Chapter 3: Problem 58
True or False The derivative of \(y=2^{x}\) is \(2^{x} .\) Justify your answer.
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In Exercises 44 and \(45,\) a particle moves along a line so that its position at any time \(t \geq 0\) is given by \(s(t)=2+7 t-t^{2}\) . Multiple Choice At which of the following times is the particle moving to the left? $$(A)t=0 \quad(\mathbf{B}) t=1 \quad(\mathbf{C}) t=2 \quad(\mathbf{D}) t=7 / 2 \quad(\mathbf{E}) t=4$$
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 $$f(x)=-(b / a) \sqrt{x^{2}-a^{2}}$$
True or False The derivative of \(y=\sqrt[3]{x}\) is \(\frac{1}{3 x^{2 / 3}} .\) Justify your answer.
Group Activity Using graphing calculators, have each person in your group do the following: (a) pick two numbers \(a\) and \(b\) between 1 and \(10 ;\) (b) graph the function \(y=(x-a)(x+b)\) ; (c) graph the derivative of your function (it will be a line with slope 2\()\) (d) find the \(y\) -intercept of your derivative a simple way to predict the \(y\) -intercept, given the values of \(a\) and \(b\) . Test your result.
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
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