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A mass on a spring vibrates horizontally on a smooth level surface (see the figure). Its equation of motion is $ x(t) = 8 \sin t, $ where $ t $ is in seconds and $ x $ in centimeters.(a) Find the velocity and acceleration at time $ t. $(b) Find the position, velocity, and acceleration of the mass at time $ t = 2\pi/3. $ In what direction is it moving at that time?

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a) $V=8 \cos (t), A_{t}=-8 \sin (t)$b) position: $$4\sqrt{3} \mathrm{cm} \quad \text{moving to the left}$$ velocity: $$-4 \mathrm{cm} / \mathrm{sec}$$acceleration: $$-4 \sqrt{3} \mathrm{cm} / \mathrm{sec}^{2}$$

01:22

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 3

Derivatives of Trigonometric Functions

Derivatives

Differentiation

Harvey Mudd College

Baylor University

Idaho State University

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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he has clear. So when you right here. So we have. Except T is equal to eight sign. So this is given. Next We're gonna find the derivative. So we get eight co sign her derivative is equal to the velocity And to find acceleration, we just have to derive it one more time. We get negative eight sign. It's a party for part B. We have the velocity function. This is equal to V. This is equal. Today we have except t, which is the position function. So worried you were gonna plug in to pie over three pretty for all three equations when we get four square root of three, which is around six point 93 centimeters. Next we're gonna plug it into our velocity function. So we get negative four centimeters per second and finally our acceleration function which give this negative six points 93 centimeters per second square. Since the velocities negative, it's going to the left

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