Chapter 2: Problem 75
Find the given higher-order derivative. $$ f^{\prime}(x)=x^{2}, \quad f^{\prime \prime}(x) $$
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Let \(f(x)=a_{1} \sin x+a_{2} \sin 2 x+\cdots+a_{n} \sin n x,\) where \(a_{1}, a_{2}, \ldots, a_{n}\) are real numbers and where \(n\) is a positive integer. Given that \(|f(x)| \leq|\sin x|\) for all real \(x\), prove that \(\left|a_{1}+2 a_{2}+\cdots+n a_{n}\right| \leq 1\)
True or False? In Exercises 137-139, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f(x)=\sin ^{2}(2 x),\) then \(f^{\prime}(x)=2(\sin 2 x)(\cos 2 x)\)
Find the equation(s) of the tangent line(s) to the parabola \(y=x^{2}\) through the given point. (a) \((0, a)\) (b) \((a, 0)\) Are there any restrictions on the constant \(a\) ?
In Exercises \(81-88\), (a) find an equation of the tangent line to the graph of \(f\) at the indicated point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. \(\frac{\text { Function }}{f(x)=\tan ^{2} x} \quad \frac{\text { Point }}{\left(\frac{\pi}{4}, 1\right)}\)
Linear and Quadratic Approximations In Exercises 33 and 34, use a computer algebra system to find the linear approximation $$P_{1}(x)=f(a)+f^{\prime}(a)(x-a)$$ and the quadratic approximation $$P_{2}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}$$ to the function \(f\) at \(x=a\). Sketch the graph of the function and its linear and quadratic approximations. $$ f(x)=\arctan x, \quad a=0 $$
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