find an equation for the tangent line to the curve at the given point. then sketch the curve and the tangent…

Explanation:

Step1: Find the derivative of the function

The function is $y = 4\sqrt{x}=4x^{\frac{1}{2}}$. Using the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, we have $y'=\frac{d}{dx}(4x^{\frac{1}{2}})=4\times\frac{1}{2}x^{\frac{1}{2}-1}=2x^{-\frac{1}{2}}=\frac{2}{\sqrt{x}}$.

Step2: Evaluate the derivative at the given point

We want to find the slope of the tangent line at the point $(1,4)$. Substitute $x = 1$ into $y'$. When $x = 1$, $y'(1)=\frac{2}{\sqrt{1}}=2$. So the slope of the tangent line $m = 2$.

Step3: Use the point - slope form to find the equation of the tangent line

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(1,4)$ and $m = 2$. Substituting these values, we get $y - 4=2(x - 1)$.

Step4: Simplify the equation

Expand the right - hand side: $y-4 = 2x-2$. Then, add 4 to both sides to get the equation of the tangent line in slope - intercept form: $y=2x + 2$.

Answer:

The equation of the tangent line is $y = 2x+2$.