Calculus

Higher-order Derivatives

Higher Order Derivatives Problem Solving

         

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

Given function f(x)f(x), let f(n)(x)f^{(n)}(x) be the nthn^{th} derivative of the function f(x)f(x) for any positive integer nn. If f(x)=ex+cosx,f(x)=e^x+\cos x, what is the value of k=1124f(k)(0)? \sum_{k=1}^{124} f^{(k)}(0)?

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

Consider the function f(x)=x2exf(x)=x^2e^{-x}. Define another function y=exf(x)y=e^xf'(x). If there exist constants aa and bb that satisfy x2yaxy+by=0x^2y''-axy'+by=0 for all real numbers xx, what is the value of a+ba+b?

f(x)f(x) is a twice differentiable function such that f(2x)=f(x+5)+(x5)2f(2x) = f(x+5) + (x-5)^2. If f(10)=abf''(10) = \frac{a}{b}, where aa and bb are coprime positive integers, what is the value of a+ba+b?

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