91Ó°ÊÓ

Evaluate. Assume \(u>0\) when In \(u\) appears. (Be sure to check by differentiating!) $$\int e^{3 x} d x$$

Calculate total cost, disregarding any fixed costs. Total cost from marginal cost. Shelly's Roadside Fruit has found that the marginal cost of producing \(x\) pints of fresh-squeezed orange juice is given by \(C^{\prime}(x)=0.000008 x^{2}-0.004 x+2,\) for \(x \leq 350\) where \(C^{\prime}(x)\) is in dollars. Approximate the total cost of producing 270 pt of juice, using 3 sub-intervals over [0,270] and the left endpoint of each sub-interval.

Determine the \(x\) -values at which the graphs of f and \(g\) cross. If no such \(x\) -values exist, state that fact. $$f(x)=9, \quad g(x)=x^{2}$$

Calculate total cost, disregarding any fixed costs. Total cost from marginal cost. Mangianello Paving, Inc., has found that the marginal cost, in dollars, of paving a road surface with asphalt is given by $$ C^{\prime}(x)=\frac{1}{6} x^{2}-20 x+1800, \quad \text { for } x \leq 80 $$ where \(x\) is measured in hundreds of feet. Use 4 sub-intervals over [0,40] and the left endpoint of each sub-interval to approximate the total cost of paving 4000 fi of road surface.

Determine these indefinite integrals. $$\int \frac{1}{x^{5}} d x$$

Evaluate using integration by parts or substitution. Check by differentiating. $$\int x^{2} \ln x^{3} d x$$

Evaluate. Assume \(u>0\) when In \(u\) appears. (Be sure to check by differentiating!) $$\int e^{7 x} d x$$

Determine the \(x\) -values at which the graphs of f and \(g\) cross. If no such \(x\) -values exist, state that fact. $$f(x)=8, \quad g(x)=\frac{1}{2} x^{2}$$

Find the area under the given curve over the indicated interval. $$y=\frac{2}{x} ; \quad[1,4]$$

Write summation notation for each expression. $$3+6+9+12+15+18$$