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Chapter 11: Problem 15
Find the indefinite integral and check your result by differentiation. $$ \int d u $$
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The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{0}^{4}\left[(x+1)-\frac{1}{2} x\right] d x $$
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 \(\frac{b-a}{2 n}\left[f\left(x_{0}\right)+2 f\left(x_{1}\right)+\cdots+2 f\left(x_{n-1}\right)+f\left(x_{n}\right)\right]\). $$ \int_{0}^{2} x^{3} d x $$
Determine which value best approximates the area of the region bounded by the graphs of \(f\) and \(g\). (Make your selection on the basis of a sketch of the region and not by performing any calculations.) \(f(x)=2-\frac{1}{2} x, \quad g(x)=2-\sqrt{x}\) (a) 1 (b) 6 (c) \(-3\) (d) 3 (e) 4
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 \(x\). $$ \frac{d R}{d x}=75\left(20+\frac{900}{x}\right) \quad x=500 $$
State whether the function is even, odd, or neither. $$ g(x)=x^{3}-2 x $$
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