Use calculus to find the slope of the tangent line

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  • #1
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I'm given f(x) = x^2 and g(x) = -x^2 + 2x - 5

And it says

Let A (a, f(a)) be the point of tangency on f(x) and B (b, g(b)) be the point of tangency on g(x). Use calculus to find the slope of the tangent line at each point. What must be true about these two slope expressions?

I know we need to use the derivatives, but do I use the original form of the derivative f(x + deltax) - f(x)/delta x or the alternate form f(x) - f(c)/x-c?

I would think to use the alternate form, because it gives us points of tangency... but when I try to use the alternate form on g(x), I can't seem to get an answer.

For f(x): x^2 - a^2 / (x-a)
(x+a)(x-a) / (x-a)
= x+a

I think I did that right. Now for g(x):
-x^2+2x-5+b^2-2b+5/ (x-b)
-x^2 + 2x + b^2 - 2b/ (x-b)

And I'm basically stuck on what to do from here. Any ideas? Or am I supposed to be using the other derivative equation?
 

Answers and Replies

  • #2
37
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f'(c) = lim{x->c} [f(x) - f(c)]/(x - c)

for f(x) = x^2 ...

f'(a) = lim{x->a} [x^2 - a^2]/(x - a)
f'(a) = lim{x->a} [(x + a)(x - a)]/(x - a)
f'(a) = lim{x->a} (x + a) = 2a


for g(x) = -x^2 + 2x - 5 ...

g'(b) = lim{x->b} [(-x^2 + 2x - 5) - (-b^2 + 2b - 5)]/(x - b)
g'(b) = lim{x->b} [b^2 - 2b + 2x - x^2]/(x - b)
g'(b) = lim{x->b} [b^2 - x^2 - 2(b - x)]/(x - b)
g'(b) = lim{x->b} [(b - x)(b + x) - 2(b - x)]/(x - b)
g'(b) = lim{x->b} [(b - x)[(b + x) - 2]]/(x - b)
g'(b) = lim{x->b} [2 - (b + x)] = 2 - 2b
 
  • #3
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Ah.. thank you, I get it now. It doesn't matter if you use the alternate form or not, heh.

The next part says use the coordinates of A and B to write an expression for the slope AB in terms of a and b.

So...

f(a) - f(b)/ (a - b)

I think that's what I'm supposed to do. The next part says to substitute things in so there's only one term to solve for. But I don't see what I can substitute.
 
  • #4
37
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something is missing from your original post ... are the two tangent lines to f and g supposed to have some relationship (like being parallel, maybe)?
 
  • #5
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skeeter said:
something is missing from your original post ... are the two tangent lines to f and g supposed to have some relationship (like being parallel, maybe)?

Oh, sorry. The graph is supposed to have one tangent line that hits both f(x) and g(x).
 
  • #6
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well ... the slopes have to be the same then, don't they?

2a = 2 - 2b
2a + 2b = 2
a + b = 1

now you have the required relationship between a and b.
 
  • #7
15
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Oh! Wow, that answer now, seems so blantant ~_~

So, a = 1-b

Substitute that into the slope equation:

(1-b) ^2 - (-b^2 + 2b -5)
1 - 2b + b^2 + b^2 - 2b + 5
= 2b^2 - 4b + 6

Oookay. Then I need to solve for the equation to get b. The thing is I get two answers so how do I know which one is correct?

To find f(b), I plug it back into 2-2b, right?
 
Last edited:
  • #8
37
1
who knows? ... maybe they're both good, and there are two different tangent lines that work.

check out each solution, one at a time.
 
  • #9
15
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Aw man, I just realized I did my math wrong.

For the slope equation, it's supposed to be

2b^2 - 4b + 6 / (1-2b)

How would I solve for b if I'm not given that it equals anything?
 
  • #10
37
1
[f(1-b) - f(b)]/[(1-b) - b] = 2 - 2b

[(1-b)^2 - b^2]/(1 - 2b) = 2 - 2b

(1 - 2b)/(1 - 2b) = 2 - 2b

1 = 2 - 2b

2b = 1

b = 1/2

since a + b = 1, a = 1/2
 
  • #11
15
0
skeeter said:
[f(1-b) - f(b)]/[(1-b) - b] = 2 - 2b

[(1-b)^2 - b^2]/(1 - 2b) = 2 - 2b

(1 - 2b)/(1 - 2b) = 2 - 2b

1 = 2 - 2b

2b = 1

b = 1/2

since a + b = 1, a = 1/2

Why would it go to b^2? o_o;

It's supposed to be g(b), and g(x) is -x^2 + 2x -5.

lol, probably from my careless typing <.<

*EDIT*

So, I have 2b^2 - 4b + 6 / (1-2b) = 2-2b
2b^2 - 4b + 6 = (2-2b)(1-2b)
2b^2 - 4b + 6 = 2 - 6b + 4b^2
2b^2 - 2b -4 = 0

Factor that out to get

(b - 2) (2b + 2)

b = 2, -1

And then plug those two points in the equation a+b = 1

Solve for a, and then plug those values of a inside f(a) = a^2?
 
Last edited:

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