All Calculus 1 Resources
Example Questions
Example Question #151 : How To Find Midpoint Riemann Sums
Utilize the method of midpoint Riemann sums to approximate the average of over the interval using three midpoints.
To find the average of a function over a given interval of values , the most precise method is to use an integral as follows:
Now for functions that are difficult or impossible to integrate,a Riemann sum can be used to approximate the value. A Riemann sum integral approximation over an interval with subintervals follows the form:
It is essentially a sum of rectangles each with a base of length equal to the subinterval length , and variable heights , which depend on the function value at a given point .
Now note that when using the method of Riemann sums to find an average value of a function, the expression changes:
We're asked to approximate the average of over the interval
The subintervals have length , and since we are using the midpoints of each interval, the x-values are
Example Question #1181 : Calculus
The derivative of an unknown function is , and a function value, , is known. Utilize the method of midpoint Riemann sums and Euler's method to approximate using three midpoints for the former and six steps for the latter.
A Riemann sum integral approximation over an interval with subintervals follows the form:
It is essentially a sum of rectangles each with a base of length and variable heights , which depend on the function value at a given point .
We're asked to approximate
So the interval is , the subintervals have length , and since we are using the midpoints of each interval, the x-values are
Now, our function is unknown--we only have its derivative, -- but we can calculate an approximation of function values using this derivative with Euler's method since we have an initial value, .
Euler's method utilizes the following generalized formula
or for the language used in this problem
Where is a step size. Given an intial value , it can in turn be used to approximate , where is the number of steps taken.
To calculate this step size, consider the range of values being evaluated and the desired number of steps.
After that, it's merely a matter of taking the steps:
And from there, putting the appropriate values back into the Riemann sum approximation:
Example Question #151 : Functions
The derivative of an unknown function is , and there's a known function value at . Utilize the method of midpoint Riemann sums and Euler's method to approximate using three midpoints and three steps.
A Riemann sum integral approximation over an interval with subintervals follows the form:
It is essentially a sum of rectangles each with a base of length and variable heights , which depend on the function value at a given point .
We're asked to approximate
So the interval is , the subintervals have length , and since we are using the midpoints of each interval, the x-values are
Now, our function is unknown--we only have its derivative -- but we can calculate an approximation of function values using this derivative with Euler's method since we have an initial value .
Euler's method utilizes the following generalized formula
or in the case of this problem
Where is a step size. Given an intial value , it can in turn be used to approximate , where is the number of steps taken.
To calculate this step size, consider the range of values being evaluated and the desired number of steps.
After that, it's merely a matter of taking the steps:
And from there, putting them back into the Riemann sum approximation:
Example Question #154 : Differential Functions
The derivative of an unknown function is , and there's a known function value at . Utilize the method of midpoint Riemann sums and Euler's method to approximate using three midpoints for the former and three steps for the latter.
A Riemann sum integral approximation over an interval with subintervals follows the form:
It is essentially a sum of rectangles each with a base of length and variable heights , which depend on the function value at a given point .
We're asked to approximate
So the interval is , the subintervals have length , and since we are using the midpoints of each interval, the x-values are
Now, our function is unknown--we only have its derivative -- but we can calculate an approximation of function values using this derivative with Euler's method since we have an initial value .
Euler's method utilizes the following generalized formula
or in the case of this problem
Where is a step size. Given an intial value , it can in turn be used to approximate , where is the number of steps taken.
To calculate this step size, consider the range of values being evaluated and the desired number of steps.
After that, it's merely a matter of taking the steps:
And from there, putting them back into the Riemann sum approximation:
Example Question #155 : Differential Functions
The derivative of an unknown function is , and there's a known function value at . Utilize the method of midpoint Riemann sums and Euler's method to approximate using three midpoints for the former and six steps for the latter.
A Riemann sum integral approximation over an interval with subintervals follows the form:
It is essentially a sum of rectangles each with a base of length and variable heights , which depend on the function value at a given point .
We're asked to approximate
So the interval is , the subintervals have length , and since we are using the midpoints of each interval, the x-values are
Now, our function is unknown--we only have its derivative -- but we can calculate an approximation of function values using this derivative with Euler's method since we have an initial value .
Euler's method utilizes the following generalized formula
or in the case of this problem
Where is a step size. Given an intial value , it can in turn be used to approximate , where is the number of steps taken.
To calculate this step size, consider the range of values being evaluated and the desired number of steps.
After that, it's merely a matter of taking the steps:
And from there, putting them back into the Riemann sum approximation:
Example Question #156 : Differential Functions
The derivative of an unknown function is , and there's a known function value at . Utilize the method of midpoint Riemann sums and Euler's method to approximate using four midpoints for the former and four steps for the latter.
A Riemann sum integral approximation over an interval with subintervals follows the form:
It is essentially a sum of rectangles each with a base of length and variable heights , which depend on the function value at a given point .
We're asked to approximate
So the interval is , the subintervals have length , and since we are using the midpoints of each interval, the x-values are
Now, our function is unknown--we only have its derivative -- but we can calculate an approximation of function values using this derivative with Euler's method since we have an initial value .
Euler's method utilizes the following generalized formula
or in the case of this problem
Where is a step size. Given an intial value , it can in turn be used to approximate , where is the number of steps taken.
To calculate this step size, consider the range of values being evaluated and the desired number of steps.
After that, it's merely a matter of taking the steps:
And from there, putting them back into the Riemann sum approximation:
Example Question #157 : Differential Functions
The derivative of an unknown function is , and there's a known function value at . Utilize the method of midpoint Riemann sums and Euler's method to approximate using four midpoints for the former and four steps for the latter.
A Riemann sum integral approximation over an interval with subintervals follows the form:
It is essentially a sum of rectangles each with a base of length and variable heights , which depend on the function value at a given point .
We're asked to approximate
So the interval is , the subintervals have length , and since we are using the midpoints of each interval, the x-values are
Now, our function is unknown--we only have its derivative -- but we can calculate an approximation of function values using this derivative with Euler's method since we have an initial value .
Euler's method utilizes the following generalized formula
or in the case of this problem
Where is a step size. Given an intial value , it can in turn be used to approximate , where is the number of steps taken.
To calculate this step size, consider the range of values being evaluated and the desired number of steps.
After that, it's merely a matter of taking the steps:
And from there, putting them back into the Riemann sum approximation:
Example Question #158 : Differential Functions
Utilize the method of midpoint Riemann sums to approximate using three midpoints.
A Riemann sum integral approximation over an interval with subintervals follows the form:
It is essentially a sum of rectangles each with a base of length and variable heights , which depend on the function value at a given point .
We're asked to approximate
So the interval is , the subintervals have length , and since we are using the midpoints of each interval, the x-values are
Example Question #152 : How To Find Midpoint Riemann Sums
Utilize the method of midpoint Riemann sums to approximate using three midpoints.
A Riemann sum integral approximation over an interval with subintervals follows the form:
It is essentially a sum of rectangles each with a base of length and variable heights , which depend on the function value at a given point .
We're asked to approximate
So the interval is , the subintervals have length , and since we are using the midpoints of each interval, the x-values are
Example Question #151 : How To Find Midpoint Riemann Sums
Utilize the method of midpoint Riemann sums to approximate using three midpoints.
A Riemann sum integral approximation over an interval with subintervals follows the form:
It is essentially a sum of rectangles each with a base of length and variable heights , which depend on the function value at a given point .
We're asked to approximate
So the interval is , the subintervals have length , and since we are using the midpoints of each interval, the x-values are