Question

for the function f(x) = √(5x), where a = 5, complete the following. a. find f(a). b. determine an equation of the line tangent to the graph of f at the point (a,f(a)) for the given value of a. a. f(5)=□ (type an integer or a simplified fraction.)

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=\sqrt{5x}=(5x)^{\frac{1}{2}}$.

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = ax^{n}$, then $y^\prime=anx^{n - 1}$. For $f(x)=(5x)^{\frac{1}{2}}$, using the chain - rule (if $y = u^{\frac{1}{2}}$ and $u = 5x$, then $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$), we have $f^\prime(x)=\frac{1}{2}(5x)^{-\frac{1}{2}}\cdot5=\frac{5}{2\sqrt{5x}}$.

Step3: Find $f^\prime(a)$

Substitute $a$ into $f^\prime(x)$. When $a = 5$, $f^\prime(5)=\frac{5}{2\sqrt{5\times5}}=\frac{5}{2\times5}=\frac{1}{2}$.

Step4: Find the equation of the tangent line

The equation of a tangent line to the curve $y = f(x)$ at the point $(a,f(a))$ is given by $y - f(a)=f^\prime(a)(x - a)$. First, find $f(5)=\sqrt{5\times5}=5$. Then, since $f^\prime(5)=\frac{1}{2}$ and $a = 5$, the equation of the tangent line is $y-5=\frac{1}{2}(x - 5)$. Expand it: $y-5=\frac{1}{2}x-\frac{5}{2}$, so $y=\frac{1}{2}x+\frac{5}{2}$.

Answer:

a. $f^\prime(5)=\frac{1}{2}$
b. $y=\frac{1}{2}x+\frac{5}{2}$