Question

If the current position of the object at time t is s(t), then the position at time h later is s(t+h). The average velocity (speed) during that additional time h is (s(t+h)-s(t))/h. If we want to analyze the instantaneous velocity at time t, this can be made into a mathematical model by taking the limit as h approaches 0, the derivative of s(t). Use this function in the model below for the velocity function v(t). The acceleration is the rate of change of velocity, so using the same logic, the acceleration function a(t) can be modeled with the derivative of the velocity function, or the second derivative of the position function a(t) = v'(t): Problem Set question: A particle moves according to the position function 2t sin(5t). Enclose arguments of functions in parentheses. For example, sin(2t). (a) Find the velocity function: v(t) = 10t cos(5t) (b) Find the acceleration function: a(t) = 10cos(5t) - 50tsin(5t) 

          If the current position of the object at time t is s(t), then the position at time h later is s(t+h). The average velocity (speed) during that additional time h is (s(t+h)-s(t))/h. If we want to analyze the instantaneous velocity at time t, this can be made into a mathematical model by taking the limit as h approaches 0, the derivative of s(t). Use this function in the model below for the velocity function v(t).

The acceleration is the rate of change of velocity, so using the same logic, the acceleration function a(t) can be modeled with the derivative of the velocity function, or the second derivative of the position function a(t) = v'(t):

Problem Set question:
A particle moves according to the position function 2t sin(5t). Enclose arguments of functions in parentheses. For example, sin(2t).

(a) Find the velocity function:
v(t) = 10t cos(5t)

(b) Find the acceleration function:
a(t) = 10cos(5t) - 50tsin(5t)
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Added by Gema O.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
James Stewart 8th Edition
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If the current position of the object at time t is s(t), then the position at time h later is s(t+h). The average velocity (speed) during that additional time h is (s(t+h)-s(t))/h. If we want to analyze the instantaneous velocity at time t, this can be made into a mathematical model by taking the limit as h approaches 0, the derivative of s(t). Use this function in the model below for the velocity function v(t). The acceleration is the rate of change of velocity, so using the same logic, the acceleration function a(t) can be modeled with the derivative of the velocity function, or the second derivative of the position function a(t) = v'(t): Problem Set question: A particle moves according to the position function 2t sin(5t). Enclose arguments of functions in parentheses. For example, sin(2t). (a) Find the velocity function: v(t) = 10t cos(5t) (b) Find the acceleration function: a(t) = 10cos(5t) - 50tsin(5t)
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Transcript

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00:01 All right, guys, so in this problem, i might be reading it wrong.
00:04 It's kind of hard to understand the whole average velocity, the position.
00:09 But so if our position function, s of t, is equal to 2t, sine of 5t, well, if we want to find the velocity function, all we have to do is take the derivative of that.
00:27 So the derivative of this, so we take the derivative of the first of the 2t, so that's 2 times the sign of 5t, plus now we leave 2t alone.
00:40 We take the derivative of 5t, which is the cosine of 5t times the derivative of what we're taking the cosine of, which is 5.
00:52 So i think that's how we're supposed to type it.
00:54 So our velocity function should be 2 times the sign of 5t plus 10 t times the cosine of 5t.
01:13 And that's our velocity function with respect to time.
01:17 All right.
01:17 So now if we want the acceleration function, we just have to take the derivative of our velocity function.
01:24 Well, the derivative of 2 sign of 5t is not negative, is just 2, okay, 2 cosine of 5t times the 5.
01:41 So that's going to be 10 cosine of 5t plus now i got to do the product rule with this.
01:51 So that's going to be 10 times the cosine of 5t plus 10t times.
02:02 Now i'm taking the derivative of cosine, which is negative sign of 5t times 5 because i've got to take the derivative of what's the chain rule with the sign.
02:15 All right.
02:16 So all of this 10 cosine, so this 10 cosine of 5t was this part here.
02:24 Now, adding, i have another 10 cosine of 5t plus, so i have a negative and a 5 and a 10, so it's actually really going to be a negative.
02:47 So it'll be minus 50t times the sign of 5t...
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