AP Calculus AB › Functions
A Riemann Sum approximation of an integral follows the form
.
Where n is number of points/subintervals used to approximate the integral.
Knowing this, imagine a modified style of Riemann Sum, such that the subintervals are not of uniform width.
Denoting a particular subinterval's width as ,
the integral approximation becomes
Which of the following parameters would give the closest integral approximation of the function:
?
The more points/intervals that are selected, the closer the Reimann sum approximation becomes to the actual integral value, so n=100 will give a closer approximation than n=50:
Now, assuming the amount of points/intervals are fixed, but the actual widths can differ, it would be prudent to have narrower intervals at those regions where the graph is steepest. Relatively flatter regions of the function do not lead to the approximation overshooting or undershooting as much as the steeper areas, as can be seen in the above figure.
Consider the function over the interval
. The steepness of this graph at a point can be found by taking the function's derivative.
The quotient rule of derivatives states:
So the derivative is
It can be seen that the slope is negative over the specified interval of . We may take the derivative once more to find the rate of change of this steepness and see if it's postive or negative:
Since this derivative is positive over the specified interval, and since the slope is negative, It means that the slope is flattening out
The intervals should in turn grow increasingly wider.
Find the derivative of the function.
To find the derivative, use the power rule which states, .
Applying the power rule to each term in the function we get,
.
Recall that the derivative of a constant is 0.
Thus, the derivative is:
Find the derivative.
Use the quotient rule to find the derivative which states,
.
Given,
the derivatives can be found using the power rule which states,
therefore,
Applying the quotient rule to our function we find the derivative to be as follows.
Simplify.
Find for the function below:
Hint: Use implicit differentiation.
Take the derivative of each term. Remember that the derivative of y should be dy/dx.
Remember to follow through with the chain rule for . Have to multiply by the derivative of the inside of the squared, which is just
.
Now isolate :
Consider the function between
and
. Find the definite integral using midpoint Riemann sums with two rectangles.
Midpoint sums are given by the formula
,
where n is the number of steps and the x values are the midpoints of the rectangles.
Since we have 2 steps, there would be 2 rectangles, from and the other one
. The heights of these rectangles are calculated at the midpoint of the two ends, which occur at
and
In our case, the rectangle midpoints are at 1.25 and 1.75, since we only have 2 steps.
Therefore, the value is
Approximate the integral of for
to
using midpoint Reimann sums and three midpoints.
The form of a Riemann sum follows:
The interval can be divided into three intervals of length
, with midpoints of
.
Therefore, the approximation of the integral is:
Using the method of midpoint Riemann sums, approximate the integral 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
.
The integral we are given is
So the interval is , the subintervals have length
, and since we are using the midpoints of each interval, the x-values are
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
Find the differential of the following equation.
To find the differential, take the derivative of each term as follows.
The derivative of anything in the form of is
so applying that rule to all of the terms yields:
Find the divergence of the function at
Hint:
Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.
To visualize this, picture an open drain in a tub full of water; this drain may represent a 'sink,' and all of the velocities at each specific point in the tub represent the vector field. Close to the drain, the velocity will be greater than a spot farther away from the drain.
What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.
We're given the function
What we will do is take the derivative of each vector element with respect to its variable
Then sum the results together:
Derivative of an exponential:
Note that u may represent large functions, and not just individual variables!
At the point