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Example Questions
Example Question #321 : Calculus
The velocity function of a particle and a position of this particle at a known time are given by . Approximate using Euler's Method and three steps.
The general form of Euler's method, when a derivative function, initial value, and step size are known, is:
In the case of this problem, this can be rewritten as:
To calculate the step size find the difference between the final and initial value of and divide by the number of steps to be used:
For this problem, we are told
Knowing this, we may take the steps to estimate our function value at our desired value:
Example Question #321 : Calculus
The position of a certain point is given by the following function:
What is the velocity at ?
Undefined
In order to find the velociy of a certain point, you must find the derivative of the position function which gives us the velocity function:
In this case, the position function is:
The velocity function is found by taking the derivative of the position function:
Finally, to find the velocity, substitute into the velocity function:
Therefore, the answer is:
Example Question #323 : How To Find Velocity
The position of is given by the following function:
Find the velocity.
Answer not listed
In order to find the velocity of a certain point, you first find the derivative of the position function to get the velocity function:
In this case, the position function is:
Then take the derivative of the position function to get the velocity function:
Then, plug into the velocity function:
Therefore, the answer is:
Example Question #321 : Calculus
The position of is given by the following function:
Find the velocity.
Answer not listed
In order to find the velocity of a certain point, you first find the derivative of the position function to get the velocity function:
In this case, the position function is:
Then take the derivative of the position function to get the velocity function:
Then, plug into the velocity function:
Therefore, the answer is:
Example Question #325 : How To Find Velocity
The position of is given by the following function:
Find the velocity.
Answer not listed
In order to find the velocity of a certain point, you first find the derivative of the position function to get the velocity function:
In this case, the position function is:
Then take the derivative of the position function to get the velocity function:
Then, plug into the velocity function:
Therefore, the answer is:
Example Question #323 : Calculus
James Bond is driving his motorcycle along the top of a train heading east at . When he started driving, 10 seconds ago, he and his motorcycle were sitting still on top of the train. He has been accelerating westward at a constant rate of . How fast is he moving relative to the ground right now?
We'll need to work backwards to solve this problem. Let's start with the information we're given.
Let's just say that westward motion is positive and eastward motion is negative. We could flip it around, but it would give us the same result, albeit with all the signs flipped.
Since the motorcycle starts out at rest relative to the train, and the train is moving at a constant speed, the initial velocity is given as
.
The motorcycle accelerates westward, so
.
At this point, we have a problem. The acceleration is in meters per second, but the initial velocity is in kilometers per hour. To convert the initial speed into meters per second, we need to multiply as follows:
Initial and final time are given as:
The aceleration is the rate of change in the velocity, so to find the velocity function, we will need to integrate and use initial conditions.
To integrate, we use the following rule:
Remember, when integrating we have to add the constant, C, to indicate that there's multiple functions that could be our answer.
Integrating then gives us:
.
To solve for C, we'll use out initial condition, .
We then need to evaluate the equation for , which gives us our answer:
At this instand, Mr. Bond is not moving relative to the ground.
Example Question #327 : How To Find Velocity
The position of is given by the following function:
Find the velocity.
Answer not listed
In order to find the velocity of a certain point, you first find the derivative of the position function to get the velocity function:
In this case, the position function is:
Then take the derivative of the position function to get the velocity function:
Then, plug into the velocity function:
Therefore, the answer is:
Example Question #322 : Calculus
The position of a particle is given by the equation . What is its velocity at ?
Differentiating the equation gives us the equation for velocity, , which, evaluated at , gives us .
To solve this, we'll need to first differentiate the position function to get the velocity function, and then plug our specific time into that velocity funtion to get the velocity at that moment. Differentiating the position equation term-by-term, using the power rule, which states that
and the constant rule, which states that the derivative of a constant is 0, we get
Evaluating at gives us
Example Question #321 : Calculus
The position of a particle is given by . Find the equation for the velocity.
Velocity is the rate of change of position, so to find the velocity, we'll need to differentiate this equation, by applying the product rule
This gives us:
Example Question #326 : Calculus
A cannonball is fired straight up into the air at time , with a velocity of . How fast is it moving and in which direction, after five seconds have passed? Assume the acceleration is due to gravity and thus equalling ?
upward
downward
upward
downward
upward
To solve this particular problem we will need to work backwards. First identify all known functions and values.
Velocity is given as,
.
Acceleration is given as,
.
The initial time and final time are given as,
.
The acceleration is the rate of change in the velocity, and in this case, it acts against the cannonball's initial motion.
To find the velocity function we will need to integrate the acceleration function and use initial conditions.
To integrate use the following rule of integration,
.
Thus,
Remember when integrating we must include the unknown constant C. To solve for C we will use the initial condition of .
So we need to evaluate the equation at , which gives us a velocity of
upward.
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