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Discuss the inverse relation of derivative and integral. Explain in your own words why for a function f, the integral of the derivative f' is the original func...

Question

Discuss the inverse relation of derivative and integral. Explain in your own words why for a function f, the integral of the derivative f' is the original function, and why the derivative of the integral of f is the original function: Use appropriate mathematical notation and use graphs and diagrams to explain your thinking:

Discuss the inverse relation of derivative and integral. Explain in your own words why for a function f, the integral of the derivative f' is the original function, and why the derivative of the integral of f is the original function: Use appropriate mathematical notation and use graphs and diagrams to explain your thinking:



Answers

Explain in words and express mathematically the inverse relationship between differentiation and integration as given by the Fundamental Theorem of Calculus.

The first part of Parliament Captain Stairs here Fix We aren't immune function on the label would be You defined the idea function to your fix. This they're definite indignantly of the X DT explosion. Then you have a prime mix or you can write less. Beware the eggs, things you do. In other words, what it does is that the idea functioning of fix the effects is actually entered a to do off function. April 6 Go to very perfect little faces and do really well fakes. And if Premix it's the idea function to you, you're defined us dp I need it.

Oh We have our number 103. You have to show the relationship between graph off F and F. In books. So let us suppose affects equal to X plus one. And so from here actually equal to F. X -1. So Evan Rose X. Being comfortable. Okay. Happened was actually able to x mine as well. No, if you draw the graph when X equals zero affects is one when X equal to minus one affects zero. And here when X equal to zero. Having diverse sexy -1. When x equal to one it is zero. So if we just draw the ground X. Y. One when -1 and -1. Sure zero comma one comma zero. This would be the graph of effect And this is the local -1. They will come on minus women and one comma zero. Okay. Okay. This is like but we consider This is a graph of f. in verse six. So graph of affection it's inverse X. R. Miller remains according to the origin are symmetric we expect with origin. Or we can say that more clearly. This will be cementing with respect to line. Why? For two weeks. Thank you.

For this problem, you want to show that the integral of f inverse of x dx is equal to X times f inverse of x minus the integral of X times the derivative of the inverse function. And then dx Now to do this, we have to apply integration by parts and and here we want to let U equal to f inverse of X. So that D V is equal to dx. And we will get differential review equal to the derivative with respect to X of f inverse of X. And then this is dx and then devi will give us V which is equal to X. So by integration by parts we have the integral of f inverse of X. Dx. This is equal to U times V minus the integral of the times do you? And that's equal to f inverse of X times x minus the integral of X times D over dx of f inverse of X. And then the X. Or that's the same as X times f inverse of x minus. The integral of X times the derivative of f inverse of X with respect to X. And then dx.

So you hama Baton A about the graph off f off extent it's invoice ticket. So get a Montepaschi if affects go function here or home arepas f inverse Order in veritable hair. OK, so you have FN wars X You don't know function Joy will basically a goose break a inverse order Head on. No, The graph of Buddha function reflects over the line. Why equals two X Okay, So how many are the exact there that the graph off if off X on f in worse X If in worse eggs reflects over the line, why equals two X and each point on each point is involved. Each point is invoiced off and for facts so that we get f in worse X. Okay, so this is basically leader on. So So I hope you are district. Thank you


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