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Example Questions
Example Question #13 : Riemann Sum: Midpoint Evaluation
Example Question #151 : Ap Calculus Bc
Example Question #1 : Trapezoidal Sums
Approximate
using the trapezoidal rule with . Round your answer to three decimal places.
The interval is 1 unit in width; the interval is divided evenly into five subintervals
units in width. They are
.
The trapezoidal rule approximates the area of the given integral by evaluating
,
where
and
.
So
Example Question #43 : Integrals
Approximate
using the trapezoidal rule with . Round your answer to three decimal places.
The interval is
units in width; the interval is divided evenly into four subintervals
units in width. They are
.
The trapezoidal rule approximates the area of the given integral by evaluating
,
where
,
,
and
.
So
Example Question #2 : Trapezoidal Sums
Approximate
using the trapezoidal rule with . Round your estimate to three decimal places.
The interval is 4 units in width; the interval is divided evenly into four subintervals
units in width - they are
.
The trapezoidal rule approximates the area of the given integral by evaluating
,
where ,
, and
.
Example Question #1 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation
Find the result:
Set . Then
, and by the chain rule,
By the fundamental theorem of Calculus, the above can be rewritten as
Example Question #2 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation
Evaluate :
By the Fundamental Theorem of Calculus, we have that . Thus,
.
Example Question #1 : Fundamental Theorem Of Calculus
Evaluate when
.
Via the Fundamental Theorem of Calculus, we know that, given a function,
.
Therefore .
Example Question #2 : Fundamental Theorem Of Calculus
Evaluate when
.
Via the Fundamental Theorem of Calculus, we know that, given a function ,
. Therefore,
.
Example Question #2 : Fundamental Theorem Of Calculus
Suppose we have the function
What is the derivative, ?
We can view the function as a function of
, as so
where .
We can find the derivative of using the chain rule:
where can be found using the fundamental theorem of calculus:
So we get
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