How do you find the original function given a point (a,b) and the equation of the line tangent to the graph of f(x) at (a,b).
For example:
The point is (4,-11) and 7x-3y=61 is the equation of the line tangent to the graph of f(x)
now you have given even less information than in your last post!
There are many many functions passing through (4,-11) tangent to that line. The simplest is, of course,
f(x) = (7x-61)/3
Even the next simplest one could be a parabola opening either up or down, and there are lots of those.
Or, it could be an exponential function, or a logarithmic function, or a sine/cosine.
I'm sure there is something you have still left out.
Actually, I'm looking for f(a) and f'(a), so i know I can just plug in the x value for f(a) which f(4) would equal -11, but how do i find the derivative in this case?
To find the original function f(x) given a point (a,b) and the equation of the tangent line, you need to follow these steps:
Step 1: Find the slope of the tangent line.
- Rewrite the equation of the line in slope-intercept form (y = mx + c), where m represents the slope.
- In this case, you have 7x - 3y = 61. Rewrite it as -3y = -7x + 61, then divide throughout by -3 to get y = (7/3)x - 61/3.
- Therefore, the slope of the tangent line is 7/3.
Step 2: Use the point-slope form to write the equation of the tangent line.
- The point-slope form is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
- Since the given point on the tangent line is (4, -11) and the slope is 7/3, substitute these values in the equation to find the equation of the tangent line.
Step 3: Differentiate the equation of the tangent line to find the derivative of f(x) at the point (a,b).
- Since the tangent line is an approximation of the graph of f(x) at the point (a,b), the derivative of f(x) at (a,b) will be the same as the slope of the tangent line.
- Therefore, differentiate the equation of the tangent line y = (7/3)x - 61/3 to find the derivative of f(x) at (a,b).
Step 4: Integrate the derivative of f(x) obtained in step 3 to find f(x).
- The derivative of f(x) represents the rate of change of f(x) at any point, and integrating it will give you the original function f(x).
- Depending on the expression obtained as the derivative in step 3, integrate it to find f(x).
Note: If you have specific values for a,b, and the equation of the tangent line, you can follow these steps to determine the original function.