All Calculus 1 Resources
Example Questions
Example Question #221 : Spatial Calculus
The position of a particle is given by the functions , . What is the velocity of the particle at time ?
Velocity of a particle can be found by taking the derivative of the position function with respect to time. Recall that a derivative gives the rate of change of some parameter, relative to the change of some other variable. When we take the derivative of position with respect to time, we are evaluating how position changes over time; i.e velocity!
To take the derivative of the position functions
,
We'll need to make use of the following derivative rule(s):
Derivative of an exponential:
Note that u may represent large functions, and not just individual variables!
Using the above properties, the velocity functions are
Example Question #222 : Spatial Calculus
Find the velocity of a particle at time if its position is given by the function
Velocity of a particle can be found by taking the derivative of the position function with respect to time. Recall that a derivative gives the rate of change of some parameter, relative to the change of some other variable. When we take the derivative of position with respect to time, we are evaluating how position changes over time; i.e velocity!
To take the derivative of the position function
We'll need to make use of the following derivative rule(s):
Derivative of an exponential:
Note that u may represent large functions, and not just individual variables!
Using the above properties, the velocity function is
Example Question #221 : Velocity
The position of particle is given by the function . What is the velocity of the particle at time ?
Velocity of a particle can be found by taking the derivative of the position function with respect to time. Recall that a derivative gives the rate of change of some parameter, relative to the change of some other variable. When we take the derivative of position with respect to time, we are evaluating how position changes over time; i.e velocity!
To take the derivative of the position function
We'll need to make use of the following derivative rule(s):
Derivative of an exponential:
Product rule:
Note that u and v may represent large functions, and not just individual variables!
Using the above properties, the velocity function is
Example Question #221 : Velocity
Find the velocity of a particle if its position function is
Velocity of a particle can be found by taking the derivative of the position function with respect to time. Recall that a derivative gives the rate of change of some parameter, relative to the change of some other variable. When we take the derivative of position with respect to time, we are evaluating how position changes over time; i.e velocity!
To take the derivative of the position function
We'll need to make use of the following derivative rule(s):
Derivative of an exponential:
Product rule:
Note that u and v may represent large functions, and not just individual variables!
Using the above properties, the velocity function is
Example Question #222 : Velocity
The position of a particle is given by the function . What is the velocity of the particle at time ?
Velocity of a particle can be found by taking the derivative of the position function with respect to time. Recall that a derivative gives the rate of change of some parameter, relative to the change of some other variable. When we take the derivative of position with respect to time, we are evaluating how position changes over time; i.e velocity!
To take the derivative of the position function
We'll need to make use of the following derivative rule:
Derivative of an exponential:
Note that u may represent large functions, and not just individual variables!
Using the above properties, the velocity function is
Example Question #226 : Spatial Calculus
The position of a particle is given by the function . What is the particle's velocity at time ?
Velocity of a particle can be found by taking the derivative of the position function with respect to time. Recall that a derivative gives the rate of change of some parameter, relative to the change of some other variable. When we take the derivative of position with respect to time, we are evaluating how position changes over time; i.e velocity!
For the position function , the velocity function is
Example Question #222 : Velocity
The position of a particle is given by the function . At what time is the particle first stationary?
Velocity of a particle can be found by taking the derivative of the position function with respect to time. Recall that a derivative gives the rate of change of some parameter, relative to the change of some other variable. When we take the derivative of position with respect to time, we are evaluating how position changes over time; i.e velocity!
To take the derivative of the position function
We'll need to make use of the following derivative rule(s):
Derivative of an exponential:
Note that u may represent large functions, and not just individual variables!
Using the above properties, the velocity function is
The particle is stationary when this velocity is zero:
Example Question #228 : Spatial Calculus
The position of a particle starting at is given by the function . At what time is the particle first stationary?
Velocity of a particle can be found by taking the derivative of the position function with respect to time. Recall that a derivative gives the rate of change of some parameter, relative to the change of some other variable. When we take the derivative of position with respect to time, we are evaluating how position changes over time; i.e velocity!
To take the derivative of the position function
We'll need to make use of the following derivative rule(s):
Note that u may represent large functions, and not just individual variables!
Using the above properties, the velocity function is:
The particle is stationary when this function equals zero:
Though there are multiple roots to this equation:
However, for the time that the particle is first stationary, we'll only consider the first positive value:
Example Question #229 : How To Find Velocity
The position of a particle starting at time is given by the function . When is the particle first stationary?
Velocity of a particle can be found by taking the derivative of the position function with respect to time. Recall that a derivative gives the rate of change of some parameter, relative to the change of some other variable. When we take the derivative of position with respect to time, we are evaluating how position changes over time; i.e velocity!
To take the derivative of the position function
We'll need to make use of the following derivative rule(s):
Derivative of an exponential:
Trigonometric derivative:
Note that u may represent large functions, and not just individual variables!
Using the above properties, the velocity function is
To find when the particle is stationary, set this equation equal to zero:
There are multiple negative values which satisfy this equation; however, since we start at , the answer is
Example Question #229 : Spatial Calculus
The position of a particle is given by the function . When is the particle first stationary?
Velocity of a particle can be found by taking the derivative of the position function with respect to time. Recall that a derivative gives the rate of change of some parameter, relative to the change of some other variable. When we take the derivative of position with respect to time, we are evaluating how position changes over time; i.e velocity!
To take the derivative of the position function
We'll need to make use of the following derivative rule(s):
Derivative of an exponential:
Note that u may represent large functions, and not just individual variables!
Using the above properties, the velocity function is
To find when the particle is stationary, set this value equal to zero:
The particle is first stationary at time