Tangent lines Suppose f(2)=2 and f′(2) =3. Let g(x) = x²f(x) a...

Question

Tangent lines Suppose f(2)=2 and f′(2) =3. Let g(x) = x²f(x) and h(x) = f(x) / x−3.

b. Find an equation of the line tangent to y = h(x) at x=2.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem dealing with tangent lines. This problem says to consider the function Q of X, which is equal to G of X, divided by the quantity of X + 4. Given that G 1 is equal to 3 and G 1 is equal to -2, find the equation for the tangent line to Y is equal to Q X at X equal to 1. We're given 4 possible choices as our answers. Choice A, we have Y is equal to 13 X plus 5. For choice B, we have Y is equal to -13 X minus 8. For choice C, we have Y is equal to -13 divided by 25, multiplied by X plus 28 divided by 25. And for choice D, we have Y is equal to -13 divided by 25, multiplied by X minus 2 divided by 25. Now we're asked to find the equation for the tangent line to Y is equal to Q of X at X equal to 1. Now, in order to find the equation for our tangent line, we will need to find the slope of our tangent line when X is equal to 1. So, in order to find the slope of that tangent line, I'm first going to need to take a derivative of Q of X with respect to X. So I'm first going to calculate Q of X. And if we look at our function Q of X, in order to take this derivative, we're going to need to use our quotient rule. So, we're going to have Q X, it's equal to, and here we call your quotient rule. We're going to have our denominator, which is X + 4. And that quantity is going to be multiplied by the derivative of our numerator, which is going to be G of X. Then we'll have minus the derivative of our denominator, which in this case the derivative of X + 4 is equal to 1, we'll have 1, multiplied by our numerator, which is G of X. And that whole quantity is divided by The denominator squared will have the quantity of X + 4 in quantity squared. So, what we need to do now is use this expression to find the slope for our tangent line. And so we recall that the slope or our tangent line at a given point is equal to the derivative at that point. So our slope for a tangent line, which we will label as M, is going to be equal to Q1, since X is equal to 1. And so this is going to be equal to the quantity of 1 + 4. In quantity, multiplied by G of 1. Minus G of 1. And that whole quantity is divided by the quantity of 1 + 4 in quantity squared. So now we need to simplify this expression. So this is going to be equal to. 1 + 4 is going to give me 5, and this will be multiplied by G of 1, but we're told that G 1 in the problem was equal to -2. So we'll have 5 multiplied by -2. Minus G of 1, which we're told in the problem is 3. And that whole quantity is divided by 5 squared. So again, simplifying this expression, this is going to be equal to -13 divided by 25. That is going to be the slope of our tangent line. Now, we could write the equation or our tangent line using our point slope form. However, we need to find the Y value of our function when X is equal to 1. So what we need to do is determine. Que of one. And that's going to be equal to the Y coordinate of our point that we're gonna use in the point slope form. So we'll have Q 1 is equal 2, and here we'll just plug that into our expression for Q of X, we'll have G of 1. Divided by the quantity of 1 + 4. Now, since G of 1 is equal to 3, this is going to be equal to 2. 3 divided by 5. And so that is our Y coordinate that we're going to use in our point slope form. So now, we can actually write the equation for our line using point slope form. So we're going to have Y minus the Y value that we just calculated, which is 3 divided by 5, and that's going to be equal to our slope, which is -13 divided by 25. That's going to be multiplying the quantity of X minus 1, since that is the value of X that we're looking for. So now we need to just simplify this expression. So we're going to distribute our slope on the right hand side. So we're gonna have Y minus 3 divided by 5, it's equal to -13 divided by 25, multiplied by X. Plus 13 divided by 25. So now we just need to add. 3 divided by 5 to both sides of this expression. And so we're gonna get Y is equal to minus 13, divided by 25. Multiplied by X Plus 28 divided by 25. And so this is the equation that we are looking for for this problem. And when we compare our equation that we calculated to the answer choices, the correct answer choice turns out to be answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

Tangent lines Suppose f(2)=2 and f′(2) =3. Let g(x) = x²f(x) and h(x) = f(x) / x−3.
b. Find an equation of the line tangent to y = h(x) at x=2.

Key Concepts

Tangent Line

Derivative

Quotient Rule

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