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Higher-order Derivatives

The first derivative is the slope of a curve, and the second derivative is the slope of the slope, like acceleration. So what's the third derivative? (Fun fact: it's actually called jerk.)

Problem Solving

         

Let \(f(t)=21 \ln (t+1)+t\) be the position at time \(t\) of point \(P\) moving along a number line. What is the acceleration of \(P\) when the speed is \(4?\)

Given function \(f(x)\), let \(f^{(n)}(x)\) be the \(n^{th}\) derivative of the function \(f(x)\) for any positive integer \(n\). If \(f(x)=e^x+\cos x,\) what is the value of \[ \sum_{k=1}^{124} f^{(k)}(0)?\]

Let \(P\) be a point moving in the \(xy\)-plane whose coordinates at time \(t\) are given by \[x(t)=2e^t\sin t, y(t)=6\cos t.\] What is the minimum value of the magnitude of the acceleration of \(P?\)

Consider the function \(f(x)=x^2e^{-x}\). Define another function \(y=e^xf'(x)\). If there exist constants \(a\) and \(b\) that satisfy \[x^2y''-axy'+by=0\] for all real numbers \(x\), what is the value of \(a+b\)?

\(f(x)\) is a twice differentiable function such that \(f(2x) = f(x+5) + (x-5)^2\). If \(f''(10) = \frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, what is the value of \(a+b\)?

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