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3. The slope of the tangent line to f(z) = Atan(z) is 1/2 when = = 2r/3. Deter- mine the equation for f(r) by solving for A...

Question

3. The slope of the tangent line to f(z) = Atan(z) is 1/2 when = = 2r/3. Deter- mine the equation for f(r) by solving for A

3. The slope of the tangent line to f(z) = Atan(z) is 1/2 when = = 2r/3. Deter- mine the equation for f(r) by solving for A



Answers

Slope of a Tangent Line The equation of the tangent line to the graph of a function $f$ at (2,6) is $y=-3 x+12$. What is $f^{\prime}(2) ?$

We want to find the slope of the tangent line of the curve, Y equals to co tangent. Yeah. A three X. Where x equals pi over 12. So we see that when we take the derivative of this, end up getting this graph right here. Um And we want to remember that when we take the derivative of the code tangent of X, it's going to be negative Costa Rican X. A negative coast again squared X. So because we get negative Costa Rican spread act, it makes sense that this is the graph that we end up with. So we're going to evaluate this With x equals high over 12. So we see that we end up getting a negative 12, which means that is the slope of our graph. Um at Pi over 12 where F a bex two eagles to co change in three exes are function.

Okay, so before we complete these examples, let's review power. What is power roll? It's just a simplified way to find the derivative of a function. So given why X. To the exponent. My derivative Weisser prime for power roll, I just bring down my exponent in front of my ex term and then subtract one from my experience. So for example, to find the derivative of why um X the third, this would just be bringing down the 3/2 times X. To the 3/2 minus one. Okay, what's 3/2 minus one? This is the same as 3/2 minus 2/2. So this will be 3-2 times ext X to the 1/2. What's anything to the rational exponents? Well, this is just going to be this three over to times square books. This is my derivative. What does my derivative mean? Well, my derivative is actually representing all the different forms of the slope Of each of the points. So let's say I wanted to find the slope at the .1, 1 I can plug in my X. Value into the derivative. So I could say 3/2 times the square root of one. What's the square one? What times itself gives me one. That will be one. So three or two times one is just three or two. So the slope of this function at this point will be 3/2. And I found that by plugging in exodus one to the derivative. I can do the same thing with B. Um To find the derivative here, looking at X. The third. Use the power will bring down the exponents checked one from the exponents. So this would be three. 3 -1 will be too. So my derivative of this function was probably um three X squared. Then to find the slope at a particular 0.11 I could just plug in the X value into the derivative three times one. The second power order of operations are going to the extent of first one times one is one. Three times one will just be three. So the slope at the .11 for this function, what do you think?

We have y is equal to co sign three x and let's go ahead and calculate the derivative of this guy. So this is negative three sine three x and were asked to find the slope of the tangent line when X is equal to 13 pi over six. So we're taking Why Prime and we're plugging in 13 pi over six. This gives us negative three sign 13 pi over two. Remember, that sign is two pi periodic so we can subtract off the two pies. And that just leaves us with pie divided by two on the inside. And we know how to calculate that high divided by sine pie divided by two is just one, so that's the slope.

Yeah. Well the theme in this problem is that we're doing the tangent line and any time we need a tangerine we need the point in the slope. And what they were doing three X squared at 13 So the order pairs at 13 The function we're talking about is three X squared. Um And to find the so called, we actually have to do is do the derivative and plug in one for X. So probably be to your benefit to actually figure out what explicit H. Is equal to. Uh And this is following a four step process. I'm probably gonna combine a few steps though because remember multiplying explosive age times X plus age is the same thing as explicit squared. So we're looking at three of X squared and then we would have had to X. H. But then when we distribute that three and there will be six X. H. And then we would end up was three H squared. So if I do the limit as a jew approaches zero, if you're curious why I'm doing that because that's the equation of the derivative. Um and I subtract off, you know, half of explicit minus F of X all over H. What you'll see is the three x squares cancel each other out. And then you can cancel out one of each of these trees. And then when you go to plug in zero for this, h you end up with six X plus zero, which is simply six X. So when we go to evaluate f prime of one, plugging in one and for this X and six times once equal to six. So now we're ready for the equation of the tangent line. I'm a big fan of point slope form or it's why minus of Y coordinate given to us in the problem equals a slope, which we just found. Uh and then expectancy exhorting and some teachers will let you leave your answer and point. So form. Other teachers might ask you to distribute that six in there, so would be six x minus six and then plus three um to give you this answer. And while I'm at it, you might also have a teacher who prefers this form where you subtract six X over, add three over. There's a lot of correct answers, though. Just pick one of these circles and green and move on. Mhm. Sure.


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