Suppose f(3) = 1 and f′(3) = 4. Let g(x) = x² + f(x...

Question

Suppose f(3) = 1 and f′(3) = 4. Let g(x) = x² + f(x) and h(x) = 3f(x).

Find an equation of the line tangent to y = h(x) at x = 3.

Accepted Answer

Below there. Today we're gonna 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. If F of 1 is equal to 2 and F 1 is equal to 6, and the function is defined as H X is equal to 4 multiplied by F of X, what is the equation of the line tangent to Y equals HX at X equals 1. Awesome. So it appears for this particular problem. We're asked to ultimately figure out what the equation of the line is, that is tangent to Y equals H X at X equals 1, given that we have our function defined as H X is equal to 4 multiplied by F of X. And also we're told that. F 1 is equal to 2 and 1 is equal to 6. So now that we know that we're ultimately trying to figure out what this equation of the line is, let's read off our multiple choice answers to see what our final answer might be, noting that they're all stating that Y is equal to some expression. So A is 4 X + 4, B is 24 X minus 16, C is 4X minus 12, and D is 24 X minus 32. Awesome. So first off, in order to help us solve this problem, we must first solve for each of one. So with that said, we need to note that H of 1 is equal to 4 multiplied by F of 1 means we're plugging in 1 in for X. So now we need to note that we need to substitute in, and let's make a note of this, we need to substitute in F of 1. So, once again, this is given to us by the problem itself, we need to state that F of 1 is equal to 2. So that will mean that H of 1 is equal to 4 multiplied by 2, which is equal to 8. We must also recall a note at this stage that the slope of a line tangent to a curve at a point is equal to the derivative of the curves equation evaluated at that specific point. So with that in mind, our next step is we need to find the value of H X. Awesome. So focusing our attention now on H of X, we need to differentiate H of X with respect to X. So when we do that, we will find that H of X is equal to 4 multiplied by F of X. And our next step now is we need to find H of 1. And when we focus our attention on H1, we could state that H of 1 is equal to 4 multiplied by F of 1. Awesome. And we could go ahead and substitute in our value, and let's make a note of this. We need to substitute in that F of 1 is equal to 6 because that's what we're told by the prime itself, that F of 1 is equal to 6. So therefore we could safely write that H. Of 1 is equal to 4 multiplied by 6, which is equal to 24. Fantastic. So now we need to note that the slope of the line tangent to Y equals HX will be or is 24. So once again, the slope of the tangent line, so M. So the slope of the tangent line 2 Y equals HFX is 24. Awesome, where M once again denotes slope. So now our next step is we need to write the equation of the tangent line using point slope form, which let us recall, let's write it up above that point slope form states that Y minus Y1 is equal to M, where once again M is the slope, multiplied by X minus X1. Fantastic. So we need to recall and note that X1 for this particular problem is equal to 1, and Y1 is equal to H of 1, which is equal to 8, and we also need to note that M is equal to H 1. Which is equal to 24. So when we plug in all these known variables into our point slope form, we will find that Y minus 8 is equal to 24, multiplied by X minus 1. And when we rearrange this equation to isolate and solve for Y all by itself using a little bit of algebra, we will find that Y is equal to 24 X minus 16. And that is our final answer. Hooray, we did it. And this corresponds to multiple choice answer letter B, which once again is Y equals 24 X minus 16. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Suppose f(3) = 1 and f′(3) = 4. Let g(x) = x2 + f(x) and h(x) = 3f(x).Find an equation of the line tangent to y = h(x) at x = 3.

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Suppose f(3) = 1 and f′(3) = 4. Let g(x) = x² + f(x) and h(x) = 3f(x).
Find an equation of the line tangent to y = h(x) at x = 3.