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An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the mass is pulled downward and then released, it vibrates vertically. The equation of motion is $ s = 2 \cos t + 3 \sin t, t \ge 0, $ where s is measured in centimeters and $ t $ in seconds. (Take the positive direction to be downward.)(a) Find the velocity and acceleration at time $ t. $(b) Graph the velocity and acceleration functions.(c) When does the mass pass through the equilibrium position for the first time?(d) How far from its equilibrium position does the mass travel?(e) When is the speed the greatest?

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00:01

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 3

Derivatives of Trigonometric Functions

Derivatives

Differentiation

Oregon State University

Idaho State University

Boston College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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it's clear. So it name right here. So velocity is the derivative of the position we have us is equal to to co sign was three sign When we derive this, we've got negative to sign less three co sign To get acceleration, we have to derive velocity Getting negative to co sign minus three. Sign per per p. We're gonna draw a graph of velocity and acceleration. So let's start from here. This is gonna be velocity an acceleration, something like this report. See, you know that the mass in equilibrium is when the acceleration is zero and we got acceleration in part A. So you get zero is equal to negative to co sign minus three sign when we get a T value. Well, arken of negative 2/3 which is around negative 0.59 We know that the tan function has a period of Hi. So it's gonna be a equilibrium when t is equal to negative 0.59 plus and pie. And we could just plug in one for and and you get 2.55 So the mass ISS equilibrium when t is equal to 2.55 party, we're gonna use the velocity function, which is the derivative of our position function. We're gonna make a equal to zero when we get a value for tea to be our 10 off three three house, which is around 0.98 to eight. We plug this in to our equation. Her distance equation for us. Really good. Around 3.6 0567 years. Known for part E. We know that speed is the absolute value of velocity. So we're gonna take us of tea. It is equal to zero, and we get our 10 of the negative 2/3 for a tea. And we know this tangerine inverse of negative two birds gives a negative number. We know that we can have a negative. So we need to write a specific equation. Just he is equal to an inverse of 2/3 waas ti pi. Since it repeats every rotation off pie and we're just gonna plug in one for Kay, we get two 0.5536 seconds

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