Chapter 5: Problem 1
Solve the differential equation. $$ \frac{d y}{d x}=x+2 $$
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In Exercises 75-77, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the area of the region bounded by the graphs of \(f\) and \(g\) is \(1,\) then the area of the region bounded by the graphs of \(h(x)=f(x)+C\) and \(k(x)=g(x)+C\) is also 1.
Find the accumulation function \(F\). Then evaluate \(F\) at each value of the independent variable and graphically show the area given by each value of \(F\). $$ F(x)=\int_{0}^{x}\left(\frac{1}{2} t^{2}+2\right) d t \quad \text { (a) } F(0) \quad \text { (b) } F(4) \quad \text { (c) } F(6) $$
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. $$ \int_{-\pi / 3}^{\pi / 3}(2-\sec x) d x $$
On the Richter scale, the magnitude \(R\) of an earthquake of intensity \(I\) is \(R=\frac{\ln I-\ln I_{0}}{\ln 10}\) where \(I_{0}\) is the minimum intensity used for comparison. Assume that \(I_{0}=1\) (a) Find the intensity of the 1906 San Francisco earthquake \((R=8.3)\) (b) Find the factor by which the intensity is increased if the Richter scale measurement is doubled. (c) Find \(d R / d I\).
Sketch the region bounded by the graphs of the algebraic functions and find the area of the region. $$ f(x)=\sqrt{3 x}+1, g(x)=x+1 $$
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