if the tangent line to the function f(x)=�ãx +a at x=16 has equation y=mx+6. find a and m. you may use the fact that f'(x)= 1/(2�ãx)
To find the values of 'a' and 'm', we need to determine the equation of the tangent line to the function f(x) = √x + a at x = 16. We also have the information that the derivative of f(x), f'(x), is equal to 1/(2√x).
To begin, let's first find the derivative of f(x). Using the given information, we have:
f'(x) = 1/(2√x)
Now, let's substitute x = 16 into f'(x) to find the slope of the tangent line at x = 16:
f'(16) = 1/(2√16) = 1/(2×4) = 1/8
So, the slope of the tangent line, m, is equal to 1/8.
Next, let's use the point-slope form of a line, y = mx + c, and plug in the point (16, f(16)) = (16, √16 + a) = (16, 4 + a). We can also use the given equation of the tangent line, y = mx + 6.
Comparing the equations, we have:
mx + 6 = y = √x + a
Since both equations are equal to y, we can equate them:
4 + a = 1/8 × 16 + 6
4 + a = 2 + 6
a = 2
Therefore, a = 2 and m = 1/8 are the values that satisfy the given conditions.
In summary:
a = 2
m = 1/8