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There are exactly two points on the curvey=tan (x)that have a tangent line with slope2(a) Find the distance between these two points_(b) Find the equation of the li...

Question

There are exactly two points on the curvey=tan (x)that have a tangent line with slope2(a) Find the distance between these two points_(b) Find the equation of the line which connects these two points.(c) What is the average rate of change f the functionf(x)=tan (x)between these two points?(d) At which point(s) on the curvey=tan (x)is the slope of the tangent line to the curve largest?

There are exactly two points on the curve y=tan (x) that have a tangent line with slope 2 (a) Find the distance between these two points_ (b) Find the equation of the line which connects these two points. (c) What is the average rate of change f the function f(x)=tan (x) between these two points? (d) At which point(s) on the curve y=tan (x) is the slope of the tangent line to the curve largest?



Answers

Let $f(x)=x^{2}-2 x+1$ a. Find the derivative $f^{\prime}$ of $f$ b. Find the point on the graph of $f$ where the tangent line to the curve is horizontal. c. Sketch the graph of $f$ and the tangent line to the curve at the point found in part (b). d. What is the rate of change of $f$ at this point?

So in this problem, were given a function two x squared were also given to X values X 00 and X one is one. And we're being asked to find a lot of different information that deals in terms of the derivative, the instantaneous rate of change. Right? So the first question that we're being asked to find is the average my A rock. My average rate of change on the interval from X one from zero to X one. So in order, find average rate of change that's basically calculating slope. So the first thing I need to do is find the two points. So the first thing that I needed to do is substitute X zero into the equation. So why equals two times zero squared, which means why will equal zero squared 00 times two is zero. So I have a point 00 This represents my ex zero y zero. When I substitute one into the equation, y equals two times one squared, one squared is one two times one is chu, which tells me I have another 10.1 to the average rate of change will be the changes of my wise over the changes of my access, so to minus zero over one minus zero is to over one, which is two. So my average rate of change on the interval from 01 is to the second question, is asking us to find the instantaneous rate of change with respect to X at a specific value. Um, at the specific value. Okay, So in this particular case, too X squared, right. So let's just get a picture of this. And that may help us to find our instantaneous rate of change. Ah, this is just a parabola that looks like this. Okay, so what we're looking at here, Um, and if I wanted to find the instantaneous rate of change for for the specific X zero, that would be right here. We can see that the tangent line that would pass through that point at zero would be would have a slope. It's a horizontal line, would have a slope of zero. So the instantaneous rate of change there would be this value. Now, if I want to find an instantaneous rate of change for any X zero. So for any other points on that, right, what I'd have to do is, I'd have to find the slope of the tangent line at that point. And so, in this particular case, the derivative of the function, the derivative of the function is why Prime is equal to four X, right? And if it's any point to find the slope of the tangent line, I would do four times whatever that X zero value is, and that's how we would be able to calculate it. Finally, if we want to find the average rate of change, which is the which is the slope of the sea can't line and the instantaneous rate of change, right? We want to be able to, uh, sketch a graph where y equals f of X together with those two lines. Right? So what we're looking at is here's my graph. Right? And what we're looking for is if I'm looking for this instantaneous rate of change, right, there's my, um there is my tangent line. So if I wanted a c can't line right, I would be using my these two values right here. So one chu is over here. And so we have this line right here. You'll notice that the slope of this line is not very close to the slow. The scope of the red line is not very close to the slope of the Green Line, but this is kind of what we have to do.

Yeah. For the given exercise, we want to um graph or solve the one over X squared here, we want to find the average rate of change. So we take F of two minus F of one and then we divide that by two minus one. So it's just gonna be divided by one is about a negative three quarters negative 3/4. However, we want to find the instantaneous rate too. So we see the f. Prime of X is going to equal negative two over Act two III. Um So then when we're making X equal to X not, it'll be this value right here and in this case X not is going to be one. So we'll get a negative to um as our answer, Which we see is very different from a negative .75.

So I go for this problem is to find the average rate of change. So given F of X equals X cubed, we know that Um the average rate of change is going to be f of two -F of one. I had to buy 2 -1, which is just once, that's seven. Now, if we take the derivative, so we have, the instantaneous rate of change will have three X squared. Um So the derivative is three X squared evaluated at one and we end up getting three, then we find the instantaneous rate of change of why with respect to act at a general point it's not. And if we graph these together, we see that this average rate of change is much greater than this instantaneous rate of change. However, this one here is much more accurate. I mean that's the one that we're going to end up starting to use. If we just had a random X not, it would be three X not squared.

Question is find the indicated quantities for fx is given The X is good in this question, we have seven parts so we will solve it one by one. The average rate of change of FX's exchanges one to to fight this is the first part. So for the first part it is saying A. There F five because we have to put the value of X. S five -F two upon 5 -2. So now we will saw this as we have to put the value of X firstly five then too, so it can be attendance three into five is where -3 into who is called upon five minus tooth maze three. So it can be a tennis after solving 63 upon three so we can save and we reduce it, it will be 21. So the answer of our first part is 21. Now come to the B part In deeper. It is given exactly same as a. so F five minus F two Divided by 5 -2. So we can see Its answer is also 20. Even because be part is exactly the same as the a part. Now come to the seaport in the C. Part, it is given F two plus edge minus F two Upon two plus H -2. So we have to put the value of X firstly to plus it then after two. So by solving it we can write it three in place of excess where it can be written as two plus H. Holy square minus three into two square. And in the denominator part two and two will be cancel out. So remaining is only edge. Now opening the bracket by using the identity. A. Plus. Behold the square. Sir it is a square plus to A B plus B squared minus three into 2 squared divided by edge. So it can be returned to us by opening the bracket by multiplying by 3 12 plus 12. Which 12 plus 12 which plus three year two square minus 12 upon edge. It means when we will solve it, 12 and 12 will be cancel out. So 12 edge plus three edges where the numerator and denominator only it. So we can take each comin from the numerator part. So it can be return is 12 plus three edge upon edge and the agent edge will be considered. So we can say in C part the final answer is 12 plus three eight. This is the answer of C. Part. Now come to the deep part of the cushion. So for deeper it is see they're exactly like see part but we have to put the value of edge which has given limit edge tends to zero. So limit edge tends to zero F two plus edge minus Function of two divided by two plus h minus two. We have already sold it it in C part so it can be written as well. Plus three is now we will when taking limit taking limit taking limit means we have to put the value of edge zero. So it can be written is 12 plus three multiplied by zero. So the final answer will be only 12. So we can say this is the answer of our depart now come to the next part which is Eva he is exactly same as the parts. So it's answer is also drilled. So we can write it limit Age tends to zero F two plus edge minus F two. In the denominator part 2-plus Edge -2. So exactly seem. So it's answer is also 12. So we can say our Ebert Ansari strip now come to the next part which is F part. So in the effort we have given exactly same D N. E. So we can say that here limit Edge tends to zero So F two plus edge Minour function of two Upon 2-plus Edge -2. So also it's answer is 12. So we can see that are F part answer is also if now next T. Which is our last word. In the cheaper it is saying to function of two Will be equal to 12 and the value of M. Is given drill and mean slope of the light. So we will use the equation of slope which is y minus violent will be equal to M X minus excellent. Now we will put the points, this is excellent and this is violent. Soviet indicted y minus 12 equals the value of M. Is given 12, 12 in two x minus two, so y minus 12 will be equals to 12 x minus 24 So it can be returning after solving y equals to 12 weeks -12. So we can say our last part, which is G. It's equation of the engine is y equals to 12 weeks -12.


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