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If h(5)=−1 and h′ (5)=4, determine the equation of the tangent line to h(x) at x=5 in the form y=mx+b⋅

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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.

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