All Calculus 1 Resources
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 to find the velocity:
Example Question #142 : How To Find Velocity
If is the position of a particle, what is the velocity at time ?
If we have the position , then the velocity is the derivative of the position:
by using the chain rule. So we plug in to get
as the velocity.
Example Question #141 : Calculus
If the acceleration of a particle at time is , and the initial velocity (i.e., at ) is , what is the velocity at ?
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 is
Example Question #143 : How To Find Velocity
If the velocity is , then what is the acceleration function, ?
To get the acceleration from , we take the derivative to get:
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 . Using the given information, find 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. 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 models the position of a frisbee as a function of time. Find the equation which models the velocity of the frisbee.
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 models the position of a frisbee as a function of time, find the velocity of the frisbee after seconds.
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 models the position of an electron as a function of time. Find , the velocity function of the electron:
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...