rightarrow 0} \frac{1}{h} \int_{x}^{x+h} \frac{d u}{u+\sqrt{u^{2}+1}}\), (ii) \(\lim _{x \rightarrow 0} \frac{1}{

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Problem 13

If \(x:=\int_{0}^{y} \frac{d t}{\sqrt{1+t^{2}}}\), find \(\frac{d^{2} y}{d x^{2}}\)

View solution

Problem 15

Let \(n \in \mathbb{N}\). Find a function \(f:[-1,1] \rightarrow \mathbb{R}\) for which \(f^{(n)}(0)\) exists, but \(f^{(n+1)}(0)\) does not. (Hint: Begin with

View solution

Problem 17

Let \(f:[a, b] \rightarrow \mathbb{R}\) be continuous and consider the function \(F:[a, b] \rightarrow \mathbb{R}\) given by \(F(x):=\int_{a}^{x} f(t) d t\) for

View solution

Key Concepts

Limit of a Function

When learning calculus, understanding the limit of a function is crucial. It is the value that a function approaches as the input approaches some value. Limits are essential when dealing with piecewise defined functions where abrupt changes occur, such as in our exercise with function f_1(x).

In the provided exercise, the limit of f_1(x) at x = c does not exist because the left-hand limit (0) does not equal the right-hand limit (1). This is called a jump discontinuity. The concept of limits also helps us understand the behavior of functions at points of discontinuity and is the backbone of concepts like continuity and differentiability.

Integrals and Antiderivatives

Integrals and antiderivatives are a fundamental part of calculus, closely tied to the concept of area under a curve. An antiderivative of a function is another function whose derivative gives back the original function. The process of finding antiderivatives is known as integration.

In our textbook exercise, we calculated the antiderivatives F_j(x) for the piecewise functions f_j(x). For instance, integrating f_1(x), which equals 1 for c < x and 0 otherwise, resulted in an antiderivative F_1(x) that is a simple linear function for c < x and constant elsewhere.

Continuity and Differentiability

Continuity refers to the property of a function to have no breaks, jumps, or holes at a certain point. A function f(x) is continuous at a point if lim f(x) as x approaches the point equals the function's value at that point. Differentiability, on the other hand, implies continuity and adds that the function has a defined derivative at that point.

In our exercise, f_3(x) is a highlight because it showcases continuous behavior (no abrupt changes) at x = c, yet it is not differentiable there, indicating a 'corner' or 'kink' in the graph. A function that is not differentiable at a point may still be continuous there, but not vice versa.

Piecewise Defined Functions

Piecewise defined functions are functions that have different expressions or rules for different intervals of the input variable. They are quite common in real-world scenarios where different conditions yield different outcomes. In calculus, these functions often present interesting challenges regarding continuity and differentiability.

The functions f_1(x), f_2(x), and f_3(x) from our exercise are all piecewise functions with different behaviors at the point x = c. Piecewise functions require us to consider limits, integrals, and derivatives piece by piece and challenge us to stitch these different pieces together to analyze the overall behavior of the function.