Worked example: Finding derivative with fundamental theorem of calculus (video) | Khan Academy
Relations between Certain Derivatives. Examples. CHAPTER IV. Successive Differentiation. 57, Definition and Notation. Though we sometimes use slope and derivatives interchangeably, there's a difference between them. The slope refers to the property of a "line". If a function is. In calculus, Leibniz's rule for differentiation under the integral sign, named after Gottfried . the interchange of a derivative and an integral (differentiation under the integral sign; i.e., Leibniz integral rule);; the change of order of .. The difference of two integrals equals the integral of the difference, and 1/h is a constant, so.
So the right hand side has the initial volume, plus a long sum of terms like f. That sum is called the integral of f with respect to t. We saw earlier that differentiating was subtracting and dividing. We've now seen that integration is just multiplying and adding.
So integration is the opposite of differentiation. We'd waste time and look a bit silly writing this every time. So instead we write it like this: The integral sign is s shaped, which can stand for 'sum' and remind us that's all it is.
Calculus: Differentials and integrals
We say we are integrating "with respect to t", because t is varying during our sum. In the way I've presented this example, V0 is a constant of integration. Integration doesn't tell you the complete answer, it only tells you how much something has changed during the process. In this case, to know the final volume in the bucket, we need to know not only the integral of the flow, but also how much was in the bucket before it started to integrate the flow.
In most cases, you will need to find the constant of integration — very often by using the initial conditions, as we did here. Perhaps now is a good time to go back to the animations above and check that integrating the velocity finding the area under the curve gives the displacement. Numerical integration If we had a set of numerical values for f t — whether experimental values or values calculated for a given mathematical function — then we could integrate just as described above: A very important practical point: This problem does not arise in multiplication.
Even better, the computation errors, being sometimes positive and sometimes negative, tend to cancel out. So numerical integration is much easier and safer than numerical differentiation.
The latter requires considerable caution and that is why my calculator doesn't have a "differentiate" button. Analytical integration This section might be shorter than you expect. We've mentioned above that integration is the opposite to differentiation: The rate at which something changes is its derivative, but You can recover that something by integrating the rate at which it changes.
So for analytical integration, we can use in reverse the tricks we established above for differentiation. Omitting constants of integration, we write The derivative of tn is ntn-1, so The integral of ntn-1 is tn provided that n is not equal to zero, because in that case the first equation gives us no information.
This is an important exception, which we'll deal with below. The exponential function One very useful integral and differential is the exponential function. So it is also its own integral. In this the derivative is not shown in red, because the function and its derivative are equal. To return to our point about numbers and physical quantities: Be careful about dimensions: Consequently, in physics, we shall often see the exponential function in equations like this one for a displacement decreasing exponentially with time: This example suggests a nice problem: And how far does it travel before coming to rest?
This situation occurs for an object moving slowly and horizontally without friction in a viscous medium. For object moving rapidly, the retarding force is approximately proportional to the square of the speed.
See the integral in car physics. If you need to, see What is a logarithm? A brief introductionbelow. Alternatively, we could say that the purple curve rises at a rate given by the red curve integration of a function means successively adding its value.
For this and later sections, you'll notice that the examples use numbers x and y, rather than displacements as a function of time. Quantities with dimensions add extra constants, as we see below, and it is easier to begin without them. Techniques for integration You need to assemble a little collection of integrals of simple functions, including those listed immediately above. There is a range of techniques for integrating more complicated expressions.
In approximate order of frequency, these are the techniques I use: Change the variables to make it more closely resemble an integral I do know.
Here are a couple of examples: Here dy is equal to dx, so this gives an integral we've seen above: Wow, that now looks much easier! This is just one of a whole family of trigonometric substitutions. This technique comes from the derivative of the product of two functions. This is getting beyond what we need for the material here, so see a book on calculus for more details.
Look up a table of integrals! These have pages and pages of integrals, which are presumably assembled "the other way round" — i. Here's one on Wikipedia.
Try an algebraic mathematical package. Where this is useful, it often makes you feel like a dill for not having seen it yourself.
Integrate numerically — for many problems, this is the only way. However, physical quantities do not, so that's another thing we'll leave to the mathematicians. Here we'll use two concrete examples to illustrate partial derivatives: Then we'll consider a surface in three spatal dimensions, f x,y.
First consider y x,t. As a specific example, let this be the displacement y of a point on a string as a function of position on the string x, and time t. So we can now think of two different derivatives. We write them differently. Imagine taking a photograph time is constant: This is not the velocity of the wave, by the way. A wave travelling on a stretched string is a standard example, which we derive and solve in this link.
Similar graphs could of course be drawn at any value of x. In the graphic below, we write these equations again in the standard notation, and set them with the time derivatives on left and space derivatives on right. Using the same layout, the animation below shows these equations as functions of x by graph and as functions of t by animation. The reason for taking second derivatives in this case is to show the solution to the wave equation for a string.
You can see this derivation and solution in detail in this linkfrom which we've borrowed the animation above. As a specific example, we could imagine that f is height or altitude of land as a function of position in the East x and North y directions, so f x,y is the shape of the landscape.
Here is a sketch. Above a large square on the x,y plane, I've drawn the outlines of a curved section of f x,ybounded by four curved lines. On one corner, at the point x,yI've drawn a small section, with width dx and depth dy. How steep is this landscape? If I were constrained to the f x plane i. But in three dimensions, the slope depends on my direction.
To quantify this, let's look at the small region in the bottom corner. I'll assume that the function f is continuous and well-behaved so that, when I make dx and dy sufficiently small, this little blue square is flat — to whatever precision I require. The approximation that f x,y is locally flat allows to write a simple equation for the change in height df.
Suppose I walk in the x East direction from x,yie. So, if I move a distance dx in the x direction, my altitude f increases by. Similarly, if I had walked in the y North direction, I would have a different slope and my height would have increased by. If necessary, revise vectors. On the sketch above, this displacement is in the NE quadrant, but by varying the sign and magnitude of dx and dy, I could choose any direction.
If my dx and dy are sufficiently small that the blue area above is approximately flat, then I can write the equation for the change in f due to arbitrary small changes dx and dy: What is a logarithm? First let's look at exponents.
- Leibniz integral rule
- The Fundamental Theorem of Calculus (FOTC)
- Calculus: differentials, integrals and partial derivatives.
So the exponent 2 or 3 in our example tells us how many times to multiply the base 10 in our example by itself. For this page, we only need logarithms to base 10, so that's all we'll discuss. In these examples, 2 is the log ofand 3 is the log of We can also have negative logarithms. Let's work out the value of 3.
Calculus II - Arc Length
This is easy enough to do, one step at a time: But what if n is not a whole number? Since the rules we have used so far don't tell us what this would mean, we can define it to mean what we like, but we should choose our definition so that it is consistent.
The definition of the logarithm of a number a to base 10 is this: In other words, the log of the number a is the power to which you must raise 10 to get the number a.
For an example of a number whose log is not a whole number, let's consider the square root of 10, which is 3. Using our definition above, we can write this as 3. The square root of 10 is Now there are a couple of questions: We leave these to mathematicians who, by the way, would be happy to give you a more rigorous treatment of exponents that this superficial account.
The conclusion of the fundamental theorem of calculus can be loosely expressed in words as: This description in words is certainly true without any further interpretation for indefinite integrals: The indefinite integral of f x is The derivative of this integral is so we see that the derivative of the indefinite integral of this function f x is f x.
Now the fundamental theorem of calculus is about definite integrals, and for a definite integral we need to be careful to understand exactly what the theorem says and how it is used. Some of the confusion seems to come from the notation used in the statement of the theorem. The first thing to notice about the fundamental theorem of calculus is that the variable of differentiation appears as the upper limit of integration in the integral: Think about it for a moment.
Unless the variable x appears in either or both of the limits of integration, the result of the definite integral will not involve x, and so the derivative of that definite integral will be zero.
Here are two examples of derivatives of such integrals. As expected, the definite integral with constant limits produces a number as an answer, and so the derivative of the integral is zero. Even though the upper limit is the variable t, as far as the differentiation with respect to x is concerned, t behaves as a constant.
So the derivative is again zero. The result is completely different if we switch t and x in the integral but still differentiate the result of the integral with respect to x. This example is in the form of the conclusion of the fundamental theorem of calculus. We work it both ways.