Explanation:
To find the equation of a tangent line to a function h(x) at a given point x = 5, we need to find both the slope of the tangent line (the derivative at x=5) and the y-intercept of the line.
Given that h(5) = -1, we do not know h(x) or the equation of the tangent line yet.
Given that h′(5) = 4, the derivative of h(x) at x=5 is 4. Therefore, the slope of the tangent line at x = 5 is 4.
Using the point-slope form of a linear equation, we know the slope is 4, and we can use the point (5,-1) to find the equation of the tangent line:
y - (-1) = 4(x - 5)
Simplifying this equation gives:
y = 4x - 21
Therefore, the equation of the tangent line to h(x) at x=5 is y = 4x - 21.