If the tangent line to y = f (x) at (9,7) passes through the point (0,6)

In summary, to find (a) f(9), we can simply plug in x=9 into the equation y=f(x) and solve for y. For (b) f'(9), we can use the slope formula to find the slope of the tangent line at (9,7), which is the same as the slope of the function at that point.
  • #1
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Homework Statement


If the tangent line to y = f (x) at (9,7) passes through the point (0,6), find the following.

(a) f (9)

(b) f ' (9)

Homework Equations



f '(x) = f(x+h) - f(x)/h ?

The Attempt at a Solution



I really have no idea how to even begin going about this one, so a shove in the right direction would be greatly appreciated.
 
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  • #2
my guess:
Since the line is tangent to f(x) at (9,7), then (9,7) must be a point on f(x). If (9,7) is on the function, then when x=9, what is y? That's f(9).

You have 2 points on the tangent line. (9,7) and (0,6). Find the slope of this line using the slope formula: rise / run. Then remember that f'(9) is simply asking what's the slope of the function at (9,7), which is the same as asking what is the slope of the tangent line.
 
  • #3
Thank you very much! I'm not sure why I had such a hard time conceptualizing what that question was asking, but your hints were correct and I got it.
 

FAQ: If the tangent line to y = f (x) at (9,7) passes through the point (0,6)

1. What is the equation of the tangent line to y = f(x) at (9,7)?

The equation of the tangent line can be found using the point-slope form: y - y1 = m(x - x1), where (x1, y1) is the given point on the tangent line and m is the slope of the tangent line. Since we know the point (9,7) is on the tangent line, we can substitute these values into the equation to get: y - 7 = m(x - 9). We still need to find the slope, which can be done by taking the derivative of f(x) and evaluating it at x = 9.

2. How do I find the slope of the tangent line to y = f(x) at (9,7)?

To find the slope of the tangent line, you will need to take the derivative of f(x) and evaluate it at x = 9. The result will be the slope of the tangent line at that point.

3. Can I use any other point on the tangent line to find its equation?

Yes, you can use any other point on the tangent line to find its equation. The point-slope form of a line allows us to find the equation of a line using any given point and the slope of the line. As long as you know the point and the slope of the tangent line, you can find its equation.

4. How do I know if the tangent line passes through the point (0,6)?

To determine if the tangent line passes through the point (0,6), you can substitute the given coordinates into the equation of the tangent line and see if the equation is true. If it is, then the tangent line does pass through the point (0,6).

5. Can I use this information to find the function f(x)?

No, this information alone is not enough to find the function f(x). In order to find the function, you will need more information such as the derivative of f(x) or other points on the graph of f(x).

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