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Chapter 6: Problem 3

Determine if each function is differentiable at \(x=1\). If it is, find the derivative. If not, explain why not. (a) \(f(x)= \begin{cases}2 x-1 & \text { if } x<1 \\ x^{2} & \text { if } x \geq 1\end{cases}\) (b) \(f(x)= \begin{cases}3 x-1 & \text { if } x<1 \\ x^{3} & \text { if } x \geq 1\end{cases}\) (c) \(f(x)= \begin{cases}3 x-2 & \text { if } x<1 \\ x^{2} & \text { if } x \geq 1\end{cases}\)

Short Answer

Expert verified
(a) The function is differentiable at \(x=1\) with a derivative of \[f'(x)= \begin{cases}2 & \text { if } x<1 \\\ 2x & \text { if } x \geq 1\end{cases}\] (b) The function is differentiable at \(x=1\) with a derivative of \[f'(x)= \begin{cases}3 & \text { if } x<1 \\\ 3x^2 & \text { if } x \geq 1\end{cases}\] (c) The function is not differentiable at \(x=1\) because the left and right-side derivatives are not equal at \(x=1\).

Step by step solution

01

Compute the left-hand derivative

Compute the derivative of the left-side function with respect to x: \[\frac{d}{dx}(2x-1)=2\]
02

Compute the right-hand derivative

Compute the derivative of the right-side function with respect to x: \[\frac{d}{dx}(x^2)=2x\]
03

Compare the derivatives at \(x=1\)

Evaluate the right-hand derivative at \(x=1\): \[2(1)=2\]. The left-hand derivative is also 2. Since the left and right-side derivatives are equal at \(x=1\), the function is differentiable at \(x=1\). The derivative is \[f'(x)= \begin{cases}2 & \text { if } x<1 \\\ 2x & \text { if } x \geq 1\end{cases}\] (b) \(f(x)= \begin{cases}3 x-1 & \text { if } x<1 \\\ x^{3} & \text { if } x \geq 1\end{cases}\)
04

Compute the left-hand derivative

Compute the derivative of the left-side function with respect to x: \[\frac{d}{dx}(3x-1)=3\]
05

Compute the right-hand derivative

Compute the derivative of the right-side function with respect to x: \[\frac{d}{dx}(x^3)=3x^2\]
06

Compare the derivatives at \(x=1\)

Evaluate the right-hand derivative at \(x=1\): \[3(1^2)=3\]. The left-hand derivative is also 3. Since the left and right-side derivatives are equal at \(x=1\), the function is differentiable at \(x=1\). The derivative is \[f'(x)= \begin{cases}3 & \text { if } x<1 \\\ 3x^2 & \text { if } x \geq 1\end{cases}\] (c) \(f(x)= \begin{cases}3 x-2 & \text { if } x<1 \\\ x^{2} & \text { if } x \geq 1\end{cases}\)
07

Compute the left-hand derivative

Compute the derivative of the left-side function with respect to x: \[\frac{d}{dx}(3x-2)=3\]
08

Compute the right-hand derivative

Compute the derivative of the right-side function with respect to x: \[\frac{d}{dx}(x^2)=2x\]
09

Compare the derivatives at \(x=1\)

Evaluate the right-hand derivative at \(x=1\): \[2(1)=2\]. The left-hand derivative is 3. Since the left and right-side derivatives are not equal at \(x=1\), the function is not differentiable at \(x=1\).

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Key Concepts

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

Piecewise Functions
When learning calculus, one of the concepts students often encounter is piecewise functions. These are functions that have different expressions for different intervals of their domain. Essentially, a piecewise function is like a mash-up of two or more functions, each

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Most popular questions from this chapter

Let \(f: I \rightarrow J, g: J \rightarrow K\), and \(h: K \rightarrow \mathbb{R}\), where \(I, J\), and \(K\) are intervals. Suppose that \(f\) is differentiable at \(c \in I, g\) is differentiable at \(f(c)\), and \(h\) is differentiable at \(g(f(c))\). Prove that \(h \circ(g \circ f)\) is differentiable at \(c\) and find the derivative.

Let \(f: I \rightarrow \mathbb{R}\), where \(I\) is an open interval containing the point \(c\), and let \(k \in \mathbb{R}\). Prove the following. (a) \(f\) is differentiable at \(c\) with \(f^{\prime}(c)=k\) iff \(\lim _{h \rightarrow 0}[f(c+h)-f(c)] / h=k\). *(b) If \(f\) is differentiable at \(c\) with \(f^{\prime}(c)=k\), then \(\lim _{h \rightarrow 0}[f(c+h)-\) \(f(c-h)] / 2 h=k\). (c) If \(f\) is differentiable at \(c\) with \(f^{\prime}(c)=k\), then \(\lim _{n \rightarrow \infty} n[f(c+1 / n)-\) \(f(c)]=k\). (d) Find counterexamples to show that the converses of parts (b) and (c) are not true.

Let \(f(x)=x^{2}\) if \(x\) is rational and \(f(x)=0\) if \(x\) is irrational. (a) Prove that \(f\) is continuous at exactly one point, namely at \(x=0\). (b) Prove that \(f\) is differentiable at exactly one point, namely at \(x=0\).

Let \(f(x)=x^{2} \sin \left(1 / x^{2}\right)\) for \(x \neq 0\) and \(f(0)=0\). (a) Show that \(f\) is differentiable on \(\mathbb{R}\). (b) Show that \(f^{\prime}\) is not bounded on the interval \([-1,1]\).

Let \(c\) be a point in the interval \(I\) and suppose \(f: I \rightarrow \mathbb{R}\). Mark each statement True or False. Justify each answer. (a) The derivative of \(f\) at \(c\) is defined by $$ f^{\prime}(c)=\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c} $$ wherever the limit exists. (b) If \(f\) is continuous at \(c\), then \(f\) is differentiable at \(c\). (c) If \(f\) is differentiable at \(c\), then \(f\) is continuous at \(c\).
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