Find an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a.
f(x) = √x+2; a=7

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Step 1: Understand that the equation of a tangent line to the graph of a function at a point (a, f(a)) is given by the formula y - f(a) = f'(a)(x - a), where f'(a) is the derivative of the function evaluated at x = a.

Step 2: Find the derivative of the function f(x) = \(\sqrt{x+2}\). Use the chain rule: if f(x) = (x+2)^{1/2}, then f'(x) = \(\frac{1}{2}\)(x+2)^{-1/2} \(\cdot\) 1.

Step 3: Simplify the derivative: f'(x) = \(\frac{1}{2\sqrt{x+2}\)}.

Step 4: Evaluate the derivative at x = a = 7: f'(7) = \(\frac{1}{2\sqrt{7+2}\)} = \(\frac{1}{2\sqrt{9}\)} = \(\frac{1}{6}\).

Step 5: Use the point-slope form of the tangent line equation with f(7) = \(\sqrt{7+2}\) = 3 and f'(7) = \(\frac{1}{6}\) to write the equation: y - 3 = \(\frac{1}{6}\)(x - 7).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point, which is determined by the derivative of the function.

Derivative

The derivative of a function at a point quantifies how the function's output changes as its input changes. It is calculated as the limit of the average rate of change of the function as the interval approaches zero. For the function f(x) = √(x + 2), the derivative will provide the slope of the tangent line at the specified point.

Point-Slope Form

The point-slope form of a linear equation is used to express the equation of a line given a point on the line and its slope. It is written as y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. This form is particularly useful for finding the equation of the tangent line once the slope and point are known.

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