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Solve the given problems by finding the appropriate derivative.Assuming that force is proportional to acceleration, show that a particle moving along the $x$ -axis, so that its displacement $x=a e^{k t}+b e^{-k t},$ has a force acting on it which is proportional to its displacement.

$$a=k^{2} x$$

Calculus 1 / AB

Chapter 27

Differentiation of Transcendental Functions

Section 8

Applications

Derivatives

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we need to show that part of what moving along the X axis has a displacement. That is equal to eight times 3 to the K. T. Plus B, times E. To the negative Casey. So let's, so the forces that none of my F deceleration by a According to Newton's 2nd law efforts, proportion of A. And uh is equal to K. Prone time. Say. So we're Kay, prime minister constant of proportionality. So we know that the acceleration is given by the double derivative of displacement. So dx DT That's equal to the 18. Uh huh. Two times A. To the K. T minus B. My name's beat me. It's in a negative Katie. Now, the second derivative, the best case square I'm E. To the K. T. Must be E to the negative Katie. So put in the value of this place. So X equal to E. To the K. T. Cosby into the -K two.

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