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Chapter 2: Problem 96

True or False? In Exercises \(93-96\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If a function is differentiable at a point, then it is continuous at that point.

Short Answer

Expert verified
The statement 'If a function is differentiable at a point, then it is continuous at that point' is true.

Step by step solution

01

Understanding Differentiability and Continuity

Differentiability at a point means that a function has a derivative at that point, or that the slope of the function is defined there. Continuity at a point, on the other hand, means that there are no jumps, breaks or holes at that point in the function. The function doesn鈥檛 skip any value on the y-axis around this point.
02

Relating Differentiability to Continuity

By definition, if a function \(f(x)\) is differentiable at \(x = c\), then the limit of the difference quotient exists at \(x = c\), i.e., \[\lim_{h \to 0} \frac{f(c+h) - f(c)}{h}\] exists. This means that the function is smooth, without any breaks, at \(x = c\). Thus, if a function is differentiable at a point, then it is continuous at that point.
03

Verification

Also, it is known, if a function is differentiable at a point, it will be continuous at that point. This can be verified by the fact that every differentiable function is continuous. Although the converse is not true; a function could be continuous at a point without being differentiable there.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Continuity
Continuity is a fundamental property of functions in calculus. We say a function is continuous at a point if it has no sudden jumps, breaks, or holes at that point. Imagine a smooth, unbroken line drawn on a graph, without lifting the pencil. If a function can be represented like this around a certain point, we consider it continuous at that point. This implies that there's no abrupt change in values. Mathematically, a function \( f(x) \) is continuous at \( x = c \) if the following conditions are met:
  • The function \( f(x) \) is defined at \( x = c \).
  • The limit \( \lim_{x \to c} f(x) \) exists.
  • The limit of the function as it approaches \( c \) is equal to the function value, \( \lim_{x \to c} f(x) = f(c) \).
This ensures that the behavior of the function around \( c \) is predictable and smooth.
Derivative
The derivative provides information about the rate of change of a function. It can be thought of as the slope of the tangent line to the curve of the function at a given point. When we say that a function is differentiable at a point, it means the derivative exists at that particular point, offering insight into how the function behaves locally. To find a derivative at \( x = c \), we substitute into the formula: \[ f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h} \]This formula calculates the slope of the tangent line at \( c \), illustrating how the function changes instantaneously.It's important to note:
  • If a function is differentiable at a point, it must be continuous at that point. This is because differentiability implies smoothness.
  • The reverse is not always true; a function may be continuous but not differentiable (like a sharp corner or cusp).
Limit
Limits are crucial in understanding the behavior of functions as they approach a certain point. The concept of a limit helps in examining what happens to function values as the input gets closer to a certain value, without necessarily reaching it. A simple example would be looking at \( \lim_{x \to c} f(x) \), where you study how \( f(x) \) behaves as \( x \) approaches \( c \). This is foundational for concepts like continuity and differentiability.Here's what you need to grasp about limits:
  • A limit exists if the function approaches a specific value, ensuring no sudden jumps or inconsistent behavior at the edge.
  • The process of taking limits is used in defining both derivatives and integrals.
By understanding limits, we gain insight into the initial and momentary behavior of functions, critical for any calculus analysis.
Difference Quotient
The difference quotient is a crucial expression used in calculus to find the derivative of a function. It forms the backbone for comprehending how functions change at an exact point. You can think of it as a way to approximate the slope of the secant line between two points on the function.For any function \( f(x) \), the difference quotient is given by: \[ \frac{f(c+h) - f(c)}{h} \]where \( h \) represents a small increment from \( c \). When \( h \) approaches 0, the difference quotient transforms into the derivative.Key points about the difference quotient:
  • It's essential in the formal definition of a derivative.
  • It provides a means to visualize and calculate the function's rate of change.
  • The limit of the difference quotient as \( h \to 0 \) gives the exact slope of the tangent line at that point.
Understanding this concept aids in grasping both differentiation and function behavior at microscopic levels.

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Most popular questions from this chapter

Think About It Let \(f\) and \(g\) be functions whose first and second derivatives exist on an interval I. Which of the following formulas is (are) true? (a) \(f g^{\prime \prime}-f^{\prime \prime} g=\left(f g^{\prime}-f^{\prime} g\right)^{\prime} \quad\) (b) \(f g^{\prime \prime}+f^{\prime \prime} g=(f g)^{\prime \prime}\)

Proof. Prove the following differentiation rules. $$ \begin{array}{l}{\text { (a) } \frac{d}{d x}[\sec x]=\sec x \tan x} \\ {\text { (b) } \frac{d}{d x}[\csc x]=-\csc x \cot x} \\ {\text { (c) } \frac{d}{d x}[\cot x]=-\csc ^{2} x}\end{array} $$

Tangent Lines Find equations of the tangent lines to the graph of \(f(x)=x /(x-1)\) that pass through the point \((-1,5) .\) Then graph the function and the tangent lines.

A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. Water is flowing into the tank at a rate of 10 cubic feet per minute. Find the rate of change of the depth of the water when the water is 8 feet deep.

Horizontal Tangent Line In Exercises \(73-76\) , determine the point(s) at which the graph of the function has a horizontal tangent line. $$ f(x)=\frac{x-4}{x^{2}-7} $$
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