Chapter 3: Problem 52
Let \(y=x^{2}+7 x-5 .\) Evaluate \(d y / d t\) when \(x=1\) and \(d x / d t=1 / 3\)
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End Behavior Model Consider the hyperbola $$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$ Show that (a) \(y=\pm \frac{b}{a} \sqrt{x^{2}-a^{2}}\) (b) \(g(x)=(b / a)|x|\) is an end behavior model for $$f(x)=(b / a) \sqrt{x^{2}-a^{2}}$$ (c) \(g(x)=-(b / a)|x|\) is an end behavior model for $$f(x)=-(b / a) \sqrt{x^{2}-a^{2}}$$
In Exercises 61 and \(62,\) use the curve \(x^{2}-x y+y^{2}=1\) Multiple Choice Which of the following is equal to \(d y / d x ?\) (A) \(\frac{y-2 x}{2 y-x} \quad\) (B) \(\frac{y+2 x}{2 y-x}\) (C) \(\frac{2 x}{x-2 y} \quad\) (D) \(\frac{2 x+y}{x-2 y}\) \((\mathbf{E}) \frac{y+2 x}{x}\)
In Exercises \(33-36,\) find \(d y / d x\) $$y=x^{1+\sqrt{2}}(1+\sqrt{2}) x^{\sqrt{2}}$$
Multiple Choice A body is moving in simple harmonic motion with position \(s=3+\sin t .\) At which of the following times is the velocity zero? (A) \(t=0\) (B) \(t=\pi / 4\) (C) \(t=\pi / 2\) (D) \(t=\pi\) (E) none of these
Explorations Let \(f(x)=\left\\{\begin{array}{ll}{x^{2},} & {x \leq 1} \\ {2 x,} & {x>1}\end{array}\right.\) \begin{array}{ll}{\text { (a) Find } f^{\prime}(x) \text { for } x<1 .} & {\text { (b) Find } f^{\prime}(x) \text { for } x>1.2} \\ {\text { (c) Find } \lim _{x \rightarrow 1}-f^{\prime}(x) .2} &{\text { (d) Find } \lim _{x \rightarrow 1^{+}} f^{\prime}(x)}\end{array} \begin{array}{l}{\text { (e) Does } \lim _{x \rightarrow 1} f^{\prime}(x) \text { exist? Explain. }} \\ {\text { (f) Use the definition to find the left-hand derivative of } f^ {}} \\ {\text { at } x=1 \text { if it exists. } } \\ {\text { (g) Use the definition to find the right-hand derivative of } f} \\ {\text { at } x=1 \text { if it exists.}} \\ {\text { (h) Does \(f^{\prime}(1)\)} \text{exist?} \text{Explain.}} \end{array}
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