#### To determine

**To explain:**

second and third derivatives f.

**To interpret : **

f" and f"' when f(x) is position function.

#### Answer

a) Acceleration of an object:

f”t=at

b) Jerk of an object:

f'"(t)=j(t)

**Explanation:**

f is position of an object.

If f=f(t) is position function of object that moves in straight line, its first derivative represents the velocity v(s) of object as a function of time:

vs=f't=dfdt

The instantaneous rate of change of velocity with respect to time is called acceleration a(t) of an object. Thus the acceleration function is the derivative of velocity function and is therefore the second derivative of the position function:

at=v't=f"(t)

In Leibnitz notation,

a=dvdt=d2fdt2

Acceleration is the change in velocity you would feel when your vehicle is speeding up or slowing down.

a'=(f")'=f'" the third derivative of the position function is the derivative of the acceleration function and is called the jerk:

j=dadt=d2fdt2

Thus the jerk j is the rate of change of acceleration, which causes an abrupt movement in a vehicle.

#### Explanation

**Given:**

Position of function of an object: y=f(x)

**Calculations:**

If f is differentiable function, then its derivative f' is the first derivative.

In Leibniz notation, f'x=dydx is the first derivative of f.

**Second derivative:**

If f is a differentiable function, then its derivative f' is also a function.f' may have a derivative of its own, denoted by f''=f".

Then f" is called second derivative of f.

In Leibniz notation,

f"(x)=d2ydx2

is second derivative of f.

**Third derivative:**

The third derivative f'" is the derivative of the second derivative, denoted by f'"=(f")'

In Leibniz notation,

f"'(x)=d3ydx3

is third derivative of f.

Now the interpretation of f" and f"' when f(x) is position function.