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Chapter 5: Problem 81
Use the functions \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the given function. \((f \circ g)^{-1}\)
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Some calculus textbooks define the inverse secant function using the range \([0, \pi / 2) \cup[\pi, 3 \pi / 2) .\) (a) Sketch the graph \(y=\operatorname{arcsec} x\) using this range. (b) Show that \(y^{\prime}=\frac{1}{x \sqrt{x^{2}-1}}\)
Use a graphing utility to graph \(f(x)=\sin x\) and \(g(x)=\arcsin (\sin x)\). (a) Why isn't the graph of \(g\) the line \(y=x ?\) (b) Determine the extrema of \(g\)
Use a computer algebra system to find the linear approximation \(P_{1}(x)=f(a)+f^{\prime}(a)(x-a)\) and the quadratic approximation \(P_{2}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}\) of the function \(f\) at \(x=a\) . Sketch the graph of the function and its linear and quadratic approximations. \(f(x)=\arctan x, \quad a=0\)
Proof Prove that
$$\tanh ^{-1} x=\frac{1}{2} \ln \left(\frac{1+x}{1-x}\right), \quad-1
Find an equation of the tangent line to the graph of the function at the given point. \(y=3 x \arcsin x, \quad\left(\frac{1}{2}, \frac{\pi}{4}\right)\)
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