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Chapter 2: Problem 97

Where are the functions \(f_{1}(x)=|\sin x|\) and \(f_{2}(x)=\sin |x|\) differentiable?

Short Answer

Expert verified
The function \(f_{1}(x)=|\sin x|\) is differentiable everywhere except at \(x = n\pi\) and the function \(f_{2}(x)=\sin |x|\) is differentiable everywhere except at \(x=0\).

Step by step solution

01

Differentiability of \(f_{1}(x)\)

Start by looking at the function \(f_{1}(x)=|\sin x|\). The function \(\sin x\) is differentiable for all real numbers. However, the absolute value function is not differentiable at zero. Therefore, the function \(f_{1}(x)=|\sin x|\) will not be differentiable whenever \(\sin x = 0\). Remembering the properties of sine function, it equals to zero at \(x = n\pi\) where \(n\) is an integer.
02

Differentiability of \(f_{2}(x)\)

Now, observe the function \(f_{2}(x)=\sin |x|\). The sine function is differentiable for all real numbers, so the issue here would be when \(|x|\) is not differentiable. Like in the first part of the problem, an absolute value function is not differentiable at zero. Therefore, \(f_{2}(x)=\sin |x|\) is not differentiable at \(x=0\).
03

Conclusion

Having analysed both functions, we can conclude that \(f_{1}(x)=|\sin x|\) is differentiable everywhere except at \(x = n\pi\), where \(n\) is an integer. The second function, \(f_{2}(x)=\sin |x|\) is differentiable everywhere except at \(x=0\).

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Most popular questions from this chapter

In Exercises \(81-88\), (a) find an equation of the tangent line to the graph of \(f\) at the indicated point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. \(\frac{\text { Function }}{y=4-x^{2}-\ln \left(\frac{1}{2} x+1\right)} \quad \frac{\text { Point }}{\left(0,4\right)}\)

True or False? In Exercises 137-139, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f(x)=\sin ^{2}(2 x),\) then \(f^{\prime}(x)=2(\sin 2 x)(\cos 2 x)\)

Linear and Quadratic Approximations The linear and quadratic approximations of a function \(f\) at \(x=a\) are \(P_{1}(x)=f^{\prime}(a)(x-a)+f(a)\) and \(P_{2}(x)=\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}+f^{\prime}(a)(x-a)+f(a)\) \(\begin{array}{llll}\text { In Exercises } & 133-136, & \text { (a) find the specified linear and }\end{array}\) quadratic approximations of \(f,\) (b) use a graphing utility to graph \(f\) and the approximations, (c) determine whether \(P_{1}\) or \(P_{2}\) is the better approximation, and (d) state how the accuracy changes as you move farther from \(x=a\). $$ \begin{array}{l} f(x)=x \ln x \\ a=1 \end{array} $$

Use the position function \(s(t)=-16 t^{2}+v_{0} t+s_{0}\) for free-falling objects. A silver dollar is dropped from the top of a building that is 1362 feet tall. (a) Determine the position and velocity functions for the coin. (b) Determine the average velocity on the interval [1,2] . (c) Find the instantaneous velocities when \(t=1\) and \(t=2\). (d) Find the time required for the coin to reach ground level. (e) Find the velocity of the coin at impact.

Linear and Quadratic Approximations In Exercises 33 and 34, use a computer algebra system to find the linear approximation $$P_{1}(x)=f(a)+f^{\prime}(a)(x-a)$$ and the quadratic approximation $$P_{2}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}$$ to the function \(f\) at \(x=a\). Sketch the graph of the function and its linear and quadratic approximations. $$ f(x)=\arccos x, \quad a=0 $$
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