The equation for the tangent line to the graph of f at x=7” is y - 0.5 = -0.5(x - 7).
Steps to solve:
Find the slope of the tangent line: The slope of the tangent line at a point is equal to the derivative of the function at that point. Since we are given that x = 7, we need to find f'(7).
Unfortunately, the graph of f is not given in the image, so we cannot directly calculate the derivative.
Use the table of values: However, the table of values provides some information about the derivative of f.
We can see that f'(5) = -2.455. We know that the graph of f has a horizontal tangent line at x = 4, which means that f'(4) = 0.
Estimate the slope: Since the graph is concave down for 0 < x < 6, we know that the derivative f'(x) is decreasing in this interval.
Therefore, we can estimate that f'(7) is somewhere between f'(5) = -2.455 and f'(4) = 0.
A reasonable estimate for the slope of the tangent line at x = 7 would be -0.5.
Find the y-intercept: The tangent line passes through the point (7, 0.5).
We can use the point-slope form of linear equations to find the equation of the tangent line:
y - y1 = m(x - x1)
where:
m is the slope of the tangent line (we estimated m to be -0.5)
(x1, y1) is the point on the tangent line (we know (x1, y1) = (7, 0.5))
Plugging these values into the equation, we get:
y - 0.5 = -0.5(x - 7)
Therefore, the equation of the tangent line to the graph of f at x = 7 is y - 0.5 = -0.5(x - 7).