Question

I have a calculus question about limits and velocity, pic included

Answer 1

Instantaneous velocity is a concept in calculus that refers to the velocity of an object at a specific instant in time. It can be found using calculus by taking the limit as the time interval approaches zero. However, in many cases, we can find precise values for instantaneous velocity without calculus.

Step-by-step explanation:

The concept being discussed in this question is instantaneous velocity, which falls under the topic of calculus in mathematics. Instantaneous velocity refers to the velocity of an object at a specific instant in time. It can be found using calculus by taking the limit as the time interval approaches zero. However, in many cases, we can find precise values for instantaneous velocity without calculus.

For example, if we have the position function of an object, we can find the instantaneous velocity by finding the derivative of the position function. The derivative gives us the rate of change of position with respect to time, which is the velocity. By evaluating the derivative at a specific time, we can determine the instantaneous velocity at that instant.

Overall, instantaneous velocity is a key concept in calculus and is used to understand the motion of objects in a precise manner.

Answer 2

To find the velocitiy of the given mass at t=4, we have to find the derivative of the function of position and then evaluate it at t=4:

`\begin{gathered} s^(\prime)(t)=8\cdot cos(4t) \\ s^(\prime)(4)=8\cdot cos(16) \\ s^(\prime)(4)=-7.66 \end{gathered}`

It means that the velocity of the mass is -7.66in/s.

To find the acceleration of the mss at t=4, we have to find the second derivative of the function and evaluate it at t=4, to do it find the derivative of the first derivative:

`\begin{gathered} s^(\prime)^(\prime)(t)=-32\cdot sin(4t) \\ s^(\prime)^(\prime)(4)=-32\cdot sin(16) \\ s^(\prime)^(\prime)(4)=9.21 \end{gathered}`

It means that the acceleration of the mass is 9.21in/s².