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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

A particle moves along a horizontal line so that its coordinate at time t is x = b 2 + c 2 t 2 , t 0 , where b and c are positive constants.

(a) Find the velocity and acceleration functions.

(b) Show that the particle always moves in the positive direction.

To determine

(a)

To find: The velocity and acceleration functions

Explanation

1) Concept:

By differentiating position function we can obtain velocity function and further by differentiating velocity function we can obtain acceleration function

2) Formula:

(i) velocity=v=dxdt

(ii) acceleration=a=dvdt

(iii) Chain rule: ddx(fgx=f'gx*g'(x)

(iv) Sum rule: ddxfx+gx=ddxfx+ddx(gx)

(v) Constant multiple rule: ddxCfx=Cddxfx

(vi) Constant function rule: ddxc=0

(vii) Power rule: ddxxn=nxn-1

(viii) Product rule: ddxfx*gx=fxddxgx+gxddx(fx)

3) Given:

Position function is x= b2+c2t2, t0

4) Calculation:

Consider the function,

x= b2+c2t2

By using formula,

v=dxdt=ddt(b2+c2t2)

By using power and chain rule,

v=12b2+c2t2ddtb2+c2t2

By using sum rule,

v=12b2+c2t2ddtb2+ddt(c2t2)

By using constant function rule and constant multiple rule,

v=12b2+c2t20+c2ddtt2

By using power rule,

v=12b2+c2t2(2c2t)

Therefore,

v=c2tb2+c2t2

Now by using formula,

a=dvdt=ddtc2tb2+c2t2=ddtc2tb2+c2t2-12

By using product rule,

a=c2tddtb2+c2t2-12+ b2+c2t2-12)ddt(c

To determine

(b)

To show: Particle always moves in the positive direction

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