All AP Calculus BC Resources
Example Questions
Example Question #1 :Applications Of Derivatives
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 #31 :Derivatives
Let
Find the first and second derivative of the function.
In order to solve for the first and second derivative, we must use the chain rule.
The chain rule states that if
and
then the derivative is
In order to find the first derviative of the function
we set
and
因为指数函数的导数是the exponential function itself, we get
And differentiatingwe use the power rule which states
As such
And so
To solve for the second derivative we set
and
因为指数函数的导数是the exponential function itself, we get
And differentiatingwe use the power rule which states
As such
And so the second derivative becomes
Example Question #15 :First And Second Derivatives Of Functions
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 #1 :Applications Of Derivatives
Find the first and second derivatives of the function
We must find the first and second derivatives.
We use the properties that
- The derivative of是
- The derivative of是
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 #259 :First And Second Derivatives Of Functions
Given the velocity function
where是real number such that,发现加速度函数
.
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 #1 :Applications Of Derivatives
The position of an object is given by the equation. What is its acceleration at t = 2?
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 #1 :Applications Of Derivatives
The equationmodels the position of an object after t seconds. What is the acceleration at 3 seconds?
If this function gives the position, the first derivative will give its speed and the second derivative will give its acceleration.
Plug in 3 for t:
Example Question #1 :Applications Of Derivatives
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:
Example Question #4 :Velocity, Speed, Acceleration
The position of an object is modeled by the equationWhat is the speed afterseconds?
If this function gives the position, the first derivative will give its speed. To differentiate, use the chain rule:. In this case,and. Sinceand, the first derivative is.
Plug infor t:
Example Question #2 :Velocity, Speed, Acceleration
A particle's position on the-axis is given by the functionfrom.
When does the particle change direction?
It doesn't change direction within the given bounds
It doesn't change direction within the given bounds
To find when the particle changes direction, we need to find the critical values of. This is done by finding the velocity function, setting it equal to, and solving for
.
Hence.
The solutions to this on the unit circle are, so these are the values ofwhere the particle would normally change direction. However, our given interval is, which does not contain. Hence the particle does not change direction on the given interval.
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