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Chapter 2: Problem 63

In Exercises 63-66, use a graphing utility to find the \(x\) -values at which \(f\) is differentiable. \(f(x)=|x+3|\)

Short Answer

Expert verified
The function \(f(x) = |x + 3|\) is differentiable for all \(x\)-values except for \(x = -3\).

Step by step solution

01

Understand the function

The function \(f(x) = |x+3|\) is an absolute value function, which means it will create a 'v' shape in its graphical representation. It will be positive whenever the input, \(x+3\), is positive, and negative whenever \(x+3\) is negative.
02

Identify the critical point

The critical point for the function is when the input to the absolute value operator, \(x+3\), is equal to zero. This is because at this point the function switches from being negative to positive (or vice versa), creating a sharp turn in the graph. This point is what's called 'non-differentiable', meaning there is no derivative at this point. In this case, the critical point is at \(x = -3\).
03

Explain differentiability

A function is differentiable at a particular \(x\)-value if there exists a derivative at that point, meaning the function has a definite slope at that point. In this function, at \(x = -3\), the function does not have a definite slope due to the sharp turn in the graph, thereby making the function non-differentiable at \(x = -3\).

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Key Concepts

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

Absolute Value Function
An absolute value function is a special type of mathematical function represented as \( f(x) = |x| \). It outputs the non-negative value of any real number \( x \). This simply means that whether you input a positive or negative number, the result will always be positive.
In the exercise given, the function is \( f(x) = |x+3| \). This function will create a graph that looks like a 'V'.
Think of the absolute value as measuring how far a number is from zero on a number line, always yielding a non-negative distance.
The behavior of absolute value graphs is very consistent:
  • For inputs where the expression inside the absolute value is positive or zero, \( f(x) = x+3 \).
  • For inputs where the expression inside is negative, \( f(x) = -(x+3) \).
Understanding this helps in analyzing and graphing the function's structure.
Critical Point
A critical point in a function is where the graph changes direction or slope. For absolute value functions, these points often occur where the expression inside the absolute value equals zero.
In \( f(x) = |x+3| \), the critical point is when \( x+3 = 0 \), solving which gives us \( x = -3 \). At this point, the graph switches from decreasing to increasing or vice versa, forming a sharp corner.
It's important to identify critical points as they provide key insights:
  • They indicate where potential changes in direction occur.
  • They can highlight areas where the function may be non-differentiable.
Critical points are crucial in calculus for understanding a function's behavior.
Graphing Utility
A graphing utility is a tool that helps visualize functions by plotting their graphs on a coordinate system.
These can be handheld calculators or software applications that provide a graphical representation of functions quickly and accurately.
In the context of \( f(x) = |x+3| \), using a graphing utility will allow you to clearly see the 'V' shape and notice where the critical point \( x = -3 \) causes a sharp turn. This is very helpful for understanding why certain points are non-differentiable.
Benefits of using a graphing utility include:
  • Immediate visualization of complex functions.
  • Ability to zoom in on areas of interest, such as critical points.
  • Aid in examining the differentiability of functions at various points.
By practicing with a graphing utility, students enhance their comprehension of both functions and calculus concepts.
Non-differentiable Points
Non-differentiable points are locations on a graph where a function does not have a derivative.
Simply put, a derivative represents the slope or rate of change at a given point on the graph. If there鈥檚 a kink or sharp point鈥攁s is often the case with absolute value functions鈥攖he derivative doesn't exist.
In \( f(x) = |x+3| \), the graph has a sharp turn at \( x = -3 \), indicating a non-differentiable point. This sharp turn prevents the function from having a single, clear slope, causing the non-differentiable nature at this point.
Key facts about non-differentiable points include:
  • They occur generally at sharp edges, cusps, or vertical tangents.
  • Functions usually cannot have a defined slope at these points.
  • Understanding non-differentiable points helps in knowing where certain rules of calculus do not apply.
Recognizing these points strengthens your grasp of function behavior and calculus applications.

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Most popular questions from this chapter

In Exercises 107-110, (a) use a graphing utility to find the derivative of the function at the given point, (b) find an equation of the tangent line to the graph of the function at the given point, and (c) use the utility to graph the function and its tangent line in the same viewing window. \(f(x)=\sqrt{x}(2-x)^{2}, \quad(4,8)\)

In Exercises 35 and 36, find an equation of the tangent line to the graph of the equation at the given point. $$ \arcsin x+\arcsin y=\frac{\pi}{2}, \quad\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) $$

In Exercises 43 and 44, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The slope of the graph of the inverse tangent function is positive for all \(x\).

Angle of Elevation A balloon rises at a rate of 3 meters per second from a point on the ground 30 meters from an observer. Find the rate of change of the angle of elevation of the balloon from the observer when the balloon is 30 meters above the ground.

Determine the point(s) at which the graph of \(y^{4}=y^{2}-x^{2}\) has a horizontal tangent.
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