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: Identify \(a\) and \(f(x)\)

Given \(f(x)=\sqrt{5x + 39}\) and \(P(5,8)\), so \(a = 5\).

Step2: Calculate \(f(a + h)\) and \(f(a)\)

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Step3: Find the limit for the slope \(m_{tan}\)

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Step4: 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 \(m=\frac{5}{16}\).

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Answer:

a. \(m_{tan}=\frac{5}{16}\)
b. \(y=\frac{5}{16}x+\frac{103}{16}\)