Position: | x(t) (sometimes wrote as s(t)) | Meters |
---|---|---|
Velocity: | xâ(t) or v(t) | Meters/Second |
Acceleration: | xâ(t) or vâ(t) or a(t) | Meters/Second^2 |
The derivative can also tell us the change of something other than motion
We just saw how the derivative can tell us the change of something but we can also have problems where the change of one thing is related to another- Related Rates!
Letâs say that a pool of water is expanding at 16Ď square inches per second and we need to find the rate of the radius expanding when the radius is 4 inches
We know that we can find the radius using A = Ďr^2
Now letâs relate our rates!
Letâs say a spherical balloon is being inflated at a rate of 10 cubic inches per second. How fast is the radius of the balloon increasing when the radius is 4 inches?
We know that the volume of a sphere is given by the formula V = (4/3)Ďr^3.
Therefore, the radius of the balloon is increasing at a rate of 10/(16Ď) inches per second when the radius is 4 inches.
To solve related rates problems in calculus, follow these steps:
Remember to always include units in your final answer and to check that your answer makes sense in the context of the problem.
Differentials are very small quantities that correspond to a change in a number. We use Îx to denote a differential.
Remember the limit definition of a derivative?
We just have to replace h with Îx and remove the limit!
Letâs say we needed a differential to approximate (3.98)^4
If a limit gives you 0/0 or â/â, then it is called âindeterminateâ and you can use
LâHospitalâs Rule to interpret it!
LâHospitalâs Rule says that we can take the derivative of the numerator and denominator and try again
Letâs say we have the limit of 5x^3 -4x^2 +1/7x^3 +2x - 6 as it approaches infinity