Calculus 1 : Spatial Calculus
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Example Questions
Example Question #101 : How To Find Velocity
The position of a ball attached to the end of a spring is given by the following function:
At what time will the velocity of the ball first be exactly zero?
The velocity will never be zero.
Since we are given a position function, our first step will be to find the velocity function. This is found by taking the derivative of position with respect to time, using the product rule for derivatives:
where in this case
yielding the following derivative, our velocity function:
To find when velocity is equal to 0, we can set the right side of the equation of zero, and proceed to solve for our time, t
From here, we can take the arctangent of both sides of the equation and solve for t:
Example Question #102 : Velocity
If the position function of an object is 
To obtain the velocity function from the position function, take the derivative.
The derivative of 
Substitute 
Example Question #103 : Velocity
The position of a bicycle is defined by the equation 
By definition, the velocity 
Thus, given
where 
Swapping in 
Example Question #104 : Velocity
The position of an atom is defined by the equation 
By definition, the velocity 
Thus, given
where 
Swapping in 
Example Question #105 : Velocity
The position of a bird is defined by the equation 
By definition, the velocity
Thus, given
where 
Swapping in 
Example Question #102 : How To Find Velocity
A given object has a position that is defined by the equation 
By definition, velocity is the first derivative of position, or 
Given,
Plugging in 
Example Question #107 : Velocity
A baseball has a position that is defined by the equation 
By definition, velocity is the first derivative of position, or 
Given,
Plugging in 
Example Question #108 : Velocity
A bird has a position that is defined by the equation 
By definition, velocity is the first derivative of position, or 
Given,
Plugging in 
Example Question #109 : Velocity
Find 
To find 
Remember the chain rule is,
The power rule is,
Also, recall that the natural log is a function who has a special derivative.
Applying these rules we get the following solution.
Example Question #110 : Velocity
What is 
We get this by using the definition of the derivative of an exponential function. Remember that the derivative of an expotential function is:
Applying this rule to the function we are able to find the first derivative.
Now we simply plug in 1.
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