Given the function f ( x ) = x 2 − 10 x + 2 f(x)=x^2-10x+2 f ( x ) = x 2 − 10 x + 2 , calculate the slope of the tangent line at x = 2 x=2 x = 2 .

in this problem, we are told, given the function f of x is equal to x squared minus 10 x plus 2, calculate the slope of the tangent line at x is equal to 2. Okay. Now when solving these types of problems, we're trying to find the slope of a tangent line. There's an equation that we can use. That the slope of the tangent line is going to be the limit as x approaches c of this. We're going to have f of x minus f of c, and then all of that is going to be divided by x minus c. So this is the equation you need. Now in this situation, c is going to be the number that we're approaching on the x axis, which I can see here is 2. That's the number that we're looking at with reference to. So what I'm gonna say is that the slope of our tangent line is going to be the limit as x approaches 2. Now what I'll have here is I'm going to have our function as well. So we can see that we have f of x, and f of x is going to be this whole thing. So we're going to have x squared minus 10x plus 2. Now at this point, I also need to subtract off f of c, and f of c is just going to be our function with 2 plugged in. So we're going to have f of 2, which is going to be 2 squared, and that's going to be minus 10 times 2 plus 2. Now 2 squared is 4. 10 times 2 is 20, and it's gonna be plus 2. So we're gonna have 4 minus 20, which is negative 16, and then plus 2, that's gonna give you negative 14. I can cancel this negative sign with that one when I plug it in for f of c, giving us this whole thing plus 14. And it's all going to be divided by x minus 2. So this is what we end up getting. Now from here, what I can do is try to combine things and simplify at the top. Because I notice here that we have a 2 and a 14 that we can add together. So I need to rewrite out this equation so we'll have that our tangent the slope of our tangent line as the for the limit as x approaches 2, that's going to be x squared minus 10 x and then 10 plus 2 or excuse me, 2 plus 14, that's going to be 16, and then we can divide this by x minus 2. Now at this point, what I can do is apply this limit. But I noticed something here. If I take this x value that we're approaching and I plug it in for x there, we're gonna get 2 minus 2, which is 0 in the bottom of this of this fraction. And we cannot have a 0 in the denominator. So we're going to need to factor to see if we can get things to cancel to hopefully give us a better equation. So rewriting this up here will have the slope of our tangent line, and that's going to be the limit as x approaches 2. We need to keep this limit consistent before we actually apply it. And then I'm going to have this x squared minus 10 x plus 16 on top, but what I need to find is a way to factor this. And the way that I can do this is by finding two numbers that will multiply to give me 16 and add to give me negative 10. Now I just need to think of various numbers that would multiply. And if I think about it well, I can multiply 28. That's going to give me 16, And 2 +8 is positive 10. So we're actually pretty close here, but the thing is is that I need to get a situation where we get a negative 10 since we have this minus sign up in here. So I can do that by getting a negative 2 and a negative 8. Because notice this is going to give me negative 2 times negative 8 where the negative signs would cancel and give me positive 16, and adding negative 2 and negative 8 would give me negative 10. So this is the pair of numbers I'm looking for. So what that means is that this thing is going to factor the top of this fraction is going to factor into x minus 2 times x minus 8, and all of that is going to be divided by x minus 2 on bottom. Now at this point, what I can do is cancel the x minus twos. So we're going to have this limit that continues. It's gonna be the limit as x approaches 2, and then we're just going to have x minus 8. Now at this point, what I can do is I can take this 2 here, and I can plug it in for x because now we have this limit broken down. We have a fraction factor, so what we're going to have is 2 minus 8. We don't have to write the limit anymore now that we've applied the limit, and 2 minus 8 is negative 6. So that right there is the slope of our tangent line at x equals 2 for this function, and that is the solution to this problem. So that is how you can find the slope of a tangent line when you're given a function and a certain value that you're looking with reference to on that graph for the x axis. So hope you found this video helpful. Thanks for watching.

2. Intro to Derivatives

Tangent Lines and Derivatives

Business Calculus

2. Intro to Derivatives

Tangent Lines and Derivatives

Struggling with Business Calculus?

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Multiple Choice

Given the function $f(x)=x^2-10x+2$ , calculate the slope of the tangent line at $x=2$ .

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Verified step by step guidance

Step 1: Recall that the slope of the tangent line to a function at a given point is given by the derivative of the function evaluated at that point. This means we need to compute f'(x), the derivative of f(x).

Step 2: Differentiate the given function f(x) = x^2 - 10x + 2. Using the power rule, the derivative of x^2 is 2x, the derivative of -10x is -10, and the derivative of the constant 2 is 0. Thus, f'(x) = 2x - 10.

Step 3: Substitute x = 2 into the derivative f'(x) to find the slope of the tangent line at that point. This means we calculate f'(2) = 2(2) - 10.

Step 4: Simplify the expression f'(2) = 2(2) - 10 to find the numerical value of the slope. This involves performing the multiplication and subtraction.

Step 5: The result of the simplification gives the slope of the tangent line at x = 2. Compare this value to the provided answer choices to identify the correct one.