In Exercises 67–72, you will explore some functions and their ...

Question

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.

68. y= (3x+2)/(2x-11), -2 ≤ x ≤ 2, x_0=1/2

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Given that g is the inverse function of f, find the equation of the tangent line to the graph of g at the point P is equal to parentheses f of x subscript 0, x subscript 0. F of X is equal to 2 X plus 5 divided by X minus 4, for X cannot equal 4, X subscript 0 is equal to 2. OK. So, it appears for this particular prompt, we're ultimately asked to determine what the equation of the tangent line is for the graph of G at the specific point that is provided to us by the prom itself, based on all the information that is provided to us by the prom itself. And once again, we're given the point equation, and we're given the expression for the function f of. X for x cannot equal 4 and x subscript 0, which I'll call x0 for short, x0 is equal to 2. So now that we know that we're ultimately trying to determine what the equation for the tangent line is, our first step that we need to take in order to solve this particular problem is we need to compute f of x0 where X0 is equal to 2. And our second step is we need to note that F of 2, where we're plugging in a value of 2 in for X, so F 2 is equal to 2 multiplied by 2 plus 5 divided by 2 minus 4. Fantastic. And when we compute this expression, so when we take 2 multiplied by 2 + 5 divided by 2 minus 4, and we write it in its most simplified form for our answer, we will find that it's equal to -9 divided by 2. Our third step now is we need to note that the point on F. Is parentheses 2, -9 divided by 2, and the symmetric point across. So once again, the symmetric, meaning it's symmetry, is the same, so the symmetric point across from Y is equal to X. Is parentheses 9 divided by 2.2 in parentheses. So, that will mean for our fourth step, the slope of the tangent to the inverse at parentheses F of X0X0. Is drum roll 1 divided by F of X0. Fantastic. So, our 5th step now is we need to compute FX, which is the first derivative of this function FX. And we need to compute FX by recalling and using the quotient rule, and when we do, we will find that FX will be equal to x minus 4 multiplied by 2 minus 2 x + 5 multiplied by 1. All divided by parentheses X minus 42. So once again, parentheses X minus 42. And when we compute this expression and write it in its most simplified form, we will find that it's equal to 13 divided by, so 13 divided by parentheses x minus 4 to 2. Fantastic. So our step number 6 is we need to evaluate F2. Fantastic, and F2 will be equal to, and once again we're plugging in a value of 2N for X, so it will be equal to -13 divided by parentheses. Parentheses of 2 minus 4. To the power of 2. So once again, parentheses 2 minus 4 to 2, and this will be equal to -13 divided by, so -13 divided by 4. Fantastic. So, with that said, Our step number 7, so this is step 7, is we need to determine the slope of the tangent 2G. So once again, the slope of the tangent 2G at parentheses -9 divided by 2,2. Is so once again is drumroll. 1 divided by, it's gonna be 1 divided by F2, which will be equal to 1 divided by 13 divided by 4, which will be equal to -4 divided by 13. Fantastic. So, step number 8 is we need to note that the equation of the tangent line at, so once again, the equation of the tangent line at parentheses X1, Y1 will be equal to -9 divided by 2.2 in parentheses. With a slope of M is equal to -4 divided by 13. Is going to be, so is, when we plug in our known variables into our in our point slope equation, Y minus 2 is equal to -4 divided by 13 multiplied by parentheses of X + 9 divided by 2. So now we need to, for this is step number 9. So for step 9, we need to expand and isolate and solve for Y. So we need to expand and solve for Y. So Y. Is equal to -4 divided by 13 X minus 4 divided by 13 multiplied by 9 divided by 2, plus 2, which will be equal to -4 divided by 13 x minus. 18 divided by 13 + 2. And when we combine like constants, so this will be step 10. So step 10 is we combine the constants, so we'll have -18 divided by, so -18. Or negative, 18 divided by 13 + 2, which will be simply equal to, when we write in its most simplified form, 8 divided by 13, so it's gonna be 8 divided by 13. So, for our final step, so this is gonna be step 11, so 11 is our final step. We can therefore write that our final answer for the equation of the tangent line will be Y is equal to -4 divided by 13 x plus 8 divided by 13, and this is our correct and final answer. Hooray, we did it. So going back to the top to recap what we've learned, we could safely say that our final answer for the equation of the tangent line once again is y is equal to -4 divided by 13 x plus 8 divided by 13. And that's it, we've officially solved for this problem. Thank you so much for watching, hopefully that helped, and I can't wait to see you in the next video. Bye.

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