Calculus 1 : Spatial Calculus
Study concepts, example questions & explanations for Calculus 1
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Example Questions
Example Question #141 : How To Find Velocity
A particle's position is defined by
What is the instantaneous velocity of the particle at 
To find the velocity function, take the derivative of the position funtion.
Now plug in 
Example Question #142 : How To Find Velocity
If 
If we have the position 
by using the chain rule. So we plug in 
as the velocity.
Example Question #141 : Calculus
If the acceleration of a particle at time 
Not enough information to find the velocity at 
This is simply an application of the fundamental theorem of calculus, since we know that 
So we have
So let's solve for 
So the velocity at 
Example Question #143 : How To Find Velocity
If the velocity is 
To get the acceleration from 
with the use of the chain rule
Example Question #145 : Spatial Calculus
A car's position is represented by the function
From this information, find the car's velocity when 
The given equation is used to find position. Based on what we know about derivatives, if we take the derivative of position with respect to time we will be finding the change in position over time, which is velocity. So the first step is to take the derivative. We will get
to represent our velocity. Next, you plug in the given t value of t=6 to find the velocity at this step in time.
Example Question #146 : Spatial Calculus
An object's position is described by the given function:
Given this information is the object's velocity constant or changing? What is the velocity when 
The given equation is used to find position. Based on what we know about derivatives, if we take the derivative of position with respect to time we will be finding the change in position over time, which is velocity. So the first step is to take the derivative. We will get
to represent our velocity. Since our function for velocity is a constant, we know that the velocity will always be 16. To confirm this, we can take the derivative of our velocity to find acceleration. The derivative of a constant is 0, therefore, there is no acceleration or change in velocity, thus velocity is constant at 16.
Example Question #147 : Spatial Calculus
A bug's position is represented by the function 
The given equation is used to find position. Based on what we know about derivatives, if we take the derivative of position with respect to time we will be finding the change in position over time, which is velocity. So the first step is to take the derivative. We will get
to represent our velocity. Next, you plug in the given t value of t=5 to find the velocity at this step in time.
Example Question #148 : Spatial Calculus
If 
Recall that velocity if the first derivative of position, so to find the equation which models the velocity, we need to take the derivative.
Recall that for any polynomial we can take the derivative as follows:
So our original function:
Becomes:
Example Question #149 : Spatial Calculus
If 
Recall that velocity if the first derivative of position, so to find the equation which models the velocity, we need to take the derivative.
Recall that for any polynomial we can take the derivative as follows:
So our original function:
Becomes:
Then, to find the velocity after 12 seconds, find f'(12)
So our answer is 448 meters per second. Perhaps a bit fast for a frisbee, but it is the correct answer in this case!
Example Question #141 : Velocity
If 
If p(t) models the position of an electron as a function of time. Find v(t), the velocity function of the electron:
Recall that velocity is the derivative of position. To find v(t), we need to find p'(t).
Remember that the derivative of cosine is negative sine, so we get...
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