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Chapter 5: Problem 4
Evaluate the expression without using a calculator. \(\arcsin \frac{1}{2}\)
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Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The slope of the graph of the inverse tangent function is positive for all \(x .\)
From the vertex \((0, c)\) of the catenary \(y=c \cosh (x / c)\) a line \(L\) is drawn perpendicular to the tangent to the catenary at point \(P .\) Prove that the length of \(L\) intercepted by the axes is equal to the ordinate \(y\) of the point \(P .\)
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. \(f(x)=\arccos x, \quad a=0\)
Find the derivative of the function. \(f(x)=\arcsin x+\arccos x\)
Proof Prove that
$$\tanh ^{-1} x=\frac{1}{2} \ln \left(\frac{1+x}{1-x}\right), \quad-1
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