Question

a. use the definition $m_{tan}=lim_{h \to 0}\frac{f(a + h)-f(a)}{h}$ to find the slope of the line tangent to the graph of f at p
b. determine an equation of the tangent line at p
$f(x)=sqrt{5x + 39},p(5,8)$
a. $m_{tan}square$
(simplify your answer. type an exact answer, using radicals as needed)

Explanation:

Step1: Find \(f(a + h)\) and \(f(a)\)

Given \(f(x)=\sqrt{5x + 30}\) and \(a = 5\), then \(f(a+h)=f(5 + h)=\sqrt{5(5 + h)+30}=\sqrt{25+5h + 30}=\sqrt{5h + 55}\), and \(f(a)=f(5)=\sqrt{5\times5+30}=\sqrt{25 + 30}=\sqrt{55}=8\).

Step2: Calculate the limit for the slope

$$
LATEXBLOCK0
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Substitute \(h = 0\) into the limit: \(\frac{5}{\sqrt{55}+\sqrt{55}}=\frac{5}{2\sqrt{55}}\)

Step3: Find the equation of the tangent - line

The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)=(5,8)\) and \(m=\frac{5}{2\sqrt{55}}\)

$$
LATEXBLOCK1
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Answer:

a. \(\frac{5}{2\sqrt{55}}\)
b. \(y=\frac{5}{2\sqrt{55}}x-\frac{25}{2\sqrt{55}}+8\)