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Chapter 4: Problem 6
Write the formula for Newton's method and use the given initial approximation to compute the approximations \(x_{1}\) and \(x_{2}\). $$f(x)=x^{2}-2 x-3 ; x_{0}=2$$
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Evaluate the following limits in terms of the parameters a and b, which are positive real numbers. In each case, graph the function for specific values of the parameters to check your results. $$\lim _{x \rightarrow 0}(1+a x)^{b / x}$$
Determine whether the following properties can be satisfied by a function that is continuous on \((-\infty, \infty) .\) If such a function is possible, provide an example or a sketch of the function. If such a function is not possible, explain why. a. A function \(f\) is concave down and positive everywhere. b. A function \(f\) is increasing and concave down everywhere. c. A function \(f\) has exactly two local extrema and three inflection points. d. A function \(f\) has exactly four zeros and two local extrema.
Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. $$a(t)=2 \cos t ; v(0)=1, s(0)=0$$
Determine the following indefinite integrals. Check your work by differentiation. $$\int\left(\csc ^{2} \theta+2 \theta^{2}-3 \theta\right) d \theta$$
Determine the following indefinite integrals. Check your work by differentiation. $$\int \frac{1+\sqrt{x}}{x} d x$$
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