Find an equation of the line tangent to the graph of f at (a, ...

Question

Find an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a.

f(x) = 1/x; a= -5

Accepted Answer

Welcome back everyone. Determine the equation of the tangent line to the curve G of X equals 2 divided by X X equals 4. We're given 4 answer choices A says Y equals -1 divided by 8 X minus 1, B-1 divided by 8 x + 1. C 1 divided by 8 X + 1 and D 1 divided by 8 X minus 1. So we want to identify the equation of the tangent line. We have to recall that the equation of the tangent line is simply the derivative of the function at a point of interest x not multiplied by X minus the point of interest X0 plus the value of the function at that point. So in this case we're given G of X, right, we're using G for our definition. So first of all, what we want to do is understand that the point of interest X not in this case is equal to 4. And if we look at the general equation, we want to identify G at 4, multiply that by x minus 4, and then add g at 4. So let's begin with G at 4, which is the slope of the tangent line, and we're going to use the definition. So we're going to say that at g at X in a general case the slope is equal to the limit as H approaches 0 of G at point X plus H minus g of X, all of that divided by H. And let's evaluate this limit. Well, we're going to get limit as H approaches 0, G of X plus H, well, this means that we are replacing the X with X plus H. We get 2 divided by X plus H, and then we're subtracting G of X, which is 2 divided by X. And then we're dividing everything by H. Now let's simplify. We're going to get our limit and essentially we're going to find the common denominator. 2 divided by X plus H. We're going to write 2 for now. Our common denominator is going to be the product of the two denominators X multiplied by X plus H. So we have to multiply 2 by X, which gives us 2 X, and then we are subtracting 2 multiplied by X plus H. So we now have our common denominator, and then we're dividing by age, so we have a double divisions, meaning we can just write a single denominator, right? Now, let's distribute the terms we're going to get limit as H approaches 0 of 2X minus 2X minus 2 H. So overall we're getting minus 2H in the numerator. And then dividing by HX multiplied by X plus H. And we can see that we can cancel out the age term. And evaluate the limit so we get negative. Applying the right substitution, we simply substitute X we substitute H equals 0. So in the denominator we get X multiplied by X, which is X2. Well done, so we have the general expression of the slope. It is equal to negative. 2 divided by x 2. Now our point of interest is x equals 4, so G at 4 is -2 divided by or squad, and that's -2 divided by 16 or 1 divided by 8. This means that our slope is negative, so we can already exclude options C and D, right? And now our final part of the equation means that we want to evaluate the G at 0.4. So our function is 2 divided by X, meaning we're dividing 2 by 4, which gives us 1/2. Returning to the original equation we get Y equals G at 0.4, which is -1 divided by 8, multiplied by x minus 4. And then we are adding G at 4, which is 1/2. And now let's distribute the terms, so -18 X. Plus Or divided by 8, this gives us plus 1/2 plus another 1/2, and of course 1/2 plus 12 gives us 1, right? So we can just say 1/8 X plus 1, which essentially means that the correct answer to this problem corresponds to the answer choice B. Thank you for watching.

Find an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a.
f(x) = 1/x; a= -5

Key Concepts

Tangent Line

Derivative

Point-Slope Form

Watch next