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Problem 10

Calculate \(\frac{d^{2} y}{d x^{2}}\). . \(y=x^{-2}+\ln x\)

Expert verified

The second derivative of the given function, expressed as \(\frac{d^2 y}{d x^2}\), is:
\[\frac{d^2 y}{d x^2} = \frac{6}{x^4} - \frac{1}{x^2}.\]

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Chapter 12

The demand and unit price for your store's checkered T-shirts are changing with time. Show that the percentage rate of change of revenue equals the sum of the percentage rates of change of price and demand. (The percentage rate of change of a quantity \(Q\) is \(\left.Q^{\prime}(t) / Q(t) .\right)\)

Chapter 12

The following graph shows the approximate value of the U.S. Consumer Price Index (CPI) from September 2004 through November \(2005.32\) The approximating curve shown on the figure is given by $$I(t)=-0.005 t^{3}+0.12 t^{2}-0.01 t+190 \quad(0 \leq t \leq 14)$$ where \(t\) is time in months ( \(t=0\) represents September 2004). a. Use the model to estimate the monthly inflation rate in July \(2005(t=10)\). [Recall that the inflation rate is \(\left.I^{\prime}(t) / I(t) .\right]\) b. Was inflation slowing or speeding up in July \(2005 ?\) c. When was inflation speeding up? When was inflation slowing? HINT [See Example 3.]

Chapter 12

Sketch the graph of the given function, indicating (a) \(x\) - and \(y\) -intercepts, (b) extrema, (c) points of inflection, \((d)\) behavior near points where the function is not defined, and (e) behavior at infinity. Where indicated, technology should be used to approximate the intercepts, coordinates of extrema, and/or points of inflection to one decimal place. Check your sketch using technology. \(f(x)=x^{2}+\frac{1}{x^{2}}\)

Chapter 12

You can now sell 50 cups of lemonade per week at \(30 \&\) per cup, but demand is dropping at a rate of 5 cups per week each week. Assuming that raising the price does not affect demand, how fast do you have to raise your price if you want to keep your weekly revenue constant? HINT [Revenue = Price \(\times\) Quantity.]

Chapter 12

The distance of the Mars orbiter from your location in Utarek, Mars is given by \(s=2(t-1)^{3}\) \(-3(t-1)^{2}+100 \mathrm{~km}\) after \(t\) seconds $(t \geq 0) .$ Obtain the extrema, points of inflection, and behavior at infinity. Sketch the curve and interpret these features in terms of the movement of the Mars orbiter.

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