All Calculus AB Resources
Example Questions
Example Question #331 :Calculus Ab
Find the velocity function from an acceleration function given by
and the condition
Acceleration is the rate of change of velocity, so we must integrate the acceleration function to find the velocity function:
The integration was performed using the following rules:
,
To find the integration constant C, we must use the initial condition given:
Our final answer is
Example Question #332 :Calculus Ab
The velocity of a particle is given by v(t). Find the function which models the particle's acceleration.
The velocity of a particle is given by v(t). Find the function which models the particle's acceleration.
To find the acceleration from a velocity function, simply take the derivative.
In this case, we are given v(t), and we need to find v'(t) because v'(t)=a(t).
To find v'(t), we need to use the power rule.
For each term, simply multiply by the exponent, and then subtract one from the exponent. Constant terms will drop out, linear terms will become constants, and so on.
So, our answer is:
Example Question #333 :Calculus Ab
The velocity of a particle is given by. Find the particle's acceleration when.
The velocity of a particle is given by. Find the particle's acceleration when.
To find the acceleration from a velocity function, simply take the derivative.
In this case, we are given, and we need to findbecause.
To find, we need to use the power rule.
For each term, simply multiply by the exponent, and then subtract one from the exponent. Constant terms will drop out, linear terms will become constants, and so on.
So, our acceleration function is:
Now, plug inforand solve.
So, our answer is.
Example Question #1 :Calculate Position, Velocity, And Acceleration
The position of a car is given by the following function:
What is the velocity function of the car?
The velocity function of the car is equal to the first derivative of the position function of the car, and is equal to
The derivative was found using the following rules:
,,,
Example Question #2 :Calculate Position, Velocity, And Acceleration
Let
Find the first and second derivative of the function.
In order to solve for the first and second derivatives, we must use the chain rule.
The chain rule states that if
and
then the derivative is
In order to find the first derivative of the function
we set
and
Because the derivative of the exponential function is the exponential function itself, we get
And differentiatingwe use the power rule which states
As such
And so
我们求解二阶导数set
and
Because the derivative of the exponential function is the exponential function itself, we get
And differentiatingwe use the power rule which states
As such
And so the second derivative becomes
Example Question #3 :Calculate Position, Velocity, And Acceleration
Find the velocity function of the particle if its position is given by the following function:
The velocity function is given by the first derivative of the position function:
and was found using the following rules:
,,,
Example Question #4 :Calculate Position, Velocity, And Acceleration
Find the first and second derivatives of the function
We must find the first and second derivatives.
We use the properties that
- The derivative ofis
- The derivative ofis
As such
To find the second derivative we differentiate again and use the product rule which states
Setting
and
we find that
As such
Example Question #5 :Calculate Position, Velocity, And Acceleration
Given the velocity function
whereis real number such that, find the acceleration function
.
We can find the acceleration functionfrom the velocity function by taking the derivative:
We can view the function
as the composition of the following functions
so that. This means we use the chain rule
to find the derivative. We haveand, so we have
Example Question #6 :Calculate Position, Velocity, And Acceleration
The position of an object is given by the equation. What is its acceleration at?
If this function gives the position, the first derivative will give its speed and the second derivative will give its acceleration.
Now plug in 2 for t:
Example Question #7 :Calculate Position, Velocity, And Acceleration
The equationmodels the position of an object after t seconds. What is its speed afterseconds?
If this function gives the position, the first derivative will give its speed.
Plug infor t:
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