Applications of primitive functions

Primitive functions are also known by the more descriptive term antiderivatives.

To be a primitive function, a function $F$F must be differentiable on an open interval. If it has derivative $F'=f$F′=f, then $F$F is said to be an antiderivative or primitive function of $f$f.

An antiderivative $F$F of a function $f$f is not unique since the derivative of $F+c$F+c where $c$c is any constant is also $f$f. We need further information in problems to determine the constant $c$c.

 

Generally, if we have a function that gives the rate of change of some quantity, often with respect to time, then an antiderivative of the function gives the quantity itself.

For example, velocity is the rate of change of displacement with respect to time. So, given velocity as a function of time, we obtain displacement as a function of time by antidifferentiation.

Similarly, acceleration is the time rate of change of velocity. So, a velocity function is obtained from an acceleration function by antidifferentiation.

The integral sign is used for antidifferentiation. Thus for displacement $s$s, velocity $v$v and acceleration $a$a, we have 

$s=\int v(t)\ \mathrm{d}t+c$s=∫v(t) dt+c

$v=\int a(t)\ \mathrm{d}t+c$v=∫a(t) dt+c

 

Example 1

Consider a body moving so that its velocity as a function of time is given by $v(t)=5t^2-2t+3$v(t)=5t2−2t+3. Given that its displacement at $t=0$t=0 is $-2$−2, find its displacement function $s(t)$s(t). What is the displacement at $t=3$t=3?

Since $s(t)=\int v(t)\ \mathrm{d}t+c$s(t)=∫v(t) dt+c, we have $s(t)=\int5t^2-2t+3\ \mathrm{d}t+c$s(t)=∫5t2−2t+3 dt+c. Hence, $s(t)=\frac{5t^3}{3}-t^2+3t+c$s(t)=5t33​−t2+3t+c. We know that $s(0)=-2$s(0)=−2. Therefore, $-2=c$−2=c and the function we seek is $s(t)=\frac{5t^3}{3}-t^2+3t-2$s(t)=5t33​−t2+3t−2.

From the displacement function, we have $s(3)=\frac{5\times27}{3}-9+9-2=43$s(3)=5×273​−9+9−2=43 

 

Example 2

A falling body near the surface of the earth has a constant acceleration of about $-9.8m/s^2$−9.8m/s2. If the body starts from rest at $t=0$t=0, find its velocity function. Given that the body begins its fall from a height of $200m$200m, find its displacement function.

 

Acceleration is the time rate of change of velocity. So, $v(t)=\int(-9.8)\ \mathrm{d}t+c$v(t)=∫(−9.8) dt+c. Hence, $v(t)=-9.8t+c$v(t)=−9.8t+c. We have $v(0)=-9.8\times0+c$v(0)=−9.8×0+c. Therefore $c=0$c=0 and the velocity function is $v(t)=-9.8t$v(t)=−9.8t.

Displacement is given by $s(t)=\int(-9.8t)\ \mathrm{d}t+k=\frac{-9.8t^2}{2}+k$s(t)=∫(−9.8t) dt+k=−9.8t22​+k. At $t=0$t=0, $s=200$s=200. So, $200=k$200=k and the displacement function is $s(t)=-4.9t^2+200$s(t)=−4.9t2+200.

 

Worked Examples

Question 1

Question 2

Question 3