{Use of Tech} Difference quotients Suppose f is differentiable...

Question

{Use of Tech} Difference quotients Suppose f is differentiable for all x and consider the function D(x) = f(x+0.01)-f(x) / 0.01 For the following functions, graph D on the given interval, and explain why the graph appears as it does. What is the relationship between the functions f and D?
f(x) = sin x on [−π,π]

f(x) = sin x on [−π,π]

Accepted Answer

Hello. In this video, we are going to be approaching the following derivative problem. Now, the problem says to consider the function F of X equal to cosine of X on the interval from 0 to 2 pi. Using the difference quotient D of X, what does the graph of D X resemble? So we're told that the function F of X is equal to cosine of X, and the difference quotient D of X is given to us as F of X. Plus 0.02 minus F of X divided by 0.02. Now, the difference quotient is very similar to that of the definition of a derivative. Recall that the definition of the derivative is defined as the limit as H approaches 0 of FX plus H minus F of X, all divided by H. If we compare this definition to the different quotient D of X, we can see that the H variable has the value of 0.02. What this means is that D of X, a different quotient, represents the derivative F of X. What this means is that we need to take the derivative of F of X to see what kind of graph D of X represents. Now, by definition, the derivative of cosine of X is negatives of X. So F of X is equal to negatives of X. What this means is that the difference quotient represents the graph of negative sine of X, and with that being said, the solution to this problem is going to be B. So I hope this video helps you in understanding how to approach this problem, and I will go ahead and see you all in the next video.

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