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Chapter 2: Problem 7
The theorem that can be used to determine the possible numbers of positive real zeros and negative real zeros of a function is called _____ _____ of _____.
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Apple juice has a pH of 2.9 and drinking water has a pH of \(8.0 .\) The hydrogen ion concentration of the apple juice is how many times the concentration of drinking water?
Website Growth The number \(y\) of hits a new website receives each month can be modeled by \(y=4080 e^{k t},\) where \(t\) represents the number of months the website has been operating. In the website's thirdmonth, there were \(10,000\) hits. Find the value of \(k,\) and use this value to predict the number of hits the website will receive after 24 months.
pH Levels In Exercises \(51-56\) , use the acidity model given by \(p \mathbf{H}=-\log \left[\mathbf{H}^{+}\right],\) where acidity \((\mathbf{p} \mathbf{H})\) is a measure of the hydrogen ion concentration \(\left[\mathbf{H}^{+}\right]\) (measured in moles of hydrogen per liter) of a solution. Compute \(\left[\mathrm{H}^{+}\right]\) for a solution in which \(\mathrm{pH}=3.2\)
Proof Prove that the complex conjugate of the product of two complex numbers \(a_{1}+b_{1} i\) and \(a_{2}+b_{2} i\) is the product of their complex conjugates.
Graphical Reasoning Sketch a graph of the function \(f(x)=x^{4} .\) Explain how the graph of each function \(g\) differs (if it does) from the graph of \(f .\) Determine whether \(g\) is even, odd, or neither. (a) \(g(x)=f(x)+2\) (b) \(g(x)=f(x+2)\) (c) \(g(x)=f(-x)\) (d) \(g(x)=-f(x)\) (e) \(g(x)=f\left(\frac{1}{2} x\right)\) (f) \(g(x)=\frac{1}{2} f(x)\) (g) \(g(x)=f\left(x^{3 / 4}\right)\) (h) \(g(x)=(f \circ f)(x)\)
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