Question

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=2 \sqrt{x} ;(4,4) $$

Short answer

The slope of the tangent line to the graph of \(f(x) = 2 \sqrt{x}\) at the point (4, 4) is \(1/2\).

Step-by-step solution

Substituting into the equation

Substitute \(f(a + h)\) and \(f(a)\) into the equation. We have \(a = 4\) and \(f(x) = 2 \sqrt{x}\), calculating these gives: \(f(a + h) = 2 \sqrt{4 + h}\) and \(f(a) = 2 \sqrt{4}\).

Solve the equation

Now that we have \(f(a + h)\) and \(f(a)\), we can substitute those into the definition of the derivative and solve for \(f'(a)\). Plug these values into the equation: \[f'(4) = \lim_{h\to 0} \[ \frac{2 \sqrt{4 + h} - 2 \sqrt{4}}{h} \]\]

Simplify the equation

Simplify the equation further for easy computation. This simplification can be achieved by rationalizing the numerator. The result is: \[f'(4) = \lim_{h\to 0} \[ \frac{2}{\sqrt{4 + h} + 2} \]\]

Compute the limit

Since we have simplified the equation, we can easily compute the limit as \(h\) approaches 0. Simply replace \(h\) with 0 in the equation. The result is: \(f'(4) = 1/2\)