All Calculus 1 Resources
Example Questions
Example Question #31 : How To Find Position
The velocity of an object is . What is the position of the object if its initial position is ?
The position is the integral of the velocity. By integrating with the power rule we can find the object's position.
The power rule is where
.
Therefore the position of the object is
.
We can solve for the constant using the object's initial position.
Therefore and .
Example Question #32 : How To Find Position
The velocity of an object is given by the equation . What is the position of the object at time if the initial position of the object is ?
The position of the object can be found by integrating the velocity. This can be done using the power rule where if
.
Using this rule we find that
.
We can find the value of using the initial position of the object.
Therefore and .
Example Question #33 : How To Find Position
The velocity of an object is given by the equation . What is the position of the object at time , if the initial position of the object is ?
The position of the object can be found by integrating the velocity. This can be done using the power rule where if
.
Using this rule to integrate the velocity gives us
.
The value of can be found by using the initial position of the object.
Therefore and .
Using the position equation we find the position at .
Example Question #34 : How To Find Position
Find the position of an object at if the velocity function is .
To determine the position of an object given the velocity function, integrate once to obtain the position function.
Substitute the value into the position function.
Example Question #31 : How To Find Position
Suppose the acceleration function is described by . What is the position when ?
Obtain the position function by integrating the acceleration twice.
Integrate again to obtain the position function.
Substitute .
Example Question #36 : How To Find Position
Find a vector perpendicular to .
By definition, any vector has a perpendicular vector . Given a vector , the perpendicular vector is .
We can verify this further by noting that the product of any vector and its perpendicular vector is equal to , or . Taking the product of and , we get:
Example Question #37 : How To Find Position
Find a vector perpendicular to .
By definition, any vector has a perpendicular vector . Given a vector , the perpendicular vector is .
We can verify this further by noting that the product of any vector and its perpendicular vector is equal to , or . Taking the product of and , we get:
Example Question #38 : How To Find Position
Find a vector perpendicular to .
By definition, any vector has a perpendicular vector . Given a vector , the perpendicular vector is .
We can verify this further by noting that the product of any vector and its perpendicular vector is equal to , or . Taking the product of and , we get:
Example Question #39 : How To Find Position
If the velocity function of a car is , what is the position when ?
To find the position from the velocity function, take the integral of the velocity function.
Substitute .
Example Question #31 : Position
For this question, remember that velocity is .
A particle's velocity is given by the function . What is the particle's position at ?
Position is the anti-derivative of velocity, so we find the anti-derivative of the given velocity function to find the position function.
The velocity function is given as , and the anti-derivative of this using the power rule is
.
We then evaluate that function at the given point to get,
.