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Chapter 2: Problem 16
Find \(f^{\prime}(x)\). $$ f(x)=a x^{3}+b x^{2}+c x+d \quad(a, b, c, d \text { constant }) $$
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You are asked in these exercises to determine whether a piecewise-defined function \(f\) is differentiable at a value \(x=x_{0}\) where \(f\) is defined by different formulas on different sides of \(x_{0} .\) You may use without proof the following result, which is a consequence of the Mean-Value Theorem (discussed in Section 4.8). Theorem. Let \(f\) be continuous at \(x_{0}\) and suppose that \(\lim _{x \rightarrow x_{0}} f^{\prime}(x)\) exists. Then \(f\) is differentiable at \(x_{0}\), and \(f^{\prime}\left(x_{0}\right)=\lim _{x \rightarrow x_{0}} f^{\prime}(x)\) Show that $$ f(x)=\left\\{\begin{array}{ll} x^{2}+x+1, & x \leq 1 \\ 3 x, & x>1 \end{array}\right. $$ is continuous at \(x=1\). Determine whether \(f\) is differen tiable at \(x=1\). If so, find the value of the derivative there Sketch the graph of \(f\).
Suppose that a function \(f\) is differentiable at \(x=0\) with \(f(0)=f^{\prime}(0)=0\), and let \(y=m x, m \neq 0\), denote any line of nonzero slope through the origin. (a) Prove that there exists an open interval containing 0 such that for all nonzero \(x\) in this interval \(|f(x)|<\left|\frac{1}{2} m x\right| .\) [Hint: Let \(\epsilon=\frac{1}{2}|m|\) and apply Definition \(1.4 .1\) to \((5)\) with \(\left.x_{0}=0 .\right]\) (b) Conclude from part (a) and the triangle inequality that there exists an open interval containing 0 such that \(|f(x)|<|f(x)-m x|\) for all \(x\) in this interval. (c) Explain why the result obtained in part (b) may be interpreted to mean that the tangent line to the graph of \(f\) at the origin is the best linear approximation to \(f\) at that point.
If an object suspended from a spring is displaced vertically from its equilibrium position by a small amount and released, and if the air resistance and the mass of the spring are ignored, then the resulting oscillation of the object is called simple harmonic motion. Under appropriate conditions the displacement \(y\) from equilibrium in terms of time \(t\) is given by $$ y=A \cos \omega t $$ where \(A\) is the initial displacement at time \(t=0\), and \(\omega\) is a constant that depends on the mass of the object and the stiffness of the spring (see the accompanying figure). The constant \(|A|\) is called the amplitude of the motion and \(\omega\) the angular frequency. (a) Show that $$ \frac{d^{2} y}{d t^{2}}=-\omega^{2} y $$ (b) The period \(T\) is the time required to make one complete oscillation. Show that \(T=2 \pi / \omega\). (c) The frequency \(f\) of the vibration is the number of oscillations per unit time. Find \(f\) in terms of the period \(T\). (d) Find the amplitude, period, and frequency of an object that is executing simple harmonic motion given by \(y=0.6 \cos 15 t\), where \(t\) is in seconds and \(y\) is in centimeters.
Find (a) \(f^{\prime \prime \prime}(2)\), where \(f(x)=3 x^{2}-2\) (b) \(\left.\frac{d^{2} y}{d x^{2}}\right|_{x=1}\), where \(y=6 x^{5}-4 x^{2}\) (c) \(\left.\frac{d^{4}}{d x^{4}}\left[x^{-3}\right]\right|_{x=1}\)
Write a paragraph that explains what it means for a function to be differentiable. Include examples of functions that are not differentiable as well as examples of functions that are differentiable.
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