Antiderivative
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Definition
Given a function , we say that a function is an antiderivative of if for all values of .
Example 1
If , the antiderivative of is where can be any real number.
Example 2
If , the antiderivative of is where can be any real number.
Physical interpretation
The derivative of distance is velocity. The derivative of velocity is acceleration
Example
A boy throws a baseball up in the air from the edge of a cliff 400 ft above the ground. The initial speed of the ball as it leaves his hand is 40 ft/sec. When does the ball reach its maximum height? Note: g (acceleration due to gravity) is 32 ft/s^{2}
Solution
We know that the acceleration is -32ft/s^{2} a(t)=dv/dt=-32
So by taking the antiderivative: v(t)= -32(t)+C
Now we need to determine what the constant C is. To do this, we set t=0.
v(0)=40 so 40=0+C, so v(t)= -32t+40
To find the maximum height reached by the ball, set v(t)=0 to find the time at which it reaches this point
0= -32(t)+40
32t=40
t= 4/5 seconds
since s'(t)=v(t), we must antidifferentiate again
s(t)= -16t^{2} +40t+D
Now we must determine what the constant D is. To do this, we set t=0
s(0)= 400
So, s(t)= -16t^{2} +40t +400
When the ball hits the ground, s(t)=0
-16t^{2} +40t+400=0
-2t^{2}+5t+50=0
2t^{2}-5t-50=0
Solve for t.