Find The \[$x\$\]-value(s) Where Each Of The Functions In Problems 12-15 Is Not Differentiable. Give A Reason Why Each Function Is Not Differentiable For Those Values Of \[$x\$\]. Do Not Use A Calculator.12. \[$f(x) = \left|x^2 -

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Introduction

In calculus, differentiability is a crucial concept that deals with the behavior of functions at specific points. A function is said to be differentiable at a point if it has a well-defined derivative at that point. However, there are instances where a function may not be differentiable at a particular point. In this article, we will explore the non-differentiable points of various functions and provide reasons why they are not differentiable at those values.

Problem 12: [$f(x) = \left|x^2 - 4\right|$

The given function is [$f(x) = \left|x^2 - 4\right|$. To find the non-differentiable points of this function, we need to analyze its behavior at different intervals.

Interval 1: [x<−2$x < -2\$

In this interval, the expression [x2−4$isalwaysnegative.Therefore,theabsolutevaluefunction\[x^2 - 4\$ is always negative. Therefore, the absolute value function \[\left|x^2 - 4\right|$ simplifies to [-(x^2 - 4)$. This is a polynomial function, and polynomials are differentiable at all points. Hence, the function [f(x)=∣x2−4∣$isdifferentiablefor\[f(x) = \left|x^2 - 4\right|\$ is differentiable for \[x < -2$.

Interval 2: [−2<x<2$-2 < x < 2\$

In this interval, the expression [x2−4$isalwaysnegative.Therefore,theabsolutevaluefunction\[x^2 - 4\$ is always negative. Therefore, the absolute value function \[\left|x^2 - 4\right|$ simplifies to [-(x^2 - 4)$. This is a polynomial function, and polynomials are differentiable at all points. Hence, the function [$f(x) = \left|x^2 - 4\right|$ is differentiable for [-2 < x < 2$.

Interval 3: [x>2$x > 2\$

In this interval, the expression [x2−4$isalwayspositive.Therefore,theabsolutevaluefunction\[x^2 - 4\$ is always positive. Therefore, the absolute value function \[\left|x^2 - 4\right|$ simplifies to [x^2 - 4$. This is a polynomial function, and polynomials are differentiable at all points. Hence, the function [f(x)=∣x2−4∣$isdifferentiablefor\[f(x) = \left|x^2 - 4\right|\$ is differentiable for \[x > 2$.

Point of Non-Differentiability: [x=2$x = 2\$

At the point [x=2$,theexpression\[x = 2\$, the expression \[x^2 - 4$ is equal to zero. Therefore, the absolute value function [∣x2−4∣$simplifiesto\[0$.Thisisaconstantfunction,andconstantfunctionsarenotdifferentiableatpointswherethefunctionvalueiszero.Hence,thefunction\[\left|x^2 - 4\right|\$ simplifies to \[0\$. This is a constant function, and constant functions are not differentiable at points where the function value is zero. Hence, the function \[f(x) = \left|x^2 - 4\right|$ is not differentiable at [$x = 2$.

Conclusion

In conclusion, the function [f(x)=∣x2−4∣$isnotdifferentiableatthepoint\[f(x) = \left|x^2 - 4\right|\$ is not differentiable at the point \[x = 2$. This is because the function value is zero at this point, and constant functions are not differentiable at points where the function value is zero.

Problem 13: [$f(x) = \left|x^2 - 4\right|^2$

The given function is [$f(x) = \left|x^2 - 4\right|^2$. To find the non-differentiable points of this function, we need to analyze its behavior at different intervals.

Interval 1: [x<−2$x < -2\$

In this interval, the expression [x2−4$isalwaysnegative.Therefore,theabsolutevaluefunction\[x^2 - 4\$ is always negative. Therefore, the absolute value function \[\left|x^2 - 4\right|$ simplifies to [-(x^2 - 4)$. Squaring this expression gives [(x^2 - 4)^2$. This is a polynomial function, and polynomials are differentiable at all points. Hence, the function [f(x)=∣x2−4∣2$isdifferentiablefor\[f(x) = \left|x^2 - 4\right|^2\$ is differentiable for \[x < -2$.

Interval 2: [−2<x<2$-2 < x < 2\$

In this interval, the expression [x2−4$isalwaysnegative.Therefore,theabsolutevaluefunction\[x^2 - 4\$ is always negative. Therefore, the absolute value function \[\left|x^2 - 4\right|$ simplifies to [-(x^2 - 4)$. Squaring this expression gives [(x^2 - 4)^2$. This is a polynomial function, and polynomials are differentiable at all points. Hence, the function [$f(x) = \left|x^2 - 4\right|^2$ is differentiable for [-2 < x < 2$.

Interval 3: [x>2$x > 2\$

In this interval, the expression [x2−4$isalwayspositive.Therefore,theabsolutevaluefunction\[x^2 - 4\$ is always positive. Therefore, the absolute value function \[\left|x^2 - 4\right|$ simplifies to [x^2 - 4$. Squaring this expression gives [(x^2 - 4)^2$. This is a polynomial function, and polynomials are differentiable at all points. Hence, the function [f(x)=∣x2−4∣2$isdifferentiablefor\[f(x) = \left|x^2 - 4\right|^2\$ is differentiable for \[x > 2$.

Point of Non-Differentiability: [x=2$x = 2\$

At the point [x=2$,theexpression\[x = 2\$, the expression \[x^2 - 4$ is equal to zero. Therefore, the absolute value function [∣x2−4∣$simplifiesto\[0$.Squaringthisexpressiongives\[02=0$.Thisisaconstantfunction,andconstantfunctionsarenotdifferentiableatpointswherethefunctionvalueiszero.Hence,thefunction\[\left|x^2 - 4\right|\$ simplifies to \[0\$. Squaring this expression gives \[0^2 = 0\$. This is a constant function, and constant functions are not differentiable at points where the function value is zero. Hence, the function \[f(x) = \left|x^2 - 4\right|^2$ is not differentiable at [$x = 2$.

Point of Non-Differentiability: [x=−2$x = -2\$

At the point [x=−2$,theexpression\[x = -2\$, the expression \[x^2 - 4$ is equal to zero. Therefore, the absolute value function [∣x2−4∣$simplifiesto\[0$.Squaringthisexpressiongives\[02=0$.Thisisaconstantfunction,andconstantfunctionsarenotdifferentiableatpointswherethefunctionvalueiszero.Hence,thefunction\[\left|x^2 - 4\right|\$ simplifies to \[0\$. Squaring this expression gives \[0^2 = 0\$. This is a constant function, and constant functions are not differentiable at points where the function value is zero. Hence, the function \[f(x) = \left|x^2 - 4\right|^2$ is not differentiable at [$x = -2$.

Conclusion

In conclusion, the function [f(x)=∣x2−4∣2$isnotdifferentiableatthepoints\[f(x) = \left|x^2 - 4\right|^2\$ is not differentiable at the points \[x = 2$ and [$x = -2$. This is because the function value is zero at these points, and constant functions are not differentiable at points where the function value is zero.

Problem 14: [$f(x) = \left|x^2 - 4\right|^3$

The given function is [$f(x) = \left|x^2 - 4\right|^3$. To find the non-differentiable points of this function, we need to analyze its behavior at different intervals.

Interval 1: [x<−2$x < -2\$

In this interval, the expression [x2−4$isalwaysnegative.Therefore,theabsolutevaluefunction\[x^2 - 4\$ is always negative. Therefore, the absolute value function \[\left|x^2 - 4\right|$ simplifies to [-(x^2 - 4)$. Cubing this expression gives [-(x^2 - 4)^3$. This is a polynomial function, and polynomials are differentiable at all points. Hence, the function [f(x)=∣x2−4∣3$isdifferentiablefor\[f(x) = \left|x^2 - 4\right|^3\$ is differentiable for \[x < -2$.

Interval 2: [−2<x<2$-2 < x < 2\$

In this interval, the expression [x2−4$isalwaysnegative.Therefore,theabsolutevaluefunction\[x^2 - 4\$ is always negative. Therefore, the absolute value function \[\left|x^2 - 4\right|$ simplifies to [-(x^2 - 4)$. Cubing this expression gives [-(x^2 - 4)^3$. This is a polynomial function, and polynomials are differentiable at all points. Hence, the function [$f(x) = \left|x^2 - 4\right|^3$ is differentiable for [-2 < x < 2$.

Interval 3: [x>2$x > 2\$

Q: What is a non-differentiable point of a function?

A: A non-differentiable point of a function is a point where the function is not differentiable. This means that the function does not have a well-defined derivative at that point.

Q: Why are non-differentiable points important?

A: Non-differentiable points are important because they can affect the behavior of a function. For example, a function may have a sharp turn or a discontinuity at a non-differentiable point.

Q: How do you find non-differentiable points of a function?

A: To find non-differentiable points of a function, you need to analyze the function's behavior at different intervals. You can use various techniques such as graphing, algebraic manipulation, and limit analysis to determine the non-differentiable points.

Q: What are some common types of non-differentiable points?

A: Some common types of non-differentiable points include:

  • Sharp turns: These occur when the function changes direction suddenly.
  • Discontinuities: These occur when the function has a gap or a jump at a particular point.
  • Points of non-differentiability: These occur when the function is not differentiable at a particular point.

Q: How do you determine if a function is differentiable at a point?

A: To determine if a function is differentiable at a point, you need to check if the function has a well-defined derivative at that point. You can use various techniques such as the definition of a derivative, the limit definition of a derivative, and the derivative rules to determine if a function is differentiable at a point.

Q: What are some examples of non-differentiable points?

A: Some examples of non-differentiable points include:

  • Absolute value functions: These functions are not differentiable at points where the expression inside the absolute value is zero.
  • Polynomial functions: These functions are not differentiable at points where the polynomial is zero.
  • Rational functions: These functions are not differentiable at points where the denominator is zero.

Q: Can you provide some examples of non-differentiable points of specific functions?

A: Yes, here are some examples of non-differentiable points of specific functions:

  • Function 1: [$f(x) = \left|x^2 - 4\right|$
    • Non-differentiable point: [$x = 2$
    • Reason: The function value is zero at this point, and constant functions are not differentiable at points where the function value is zero.
  • Function 2: [$f(x) = \left|x^2 - 4\right|^2$
    • Non-differentiable points: [x=2$and\[x = 2\$ and \[x = -2$
    • Reason: The function value is zero at these points, and constant functions are not differentiable at points where the function value is zero.
  • Function 3: [$f(x) = \left|x^2 - 4\right|^3$
    • Non-differentiable points: [x=2$and\[x = 2\$ and \[x = -2$
    • Reason: The function value is zero at these points, and constant functions are not differentiable at points where the function value is zero.

Conclusion

In conclusion, non-differentiable points are an important concept in calculus that can affect the behavior of a function. By understanding the different types of non-differentiable points and how to determine them, you can better analyze and work with functions in various mathematical and real-world applications.