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Chapter 11: Problem 16
Find the indefinite integral and check your result by differentiation. $$ \int d r $$
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Find the change in cost \(C\), revenue \(R\), or profit \(P\), for the given marginal. In each case, assume that the number of units \(x\) increases by 3 from the specified value of \(x\). $$ \frac{d P}{d x}=\frac{400-x}{150} \quad x=200 $$
Use the Trapezoidal Rule with \(n=8\) to approximate the definite integral. Compare the result with the exact value and the approximation obtained with \(n=8\) and the Midpoint Rule. Which approximation technique appears to be better? Let \(f\) be continuous on \([a, b]\) and let \(n\) be the number of equal subintervals (see figure). Then the Trapezoidal Rule for approximating \(\int_{a}^{b} f(x) d x\) is \(\frac{b-a}{2 n}\left[f\left(x_{0}\right)+2 f\left(x_{1}\right)+\cdots+2 f\left(x_{n-1}\right)+f\left(x_{n}\right)\right]\). $$ \int_{0}^{2} x^{3} d x $$
Use the Midpoint Rule with \(n=4\) to approximate the area of the region bounded by the graph of \(f\) and the \(x\) -axis over the interval. Compare your result with the exact area. Sketch the region. $$ f(x)=x^{2}-x^{3} $$ $$ [0,1] $$
The rate of change of mortgage debt outstanding for one- to four-family homes in the United States from 1998 through 2005 can be modeled by \(\frac{d M}{d t}=5.142 t^{2}-283,426.2 e^{-x}\) where \(M\) is the mortgage debt outstanding (in billions of dollars) and \(t\) is the year, with \(t=8\) corresponding to \(1998 .\) In 1998 , the mortgage debt outstanding in the United States was \(\$ 4259\) billion. (Source: Board of Governors of the Federal Reserve System) (a) Write a model for the debt as a function of \(t\). (b) What was the average mortgage debt outstanding for 1998 through \(2005 ?\)
A deposit of \(\$ 2250\) is made in a savings account at an annual interest rate of \(6 \%\), compounded continuously. Find the average balance in the account during the first 5 years.
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