practice another\nconsider the following function.\nf(x) = √x, (4, 2)\n(a) find an equation of the tangent…

Explanation:

Step1: Find the derivative of $f(x)$

The derivative of $y = \sqrt{x}=x^{\frac{1}{2}}$ using the power - rule $(x^n)'=nx^{n - 1}$ is $f'(x)=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}}$.

Step2: Evaluate the derivative at $x = 4$

Substitute $x = 4$ into $f'(x)$. $f'(4)=\frac{1}{2\sqrt{4}}=\frac{1}{4}$. This is the slope $m$ of the tangent line.

Step3: Use the point - slope form of a line

The point - slope form is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(4,2)$ and $m=\frac{1}{4}$. $y - 2=\frac{1}{4}(x - 4)$.

Step4: Simplify the equation

Expand the right - hand side: $y - 2=\frac{1}{4}x - 1$. Add 2 to both sides to get $y=\frac{1}{4}x+1$.

Answer:

$y=\frac{1}{4}x + 1$