An object moves along the y axis marked in feet so that its...

An object moves along the y-axis (marked in feet) so that its position at time x (in seconds) is f(x) = x^3 - 6x^2 + 9x. a) Find the instantaneous velocity function v. b) Find the velocity at x = 2 and x = 5 seconds. c) Find the time(s) when the velocity is zero.

Transcript

00:01 In this question we have given an object moves along the y -axis so that the position at times x in second is given as f of x is equal to x -cube minus 6x square plus 9x.

00:21 And using this function we have to find the instantaneous velocity in part a, in part b.

00:30 We have to find the velocity at x is equal to 2 and x is equal to 5 seconds and in part c we have to find when the velocity of the object is 0 we have to find time for that as we know that v of x that is velocity is derivative of the position function so it can be written as v of x is equal to derivative of the function with respect to x.

01:18 So consider position function f of x is equal to x cube minus 6x square plus 9x for the part a.

01:40 So if we derivative this function with respect to x we will get it as v of x is equal to to derivative of x cubed minus 6 x square plus 9x with respect to x.

01:57 So it implies v of x is equal to 3x square minus 12x plus 9.

02:12 Here we have used derivative of x raised to the power n with respect to x is equal to n times x x x to the power n minus 1 so this is the instantaneous velocity of this position vector now in part b we have to find the velocity when x is equal to 2 seconds so we have just now obtained the velocity function as 3 x square minus 12 x plus 9 so if we substitute x is equal to 2 in this equation we will get it as velocity at x is equal to 2 is equal to 3 multiplied with 2 raised to the part 2 minus 12 multiplied with 2 plus 9.

03:08 It implies velocity at x is equal to 2 second is equals to 3 times 4 minus 24 plus 9.

03:21 On simplifying it, we get it as velocity at x is equal to 2 is equal to minus 3.

03:29 We have given position in feet and time in second.

03:33 So the unit will be feet per second.

03:36 Now we have to find velocity at x is equals to 5 seconds.

03:42 So again, v of x is equal to 3x square minus 12x plus 9.

03:47 Therefore, v at x is equal to 5 is equal to 3 times pi raised to the power 2 minus 12 multiplied with 5 plus 9.

04:03 Therefore, on simplifying it, we will get it as 3 times 25 minus 60 plus 9.

04:13 Therefore, we get velocity at x is equal to 5 is equal to 20.

04:19 Now in part c we have to find when v...

Question

An object moves along the y-axis (marked in feet) so that its position at time x (in seconds) is f(x) = x^3 - 6x^2 + 9x. a) Find the instantaneous velocity function v. b) Find the velocity at x = 2 and x = 5 seconds. c) Find the time(s) when the velocity is zero.

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