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Chapter 6: Problem 37
Find all functions \(f(t)\) with the following property: $$f^{\prime}(t)=t^{3 / 2}$$
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Use a Riemann sum to approximate the area under the graph of \(f(x)\) on the given interval, with selected points as specified. \(f(x)=x^{3} ; 1 \leq x \leq 3, n=5\), left endpoints
Heat Diffusion Some food is placed in a freezer. After \(t\) hours, the temperature of the food is dropping at the rate of \(r(t)\) degrees Fahrenheit per hour, where \(r(t)=12+4 /(t+3)^{2} .\) (a) Compute the area under the graph of \(y=r(t)\) over the interval \(0 \leq t \leq 2\). (b) What does the area in part (a) represent?
A property with an appraised value of $$\$ 200,000$$ in 2008 is depreciating at the rate \(R(t)=-8 e^{-.04 t}\), where \(t\) is in years since 2008 and \(R(t)\) is in thousands of dollars per year. Estimate the loss in value of the property between 2008 and 2014 (as \(t\) varies from 0 to 6 ).
Given \(\int_{0}^{1} f(x) d x=3.5\) and \(\int_{1}^{4} f(x) d x=5\), find \(\int_{0}^{4} f(x) d x\).
Use a Riemann sum with \(n=4\) and left endpoints to estimate the area under the graph of \(f(x)=4-x\) on the interval \(1 \leq x \leq 4\). Then repeat with \(n=4\) and midpoints. Compare the answers with the exact answer, \(4.5\), which can be computed from the formula for the area of a triangle.
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