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Answer 1

By defining kinematics and derivative relations we can find which statements are true or false:

         a) False. The derivative is the instantaneous velocity.

        b) True. The second drift is the instantaneous acceleration.

        c) False the derivative is

        d) False the derivative is

a) The velocity is defined with the variation of the position with respect to time.

           $v= \frac{dx}{dt}$  

Wher x is the position and t the time.

In the change of the average velocity is the average value of the velocity in an interval

           $v = \frac{v_f - v_o}{t}$  

We can see that the derivative is the speed in a very small timet, that is, the speed instantaneous. Therefore the statement is False.

 

The prime derivative of the position is the instantaneous velocity, not the average velocity.

b) Acceleration is defined as the change in position with respect to time..

           $a = \frac{dv}{dt}$  

Let's use the chain rule.

          a = $\frac{d}{dt} \frac{dx}{dt}$  

          a = $\frac{d^2 x}{dt^2}$  

Therefore the second  derivative of the position is the instantaneous acceleration.

The statement is True.

Questions c and d ask the derivative of a function

             f (x) = 2 x aⁿ

c) derivative with respect  of x

            $\frac{df}{dx} = 2 a^n$

The answer is False.

d) derivative with respect to a.

            $\frac{df}{da} = 2n \ x a^{n-1}$

Answer d is false

In conclusion using the definition of kinematics and derivative relations we can find which statements are true.

        a) False. The derivative is the instantaneous velocity.

        b) True. The second drift is the instantaneous acceleration.

        c) False the derivative is:   $\frac{df}{dx} = 2 a^n$

        d) False the derivative is:   $\frac{df}{da} = 2n \ x a^{n-1}$

Learn more about the relationship between derivatives and kinematics here:   brainly.com/question/15344251