Ap Calc Reading Derivative Graphs Fundamental Theorem of Calculus
AP Calculus Examination Review: Fundamental Theorem of Calculus
The Central Theorem of Calculus (FTC) is one of the most of import mathematical discoveries in history. Y'all might think I'm exaggerating, only the FTC ranks up there with the Pythagorean Theorem and the invention of the numeral 0 in its elegance and wide-ranging applicability. In this article I will explain what the Fundamental Theorem of Calculus is and show how information technology is used.
What is the Fundamental Theorem of Calculus?
Although the main ideas were floating around beforehand, it wasn't until the 1600s that Newton and Leibniz independently formalized calculus — including the Fundamental Theorem of Calculus.
Gottfried Wilhelm Leibniz (left) and Sir Isaac Newton (right). (Image of Leibniz courtesy of Ad Meskens)
The FTC establishes a directly link between two different branches of mathematics: analysis and geometry. Analysis deals with backdrop of functions and rates of alter, while geometry can be used to measure things nigh shapes.
Informally, the FTC states that the expanse under a curve y = f(ten) (a geometric measurement) tin can be institute using an antiderivative of the function (an analytic tool). This correspondence between area and antiderivatives is what makes the theorem so important and useful.
Argument of the Theorem (Definite Integral Form)
There are really two dissimilar forms of the Cardinal Theorem of Calculus. Hither is the one that is used most oft.
If f is continuous on a airtight interval [a, b], then
In the in a higher place formula, F is antiderivative for f. That is, F is a function whose derivative is equal to f. Using mathematical notation, we would say that F '(x) = f(ten).
Using the FTC
The Fundamental Theorem of Calculus provides a powerful tool for evaluating definite integrals. Here are the steps:
- Find an antiderivative for the integrand, using appropriate integration formulas.
- Plug the upper limit (b) and lower limit (a) of integration into the antiderivative F.
- Subtract to find the last answer: F(b) – F(a).
Example
In the following example, nosotros work out a definite integral using the FTC. Remember, cos x is the derivative of sin x. So sin ten is the antiderivative of cos x.
Applications of the FTC
Any time a definite integral needs to be evaluated, the Primal Theorem of Calculus can come up to the rescue. One of the most mutual applications you'll see on the AP Calculus exams is expanse nether a curve.
Area Under a Curve
If a office f(ten) is nonnegative on an interval [a, b], and so the area of the region under the curve tin be computed by a definite integral. The limits of integration, a and b, specify the left and right boundaries of the region. The bottom boundary is the x-axis, and the acme boundary is the graph of f(ten) itself.
Allow'south see how it works in one of the simplest cases, f(10) = ten. The graph is a diagonal line through the origin.
Suppose b > 0. We will find the area under y = f(10) = x betwixt x = 0 and 10 = b.
First set up the definite integral that computes the expanse. Then, according to the Fundamental Theorem of Calculus, nosotros only need to find an antiderivative for f(x) = 10. Yous can verify that F(x) = 10 2/ii does the play tricks.
Areas and Antiderivatives
At present allow's endeavor this same example (area under f(x) = x on [0, b]) but in a different manner. By working the problem out using a different method, I hope to testify yous the remarkable connexion between areas and antiderivatives.
Notice that the region under this particular function is but a right triangle. You should know how to detect the area of whatsoever triangle, using Area = (1/two)×(base of operations)×(height). In this case, the base and height are the same: both are equal to b.
Therefore, by geometry we get Area = (one/2)×b×b = b 2/2. The same result as we got above!
The fact that we get the aforementioned answer in this instance might not be too surprising. However, the real power of the Central Theorem of Calculus is that this link between areas and antiderivatives is truthful every single time. No matter how complicated the function is, yous tin observe the area under the curve but using calculus.
The FTC and Accumulation Functions
There is a second office to the Central Theorem of Calculus. It involves so-chosen accumulation functions. These are functions defined by a definite integral in which the upper limit of integration is the variable.
What is an Aggregating Function?
It'southward helpful to call back of an aggregating part as an "area then far" function. For whatever input x, the value of F(x) is the area nether f from a to x. As ten increases, more of the area gets "painted."
For case, the accumulation function for f(x) = x with left endpoint a = 0 is F(10) = x two/2. This is because the surface area under y = x on the interval [0, x] is equal to the area of the triangle with base of operations and height both each to x.
The 2nd Cardinal Theorem of Calculus
The second part of the FTC states that the accumulation part is simply a item antiderivative of the original function. Equivalently, the derivative of an accumulation role for a function f is equal to f(x) itself.
Hither's the formal statement of the theorem.
If f is continuous on a closed interval [a, b], and so
Using the Second Key Theorem
On the AP Calculus test, yous may be asked to take the derivative of an accumulation function. The FTC is only the right tool for the job. These problems can range from very piece of cake to much more challenging, so permit's run into a couple piece of cake examples first.
Example Problem i
In this problem, the FTC can exist practical directly. Notice how the variable t gets swapped out and becomes x.
Example Problem 2
This time, we demand to opposite the order of integration starting time. Call up, this introduces a negative.
The Chain Rule and the FTC
In the to a higher place examples, we could apply the theorem because one of the limits of integration happened to be a single variable 10. What if i or both of the limits of integration are functions of x? This situation tin be handled with a more than general formula that involves the Chain Rule.
Suppose u and v are both differentiable functions of x. Then
Hither's how it works:
Example Problem 3
Solution:
Final Thoughts
As with any topic on the AP Calculus exams, in that location is a lot to understand nigh the Fundamental Theorem of Calculus. However I hope that after having read this article, you take a much better understanding.
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